Theorem of the heart for Weibel's homotopy $K$-theory

THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y ALEXANDER I. EFIMO V Abstract. In this paper we prov e the theorem of the heart for W eib el’s homotop y K - theory K H. Namely , if C is a small stable 8 -category with a b ounded t -structure, then the realization functor D b p C ♡ q Ñ C induces an equiv alence of spectra K H p C ♡ q „ Ý Ñ K H p C q . In a certain sense this result is dual to the Dundas-Goo dwillie-McCarthy theorem. W e deduce the d ´ evissage theorem for K H of ab elian categories, also on the level of spectra (in all degrees). More generally , we prov e these results for dualizable categories with nice t -structures and for the so-called coherently assembled ab elian categories. The pro of is heavily based on another new result, which is a muc h stronger version of Barwic k’s theorem of the heart. Its sp ecial case states the following: if C is a small stable category with a bounded t -structure, such that for some n ě 1 the realization functor induces isomorphisms on Ext ď n b et ween the ob jects of C ♡ , then the map K j p C ♡ q Ñ K j p C q is an isomorphism for j ě ´ n ´ 1 , and a monomorphism for j “ ´ n ´ 2 . Moreov er, w e prov e that these estimates are sharp, ev en for dg categories ov er a ﬁeld. In particular the naiv e K -theoretic theorem of the heart fails for K ´ 3 . Contents 0. In tro duction 2 1. Coheren tly assembled ab elian categories and exact categories 11 1.1. Reminder on exact categories 11 1.2. Accessibly exact categories 14 1.3. Grothendiec k exact categories 17 1.4. Criterion for an exact category to be locally κ -coherent 18 1.5. Presen table stable en velopes 19 1.6. Reminder on compactly assembled and dualizable categories 21 1.7. Coheren tly assembled exact categories 23 1.8. Coheren tly assembled ab elian categories 28 1.9. The situation of d´ evissage 32 2. Compactly assembled t -structures on dualizable categories 35 2.1. Reminder on t -structures 35 2.2. t -structures compatible with ﬁltered colimits 36 2.3. Compactly assembled t -structures 38 1 2 ALEXANDER I. EFIMOV 2.4. General constructions of t -structures 44 2.5. Suﬃcien t conditions for left t -exactness 48 2.6. Filtered colimits of dualizable t -categories 49 2.7. Dual t -structures 50 3. Reﬁned K -theoretic theorem of the heart for dualizable categories 51 4. Co connectivit y estimates 62 4.1. Estimates for K -groups 62 4.2. Estimates for higher nil groups 68 5. Theorem of the heart for K H 70 6. D ´ evissage theorems for K -theory and homotop y K -theory 75 7. Sharpness of the estimates 76 7.1. Estimates for theorems of the heart and d ´ evissage 76 7.2. Estimates for higher nil groups of abelian categories 79 8. Examples of compactly assem bled t -structures and coherently assem bled ab elian categories 84 8.1. Shea v es on locally compact Hausdorﬀ spaces 84 8.2. Nuclear mo dules o ver Z p 85 8.3. Chase criterion 87 8.4. Concluding remarks 89 References 90 0. Introduction In this paper we study the non-connectiv e K -theory and W eib el’s homotop y K -theory K H of small stable 8 -categories with bounded t -structures, and more generally for du- alizable categories with suitably deﬁned nice t -structures. In the latter case we consider the in v arian ts K cont and K H cont as deﬁned in [E24]. T o ﬁx the ideas, in the introduction w e give the form ulations of our main results in the context of small categories, mentioning the more general versions for dualizable categories and the so-called coherently assem bled ab elian categories (Deﬁnition 1.41). F or brevity w e will sa y “stable category” instead of “stable 8 -category”. Originally , W eib el’s deﬁnition of K H was giv en in the Z -linear context [W ei89, W ei13, T ab15]. The natural generalization for stable categories is explained for example in [E25c, Section 8, Corollary 8.3]. F or a small stable category C the spectrum K H p C q is giv en b y THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 3 the geometric realization (0.1) K H p C q “ colim r n sP ∆ op K p C b P erf p S r x 1 , . . . , x n sqq , where for n ě 0 w e put S r x 1 , . . . , x n s “ S r x 1 s b ¨ ¨ ¨ b S r x n s , S r x s “ Σ 8 p N ` q , N ` “ N \ t˚u . F or a small ab elian category A we put K H p A q “ K H p D b p A qq . Our main result is the follo wing theorem, whic h is a special case of Theorem 5.1. Theorem 0.1. L et C b e a smal l stable c ate gory with a b ounde d t -structur e p C ě 0 , C ď 0 q . Then the r e alization functor D b p C ♡ q Ñ C induc es an e quivalenc e of sp e ctr a K H p C ♡ q „ Ý Ñ K H p C q . This theorem can b e considered as a dual version of the Dundas-Go o dwillie-McCarth y theorem [DGM13, Theorem 7.0.0.2], see b elow for an explanation of this analogy . W e also deduce the follo wing d ´ evissage theorem for K H , which is a special case of Theorem 6.1 (ii). Theorem 0.2. L et F : B Ñ A b e an exact ful ly faithful functor b etwe en smal l ab elian c ate gories such that F p B q gener ates A via extensions (e quivalently, F p B q Ă A satisﬁes the assumptions of Quil len ’s D´ evissage the or em). Then F induc es an e quivalenc e of sp e ctr a K H p B q „ Ý Ñ K H p A q . The pro of of Theorem 0.1 is based on the follo wing muc h stronger version of Barwic k’s theorem of the heart [Bar15, Theorem 6.1]. This is a sp ecial case of Theorem 4.1. Theorem 0.3. L et C and D b e smal l stable c ate gories with b ounde d t -structur es p C ě 0 , C ď 0 q r esp. p D ě 0 , D ď 0 q . L et n ě 1 b e an inte ger and supp ose that we have an exact and t -exact functor F : C Ñ D , such that the fol lowing c onditions hold: ‚ the image F p C q gener ates D as an idemp otent-c omplete stable sub c ate gory; ‚ for x, y P C ♡ the map Ext i C p x, y q Ñ Ext i D p F p x q , F p y qq is an isomorphism for i ď n ´ 1 , and a monomorphism for i “ n. Then the induc e d map K j p C q Ñ K j p D q is an isomorphism for j ě ´ n, and a monomor- phism for j “ ´ n ´ 1 . W e men tion the following immediate corollaries, whic h are again sp ecial cases of more general results for large categories (Corollary 3.2 and Theorem 6.1 (i)). 4 ALEXANDER I. EFIMOV Corollary 0.4. L et C b e a smal l stable c ate gory with a b ounde d t -structur e p C ě 0 , C ď 0 q . Supp ose that for some n ě 1 the maps Ext i C ♡ p x, y q Ñ Ext i C p x, y q ar e isomorphisms for i ď n and x, y P C ♡ (this always holds for n “ 1 ). Then the map K j p C ♡ q Ñ K j p C q is an isomorphism for j ě ´ n ´ 1 , and a monomorphism for j “ ´ n ´ 2 . Note that in the situation of the ab o ve corollary the maps Ext n ` 1 C ♡ p x, y q Ñ Ext n ` 1 C p x, y q are automatically monomorphisms for x, y P C ♡ . Corollary 0.5. L et F : B Ñ A b e a ful ly faithful exact functor b etwe en smal l ab elian c ate gories, such that F p B q gener ates A via extensions. Then the induc e d map K j p B q Ñ K j p A q is an isomorphism for j ě ´ 1 , and a monomorphism for j “ ´ 2 . W e note that the ab ov e results are new only when the hearts are non-no etherian (and non-artinian). Namely , if C ♡ is no etherian, then w e hav e K ă 0 p C q “ 0 b y a theorem of Antieau-Gepner-Heller [AGH19, Theorem 3.6], and we hav e K p C q » K H p C q (more precisely , K p C q » K p C b Perf p S r x 1 , . . . , x n sqq for all n ě 0 ). W e prov e that the estimates in Corollaries 0.4 and 0.5 are sharp (Theorem 7.1), ev en if we are dealing with small dg categories resp. small ab elian categories ov er a ﬁeld. In particular w e obtain that the naive K -theoretic theorem of the heart fails for K ´ 3 , which is a m uch more precise v ersion of a result of Ramzi-Sosnilo-Winges [RSW24, Theorem B]. W e now explain a v ery surprising analogy with the familiar results on connectiv e E 1 - ring sp ectra. In tuitiv ely the notion of a t -structure is dual to the notion of a weigh t structure [Bon10, Sos19]. In particular the b ounded t -structures are “dual” to b ounded w eigh t structures. Consider an idemp oten t-complete stable category C with a b ounded w eigh t structure p C w ě 0 , C w ď 0 q . F or simplicit y w e assume that C is generated by a single ob ject G (this assumption is harmless, it will mak e no essential diﬀerence in what follo ws). In this case we ma y assume that G is of weigh t zero. Putting R “ End C p G q (meaning the sp ectrum of endomorphisms), we obtain C » P erf p R q , and R is a connective E 1 -ring. Con v ersely , for a connectiv e E 1 -ring R the category P erf p R q has a unique bounded w eight structure such that R has weigh t zero [Sos19]. The follo wing statement is well-kno wn. It is due to W aldhausen [W al78] for simplicial rings and connective K -theory , and the general case is pro v en b y Land and T amme [L T19] (the case n “ 0 follo ws from the proof of [L T19, Corollary 3.5]). Prop osition 0.6. [W al78, Prop osition 1.1] [L T19, Lemma 2.4] L et n ě 0 b e an inte ger and let f : R Ñ S b e a map of c onne ctive E 1 -rings, such that the map π i p R q Ñ π i p S q is an isomorphism for i ď n ´ 1 , and an epimorphism for i “ n. If n “ 0 , we r e quir e that the ide al ker p π 0 p f qq Ă π 0 p R q is nilp otent. Then the induc e d map K j p R q Ñ K j p S q is an isomorphism for j ď n, and an epimorphism for j “ n ` 1 . THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 5 It is clear from the formulations that our Theorem 0.3 for some n ě 1 is “dual” to Prop osition 0.6 for n ´ 1 : b oth in the assumptions and in the conclusions one simply replaces π j with π ´ 1 ´ j for the relev an t sp ectra, and one replaces epimorphisms with monomorphisms. Contin uing this analogy , Corollary 0.4 for some n ě 1 is “dual” to the sp ecial case of Prop osition 0.6 for the same n when S “ π 0 p R q and π i p R q “ 0 for 1 ď i ď n ´ 1 . F urther, Corollary 0.5 is “dual” to the case n “ 0 and R “ π 0 p R q , S “ π 0 p S q . W e do not hav e any conceptual explanation for wh y such a duality takes place, but we note that in b oth con texts the (co)connectivity estimates are sharp. No w w e explain why from this p oint of view our Theorem 0.1 is “dual” to the Dundas- Go o dwillie-McCarth y theorem. W e recall some notation. W e denote b y Cat perf the cate- gory of idempotent-complete stable categories and consider the universal ﬁnitary lo calizing in v arian t U loc : Cat perf Ñ Mot loc [BGT13, E25c]. The category Mot loc is naturally sym- metric monoidal, and moreo v er it is rigid b y [E25c, Theorem 0.1] (we mean rigidit y in the sense of Gaitsgory and Rozenblyum [GaiRoz17, Deﬁnition 9.12]). W e also consider the univ ersal ﬁnitary A 1 -in v arian t lo calizing in v ariant U A 1 loc : Cat perf Ñ Mot loc , A 1 , so Mot loc , A 1 is a quotient of Mot loc b y the co complete ideal generated b y the reduced motiv e of the ﬂat aﬃne line r U loc p S r x sq . By [E25c, Theorem 8.1] this is a smashing lo calization. More pre- cisely , w e ha ve the idempotent E 8 -algebra in Mot loc , giv en by the geometric realization A “ | U loc p ∆ ‚ q| , where ∆ n “ Sp ec S r x 1 , . . . , x n s . W e hav e an equiv alence Mot loc , A 1 » Mo d - A Ă Mot loc . The inclusion functor i ˚ : Mot loc , A 1 Ñ Mot loc has a left adjoint i ˚ and a right adjoin t i ! , giv en by i ˚ p M q “ M b A, i ! p M q “ Hom p A, M q , where Hom is the in ternal Hom in Mot loc . W e also denote b y Γ : Mot loc , A 1 Ñ Sp the functor of global sections, i.e. Γ “ Hom p 1 , ´q , where 1 is the unit ob ject A. By [E25c, Theorem 9.3] (whose proof of course uses the DGM theorem) for a connective E 1 -ring R we hav e a natural equiv alence of sp ectra (0.2) Γ p i ! p U loc p R qqq » K inv p R q “ Fib er p K p R q Ñ TC p R qq , where U loc p R q “ U loc p P erf p R qq , and TC p R q is the top ological cyclic homology [BHM93, NS18]. The map K p R q Ñ TC p R q is the cyclotomic trace. In view of the equiv alence (0.2), the DGM theorem states that w e ha v e an equiv alence Γ p i ! p U loc p R qqq „ Ý Ñ Γ p i ! p U loc p π 0 p R qqqq . More precisely , [DGM13, Theorem 7.0.0.2] states a more general version for a quotien t of π 0 p R q b y a nilp otent ideal, but this is automatic by [L T19, Corollary 3.5]. 6 ALEXANDER I. EFIMOV No w, by [E25c, Corollary 8.3] for C P Cat perf w e hav e Γ p i ˚ p U loc p C qqq » K H p C q . Therefore, the statement of Theorem 0.1 is “dual” to the DGM theorem if w e contin ue the ab o ve analogy . This was in fact our motiv ation for wh y theorem of the heart should hold for K H . The original goal of this pap er was muc h more modest: to pro ve the generalizations of Barwic k’s theorem of the heart [Bar15, Theorem 6.1] and Quillen’s D ´ evissage theorem [Qui73, Theorem 4] for large categories. Ho wev er, the metho ds which w e dev elop ed allo wed for muc h b etter results. W e brieﬂy explain whic h large abelian categories w e consider, and whic h t -structures on large categories are “nice” in this con text. A Grothendieck abelian category A is c oher ently assemble d if the colimit functor colim : Ind p A q Ñ A has an exact left adjoint ˆ Y : A Ñ Ind p A q . Recall that the mere existence of this left adjoin t means that the category is compactly assem bled [Lur18, Deﬁnition 21.1.2.1] (since A is an ordinary category , this means that A is con tinuous in the sense of Jo yal and Johnstone [JJ82]). In our situation this is equiv alent to the condition that A satisﬁes the Grothendieck’s axiom (AB6) [Gro57]. As explained in [E24, Prop osition E.8], if A is a compactly generated Grothendieck ab elian category , then A is coherently assembled if and only if A is lo cally coheren t (i.e. equiv alent to an ind-completion of a small abelian category). Recall that if A is a Grothendieck abelian category , then its unsep ar ate d derive d c ate gory ˇ D p A q is identiﬁed with the (dg nerve of ) the dg category Ch p Inj A q of unbounded complexes of injectiv e ob jects of A , see [Kr05, Kr15] and [Lur18, Section C.5.8]. If A is moreov er coheren tly assem bled, then the category ˇ D p A q is dualizable (Prop ositions 1.26 and 1.34). Hence, we can deﬁne the contin uous K -theory of A : K cont p A q “ K cont p ˇ D p A qq . If A is lo cally coherent, then ˇ D p A q » Ind p D b p A ω qq b y [Kr15, Theorem 4.9], hence K cont p A q » K p A ω q . No w, if C is a dualizable (presentable stable) category , then a t -structure p C ě 0 , C ď 0 q is called c omp actly assemble d if it is accessible, compatible with ﬁltered colimits and the functor ˆ Y : C Ñ Ind p C q (left adjoint to colim ) is t -exact. In this case the heart C ♡ is automatically coheren tly assem bled. W e further imp ose a condition that the t -structure is c ontinuously b ounde d , whic h means that C is generated b y C b (equiv alen tly , by C ♡ ) as a localizing sub category . F or brevit y we say that C is a dualizable t -category if it is a dualizable category equipp ed with a compactly assembled con tinuously bounded t - structure. THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 7 F or a dualizable t -category C w e hav e a (strongly contin uous) realization functor ˇ D p C ♡ q Ñ C , whic h follo ws from the univ ersal prop erty of ˇ D p A q explained in [Lur17b], and also from the univ ersal property of ˇ D p A q ě 0 from [Lur18, Proposition C.5.5.20, Theo- rem C.5.8.8]. If C is compactly generated, then the t -structure on C induces a bounded t -structure on C ω , and the functor ˇ D p C ♡ q Ñ C is identiﬁed with the ind-completion of the realization functor D b p C ω , ♡ q Ñ C ω . With this in mind, it is clear that the ab ov e state- men ts for small stable categories with b ounded t -structures can at least b e form ulated more generally for dualizable t -categories. Ho w ev er, one has to b e cautious with generalizations to the dualizable context. Namely , for a small stable category D with a b ounded t -structure w e hav e K ´ 1 p D q “ 0 by a theorem of An tieau-Gepner-Heller [AGH19, Theorem 2.35]. How ever, for a dualizable t - category C in general w e ha ve K cont ´ 1 p C q ‰ 0 , even if the realization functor ˇ D p C ♡ q Ñ C is an equiv alence (Corollary 8.2). This is a very rare example of a situation when a reasonable statemen t ab out dualizable categories fails in general, but holds for compactly generated categories. W e mention the follo wing natural questions which are not studied in this paper. The ﬁrst one is somewhat v ague, and it is motiv ated b y the w ork of Levy and Sosnilo [LS25] and by the abov e analogy (dualit y) with w eight structures. Question 0.7. L et C b e a dualizable c ate gory with a t -structur e p C ě 0 , C ď 0 q satisfying the fol lowing c onditions: the t -structur e is ac c essible, c omp atible with ﬁlter e d c olimits, C is gener ate d by C ♡ as a lo c alizing sub c ate gory, and for some n ě 0 the functor ˆ Y r´ n s : C Ñ Ind p C q is left t -exact. The (dualizable versions of ) the ab ove r esults de al with the c ase n “ 0 . Is ther e a r e asonable gener alization to the c ase n ą 0? The following question is motiv ated by the ab o ve analogy with connective E 1 -rings and b y the functor TR (topological restriction) with co eﬃcients in a bimo dule [LM12, KMN23]. W e in tro duce some non-standard notation for the “dual” of the familiar Go o dwillie to wer [Go o03]. Let Φ : C Ñ D b e a (not necessarily exact) functor b et ween ω 1 -presen table stable categories, commuting with ω 1 -ﬁltered colimits. F or n ě 0 denote by P _ n p Φ q : C Ñ D the univ ersal p olynomial functor of degree ď n, commuting with ω 1 -ﬁltered colimits, with a map P _ n p Φ q Ñ Φ (its existence follows from the adjoin t functor theorem). W e sa y that the c o analytic appr oximation of Φ is the colimit lim Ý Ñ n P _ n p Φ q (computed ob ject wise). Question 0.8. L et C b e a dualizable c ate gory (without a t -structur e), and let F : C Ñ C b e a c olimit-pr eserving endofunctor. We c onsider the tensor algebr a T p F q “ À n ě 0 F ˝ n as a monad on C , and take the c ate gory of mo dules Mo d T p F q p C q , which is dualizable by [E24, 8 ALEXANDER I. EFIMOV Prop osition C.1] . Consider the functor (0.3) F un L p C , C q Ñ Sp , F ÞÑ r K cont p Mo d T p F q p C qq “ Cone p K cont p C q Ñ K cont p Mo d T p F q p C qqq . We denote by TR _ p C , ´q : F un L p C , C q Ñ Sp the c o analytic appr oximation of (0.3) . Now supp ose that C is a dualizable t -c ate gory and F : C Ñ C is (c olimit-pr eserving and) left t -exact. Is it true that the map TR _ p C , F q Ñ r K cont p Mo d T p F q p C qq is an e quivalenc e of sp e ctr a? W e give a brief o v erview of the structure of the paper. In Section 1 w e introduce and study lo cally κ -coheren t exact 8 -categories, fo cusing on coheren tly assem bled exact 8 -categories. W e are mostly interested in the presentable stable en v elop e ˇ St p E q when E is coherently assembled, and in the prop erties of the category ˇ St p E q , suc h as dualizability . In particular, we deﬁne the contin uous K -theory of such exact 8 - categories b y putting K cont p E q “ K cont p ˇ St p E qq . Our approac h in this section is somewhat minimalistic, in particular w e mostly a void the non-coherently assembled case. W e reduce most of the non-trivial statemen ts to the familiar results ab out small exact 8 -categories and their (small) stable env elop es. Among other things, we explain a natural v ersion of the situation of d´ evissage for coheren tly assem bled ab elian categories in Subsection 1.9. The results of this section are used in the pro ofs of our main theorems, esp ecially in Section 3. In Section 2 we in tro duce and study compactly assem bled t -structures on dualizable categories, mostly focusing on contin uously b ounded t -structures. In particular, in Sub- section 2.4 w e explain a general method of constructing a nice t -structure via a functor from a coherently assem bled ab elian category , based on a similar result for small categories. This metho d allo ws to construct man y auxiliary categories with t -structures, which are used essen tially in all the pro ofs of the main theorems. In particular, w e consider the pre- sen table stable en velopes ˇ St p C r 0 ,m s q with natural t -structures with the same heart. Here C r 0 ,m s “ C ě 0 X C ď m is considered as an exact 8 -category with the induced exact structure from C . Again, our approach in this section is minimalistic, and some statemen ts can b e generalized to the case when C is merely presen table stable, the t -structure is compatible with ﬁltered colimits and right complete, and the Grothendieck prestable category C ě 0 is an ticomplete in the sense of [Lur18, Deﬁnition C.5.5.4]. THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 9 In Section 3 we pro ve reﬁned theorem of the heart in the con text of dualizable cate- gories (Theorem 3.1). It states that for a dualizable t -category C and for m ě 0 the sp ectrum Cone p K cont p C r 0 ,m s q Ñ K cont p C qq is p´ m ´ 3 q -co connective. This in particular implies the abov e Corollary 0.4. One of the key ingredien ts is the surjectivity of the map K cont ´ m ´ 2 p C r 0 ,m s q Ñ K ´ m ´ 2 p C q , whic h is pro ved by a v ery non-trivial argument. In particular, it requires some careful analysis of the (compactly generated) iterated Calkin categories and natural t -structures on them. Another important ingredient is the construction of certain short exact sequences of dualizable categories (Prop osition 3.7), which relate the categories ˇ St p C r 0 ,m s q for v arious m. In Section 4 we prov e Theorem 4.1 on the co connectivity estimates, whic h is a gener- alization of Theorem 0.3. One of the main ideas is a v ery abstract construction: given a strongly contin uous exact functor F : C Ñ D b et w een dualizable categories suc h that F p C q generates D , there is a canonical sequence C “ C 0 F 0 Ý Ñ C 1 F 1 Ý Ñ C 2 . . . in Cat dual st , suc h that lim Ý Ñ n C n » D and eac h monad F R n ˝ F n on C n is obtained as a so-called deformed tensor algebra. If C and D are dualizable t -categories and F is t -exact, then all C n are also naturally dualizable t -categories and the transition functors are t -exact. A further analysis of deformed tensor algebras allows to ev entually reduce Theorem 4.1 to Theorem 3.1 (reﬁned theorem of the heart). W e also deduce similar results for higher nil groups (Corollaries 4.11 and 4.12). In Section 5 we pro v e theorem of the heart for K H cont for dualizable t -categories (The- orem 5.1), whic h is a generalization of Theorem 0.1. Here w e brieﬂy explain a more precise statemen t, for simplicity restricting to the case of small categories. F or k ě 0 denote b y U k “ Fil k K H the k -th term of the standard ﬁltration on K H , namely U k p C q is obtained b y taking the colimit o v er r n s P ∆ op ď k in the right hand side of (0.1). Then the lo calizing in v arian ts U k form a direct sequence with U 0 p C q “ K p C q and lim Ý Ñ k U k p C q » K H p C q . W e pro v e b y induction on m ě 0 that if F : C Ñ D satisﬁes the assumptions of Theorem 0.3 for some n ě 1 , then the map of sp ectra τ ě c Cone p U k p F qq Ñ τ ě c Cone p U l p F qq is null-homotopic for c ě ´ 2 m p n ´ 1 q ` 2 k ´ 1 , l ´ k ě 2 m ´ 1 . The case n “ 2 directly implies Theorem 0.1. In Section 6 we deduce d´ evissage theorems for K H cont and K cont of coherently assem- bled ab elian categories, which are generalizations of Theorem 0.1 and Corollary 0.5. 10 ALEXANDER I. EFIMOV In Section 7 w e pro v e that our estimates in Corollaries 0.4 and 0.5 are sharp (Theorem 7.1), working ov er a ﬁeld k . The constructions are based on the exact category E of vector bundles on a (pro jectiv e) cuspidal cubic curv e X ov er k , and on the ab elian category eﬀ p E q of eﬀaceable functors E op Ñ Ab . Essen tially w e only use that X is prop er and N K 0 p X q ‰ 0 . W e also prov e the sharpness of the estimates for higher nil groups (Theorem 7.3), also giving a non-trivial example of a computation of D K p A q “ Fiber p K p A q Ñ K H p A qq for an ab elian category A . In Section 8 w e study examples of compactly assem bled t -structures and coherently assem bled ab elian categories. In particular, w e consider shea ves on lo cally compact Haus- dorﬀ spaces (Prop osition 8.1). As a sp ecial case, we obtain that the ab elian category A of shea v es of vector spaces on the real line is coheren tly assem bled, and w e ha v e K cont ´ 1 p A q – Z . Therefore, already the Schlic hting’s v anishing theorem for K ´ 1 of small ab elian categories [Sc hl06, Theorem 6] do es not generalize to coherently assembled ab elian categories. W e con- sider the category of n uclear solid modules o v er Z p (the original v ersion due to Clausen and Sc holze [CS20]), and sho w that it has a natural compactly assem bled con tinuously b ounded t -structure (Prop osition 8.3). W e also pro ve a generalization of Chase criterion [Ch60] of coherence of associative unital rings. This is Theorem 8.5. One w a y to state it is as follows: a dualizable Ab -mo dule A in Pr L is a coherently assembled abelian category if and only if the category of ﬂat ob jects in A _ has inﬁnite products. Here an ob ject x P A _ is ﬂat if the functor e v p´ , x q : A Ñ Ab is left exact. W e recall that an Ab -mo dule in Pr L is the same thing as a presen table additive (ordinary) category A . By the forthcoming w ork of Levy , Liang and Sosnilo [LLS], A is dualizable if and only if A is a Grothendieck ab elian category satisfying (AB6) and (AB4*). Equiv alen tly b y Roos’ theorem [Ro o65] this means that A is equiv alen t to a category of almost modules, i.e. A » Mo d - R { Mo d - p R { I q , where R is an asso ciativ e unital ring and I Ă R is a tw o-sided ideal such that I 2 “ I . Finally , in Subsection 8.4 w e mention some example and questions which are not cov ered by this pap er. Con v en tion. We wil l fr e ely use the the ory of 8 -c ate gories as develop e d in [Lur09, Lur17a] . We wil l simply say “c ate gory” inste ad of 8 -c ate gory. If the c ate gory is 0 -trunc ate d (i.e. the morphism sp ac es ar e discr ete), we wil l say “or dinary c ate gory”. Sinc e a stable c ate gory C is natur al ly enriche d over sp e ctr a, for x, y P C we denote by Hom C p x, y q the sp e ctrum of morphisms. We denote by Sp the c ate gory of sp e ctr a, and by Sp ě 0 the c ate gory of c onne ctive sp e ctr a. As we alr e ady mentione d ab ove, we denote by Cat perf the c ate gory of idemp otent-c omplete smal l stable c ate gories and exact functors. Thr oughout this p ap er we use the homolo gic al gr ading, unless otherwise state d. THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 11 Ac kno wledgemen ts. I am grateful to Dmitry Kaledin, Ishan Levy , Maxime Ramzi and Vladimir Sosnilo for useful discussions. This w ork w as p erformed at the Steklov Interna- tional Mathematical Cen ter and supp orted b y the Ministry of Science and Higher Education of the Russian F ederation (agreemen t no. 075-15-2025-303). I was partially supp orted by the HSE Univ ersity Basic Researc h Program. 1. Coherentl y assembled abelian ca tegories and exact ca tegories 1.1. Reminder on exact categories. W e recall some material from [Bar15, Kle22, SW26] on small exact categories. Deﬁnition 1.1. [Bar15, Section 3] An exact c ate gory is an additive c ate gory E with a choic e of two sub c ate gories p in E , pr E q , c onsisting of (exact) inclusions r esp. (exact) pr oje ctions, such that the fol lowing c onditions hold. (i) F or x P E , the map 0 Ñ x is an inclusion and the map x Ñ 0 is a pr oje ction. (ii) The class of inclusions is stable under pushouts along al l maps, and the class of pr oje ctions is stable under pul lb acks along al l maps. (iii) F or a squar e x y x 1 y 1 i p q j in E the fol lowing ar e e quivalent: (a) i is an inclusion, p is a pr oje ction and the squar e is c o c artesian; (b) j is an inclusion, q is a pr oje ction and the squar e is c artesian. Such a squar e is c al le d an exact squar e. If x 1 “ 0 , we wil l say that x Ñ y Ñ z is a short exact se quenc e in E . W e will omit the sub categories in E and pr E from the notation, assuming that they are chosen. An additiv e functor F : E Ñ E 1 b et ween exact categories is called exact if it preserv es short exact sequences, or equiv alen tly exact squares. W e denote b y F un ex p E , E 1 q the (additiv e) category of exact functors E Ñ E 1 . W e denote b y Exact the category of small exact categories and exact functors. W e sa y that an exact functor b et w een exact categories F : E Ñ E 1 reﬂects exactness if for any square in E , if its image is exact in E 1 , then it is exact in E . If w e denote b y Add the category of small additive categories and additive functors, then the forgetful functor Exact Ñ Add has a left adjoint A ÞÑ A add . Here A add is the exact category p A , in A , pr A q , where in A consists of split monomorphisms and pr A consists of split epimorphisms. W e call this a split exact structure on A . Note that if E is a small 12 ALEXANDER I. EFIMOV exact category , then the iden tity functor E add Ñ E is exact, but it does not reﬂect exactness in general. Note that any stable category C can b e considered as an exact category with in C “ pr C “ C . F or stable categories exactness of a functor in the abov e sense is equiv alent to the exactness in the usual sense. Denoting b y Cat ex the category of small stable categories and exact functors, we obtain a fully faithful functor Cat ex ã Ñ Exact . By [Kle22, Prop osition 4.22, Corollary 4.24] it has a left adjoint, whic h we denote by St : Exact Ñ Cat ex st . The category St p E q is called a stable env elop e of the category E . The following is due to Klemenc. Prop osition 1.2. [Kle22, Prop ositions 4.17, Proposition 4.25] L et E b e a smal l exact c ate- gory, and denote by j : E Ñ St p E q the universal exact functor. Then j is ful ly faithful and r eﬂe cts exactness, and its essential image is close d under extensions. W e refer to [SW26] for further details. W e will need the follo wing more precise state- men t ab out the stable env elop e of an exact category E . First, consider the stable category St p E add q » St add p E q with the univ ersal additiv e functor Y : E Ñ St p E add q . W e hav e Ind p St p E add qq » F un add p E op , Sp q , where the sup erscript “add” means that we consider the additiv e functors. W e identify St p E add q with the full sub category of this category of functors, and w e ha ve Y p x q “ Hom E p´ , x q . Clearly , w e ha v e a quotien t functor St p E add q Ñ St p E q . Its k ernel is denoted b y Ac p E q , and it is generated as a stable subcategory b y the ob jects of the form E p f q P Ac p E q , where x f Ý Ñ y Ñ z is a short exact sequence in E and (1.1) E p f q “ Cone p Cone p Y p f qq Ñ Y p z qq . As explained for example in [SW26, Prop osition 3.9], the asso ciated functor E op Ñ Sp tak es v alues in Sp ♡ » Ab , and w e ha ve E p f q – coker p π 0 Y p y q Ñ π 0 Y p z qq . Suc h functors E op Ñ Ab are called eﬀaceable, and they form an ordinary additive category , denoted by eﬀ p E q . This category is in fact ab elian, more precisely the follo wing holds. Prop osition 1.3. [Nee21, Lemma 1.2] [E24, Prop osition G.7] [SW26, Corollary 3.12] The c ate gory Ac p E q has a b ounde d t -structur e, which is induc e d by the standar d (Postnikov) t -structur e on F un add p E op , Sp q . We have Ac p E q ♡ » eﬀ p E q . Remark 1.4. If E is an or dinary exact c ate gory, then the ab ove c ate gory Ac p E q is denote d by Ac b p E q in [Nee21] , sinc e it is the c ate gory of b ounde d acyclic c omplexes. This should not le ad to c onfusion, b e c ause we wil l not c onsider the unb ounde d version. THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 13 W e also recall a more general class of functors. Recall that for an exact category E , an additiv e functor F : E op Ñ Ab is called we akly eﬀac e able if for any x P E and for an y α P F p x q there exists a pro jection f : y Ñ x in E suc h that F p f qp α q “ 0 . W e denote b y Eﬀ p E q the category of w eakly eﬀaceable functors E op Ñ Ab . It is easy to see that we ha v e eﬀ p E q Ă Eﬀ p E q . Moreov er, by [SW26, Prop osition 3.9] an additiv e functor E op Ñ Ab is eﬀaceable if and only if it is ﬁnitely presented and w eakly eﬀaceable. It is easy to see (assuming Proposition 1.3) that the category Eﬀ p E q is ab elian and lo cally coherent, and its full sub category of ﬁnitely presen ted ob jects coincides with eﬀ p E q . Indeed, it suﬃces to observ e that an y map from a ﬁnitely presented additive functor F : E op Ñ Ab to a weakly eﬀaceable functor G : E op Ñ Ab factors through an eﬀaceable functor. W e will use the following notation. If C is a stable category , then for x, y P C and n P Z w e put Ext n C p x, y q “ π ´ n Hom C p x, y q . F or a small exact category E and for x, y P E we put Ext n E p x, y q “ Ext n St p E q p j p x q , j p y qq , where j : E Ñ St p E q is the univ ersal exact functor. By Prop osition 1.2 we ha ve Ext ´ n E p x, y q “ π n Map E p x, y q for n ě 0 . Also, the abelian group Ext 1 E p x, y q classiﬁes extensions of x by y in E . The following is a w ell-kno wn observ ation, but we are not a ware of a precise reference. Prop osition 1.5. L et C b e a stable c ate gory. L et E Ă C b e a smal l ful l additive sub c ate gory close d under extensions. We c onsider E as an exact c ate gory with the induc e d exact struc- tur e fr om C . Consider the exact functor St p E q Ñ C induc e d by the inclusion E Ñ C . Then for x, y P E the induc e d map Ext n E p x, y q Ñ Ext n C p x, y q is an isomorphism for n ď 1 , and a monomorphism for n “ 2 . Pr o of. The isomorphisms for n ď 1 follo w from Prop osition 1.2 and from the description of Ext 1 via extensions. Now take an element α P Ext 2 E p x, y q . By [SW26, Lemma 4.3] there exists a pro jection p : z Ñ x suc h that w e ha ve p ˚ p α q “ 0 P Ext 2 E p z , y q . Putting w “ Fib er p α q P E , we see that α “ β ˝ γ , where β P Ext 1 E p w , y q , γ P Ext 1 E p x, w q . If the image of α in Ext 2 C p x, y q is zero, then the map w Ñ y r 1 s in C factors through z , hence β is contained in the image of the map Ext 1 E p z , y q Ñ Ext 1 E p w , y q . But this exactly means that α “ 0 . This shows the injectivit y of the map Ext 2 E p x, y q Ñ Ext 2 C p x, y q . □ W e recall the following class of exact sub categories. The condition in the follo wing deﬁnition already appears in [Kel96]. Deﬁnition 1.6. [SW26, Deﬁnition 1.1] L et E b e an exact c ate gory. A ful l additive sub c ate gory E 1 Ă E is c al le d left sp e cial if E 1 is close d under extensions in E , and for any pr oje ction p : y Ñ x in E with x P E 1 , ther e exists z P E 1 and a map f : z Ñ y , such that the c omp osition p ˝ f : z Ñ x is a pr oje ction in E . 14 ALEXANDER I. EFIMOV Theorem 1.7. [SW26, Theorem 4.5] L et E b e a smal l exact c ate gory, and let E 1 Ă E b e a left sp e cial ful l sub c ate gory. We c onsider E 1 as an exact c ate gory with an induc e d exact structur e. Then the functor St p E 1 q Ñ St p E q is ful ly faithful. F or completeness we mention the follo wing statement, which might b e not in the litera- ture. In the case of ordinary exact categories this is [BS01, Theorem 2.8], with a diﬀeren t pro of. Prop osition 1.8. L et E b e an idemp otent-c omplete exact c ate gory. Then the c ate gory St p E q is also idemp otent-c omplete. Pr o of. Denote b y St p E q Kar the idempotent completion. The category St p E add q is idemp oten t-complete by [Bon10, Theorem 5.3.1, Proposition 6.2.1], so w e hav e a short exact sequence in Cat perf : 0 Ñ Ac p E q Ñ St p E add q Ñ St p E q Kar Ñ 0 . By Prop osition 1.3 the category Ac p E q has a b ounded t -structure, hence by [AGH19, Theorem 2.35] w e hav e K ´ 1 p Ac p E qq “ 0 . The long exact sequence giv es the surjectivit y of the map K 0 p St p E add qq Ñ K 0 p St p E q Kar q . By [Th97, Theorem 2.1] this implies that St p E q is idemp oten t-complete. □ 1.2. Accessibly exact categories. T o deal with large exact categories w e will need the fol- lo wing basic construction. W e will use the follo wing notation: if A is a small category and κ is a regular cardinal, then w e write the ob jects of Ind κ p A q as “lim Ý Ñ i P I ” x i , where I is a κ -directed p oset and we are assuming a functor I Ñ A , i ÞÑ x i . One can more generally tak e I to b e a κ -ﬁltered 8 -category , but recall that b y [Lur09, Prop osition 5.3.1.18] for suc h I there exists a κ -directed poset J and a coﬁnal functor J Ñ I . Prop osition 1.9. L et κ ď λ b e a r e gular c ar dinals. L et E b e a smal l exact c ate gory. Then the c ate gory Ind κ p E q λ has the fol lowing exact structur e. A morphism f in Ind κ p E q λ is an inclusion r esp. a pr oje ction if it is a λ -smal l κ -dir e cte d c olimit of inclusions r esp. pr oje ctions in E . T aking the union over λ, we obtain an exact structur e on Ind κ p E q . The (ful ly faithful) functor Ind κ p E q Ñ Ind κ p St p E qq pr eserves and r eﬂe cts exactness. Its essential image is close d under extensions. Pr o of. Consider the stable env elop e St p E q , and denote b y j 1 : E Ñ St p E q the universal exact functor. By Prop osition 1.2 j 1 is fully faithful, reﬂects exactness, and its ess en tial image is closed under extensions in St p E q . Consider the stable category C “ Ind κ p St p E qq λ . Then we hav e a fully faithful functor j : Ind κ p E q λ Ñ C . THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 15 W e claim that the essential image of j is closed under extensions. Indeed, consider a coﬁb er sequence of the form j p x q Ñ z Ñ j p y q in C . By [Lur09, Prop osition 5.3.5.15] and its pro of the morphism j p y q Ñ j p x qr 1 s is a λ -small κ -directed colimit of morphisms j 1 p y i q Ñ j 1 p x i qr 1 s , where x i , y i P E , i P I . Hence, z – “lim Ý Ñ i P I ” Fib er p j 1 p y i q Ñ j 1 p x i qr 1 sq P C is contained in the essential image of j, since eac h ob ject Fib er p j 1 p y i q Ñ j 1 p x i qr 1 sq is con tained in the essen tial image of j 1 . This shows that the essen tial image of j is closed under extensions. Consider the induced exact structure on Ind κ p E q λ . The same argument as abov e sho ws that short exact sequences in this category are exactly the λ -small κ -directed colimits of short exact sequences in E . Finally , the assertions about the functor Ind κ p E q Ñ Ind κ p St p E qq follo w from the abov e. □ Deﬁnition 1.10. We deﬁne a lo c al ly κ -c oher ent exact c ate gory to b e a c ate gory of the form Ind κ p E q , wher e E is a smal l exact c ate gory and κ is a r e gular c ar dinal. The exact structur e is describ e d in Pr op osition 1.9. F or κ “ ω we say “lo c al ly c oher ent” inste ad of “lo c al ly ω -c oher ent”. A n ac c essibly exact c ate gory is a lo c al ly κ -c oher ent exact c ate gory for some κ. W e will consider the accessible stable categories as accessibly exact categories in the natural wa y . Prop osition 1.11. L et E b e a smal l exact c ate gory and let κ b e a r e gular c ar dinal. (i) The inclusion functor Ind κ p E q Ñ Ind p E q pr eserves and r eﬂe cts exactness. Its es- sential image is close d under extensions. (ii) L et E 1 Ă E b e an additive sub c ate gory close d under extensions, e quipp e d with the induc e d exact structur e fr om E . Then the functor Ind κ p E 1 q Ñ Ind κ p E q pr eserves and r eﬂe cts exactness. Its essential image is close d under extensions. Pr o of. (i) If E is stable with the standard exact structure, then the statemen t is eviden t. The general case reduces to the stable case using Prop osition 1.9. Namely , we hav e a comm utativ e square Ind κ p E q Ind p E q Ind κ p St p E qq Ind p St p E qq , and b oth vertical functors are fully faithful, preserv e and reﬂect exactness, and their essen tial images are closed under extensions. The same holds for the low er horizon tal functor by the ab o ve sp ecial case. Hence, the same holds for the upp er horizon tal functor. 16 ALEXANDER I. EFIMOV (ii) The pro of is similar, but a little caution is required: the functor St p E 1 q Ñ St p E q do es not hav e to be fully faithful. Instead, w e ﬁrst observ e that w e can replace E by St p E q , by Prop osition 1.9. Assuming that E is stable, the same argumen t as in loc. cit. sho ws that the essential image of Ind κ p E 1 q in Ind κ p E q is closed under extensions. It also sho ws that this functor reﬂects exactness. □ The following deﬁnition is essen tially tautological, but we sp ell it out for completeness. Deﬁnition 1.12. L et E b e an ac c essibly exact c ate gory and let κ b e a r e gular c ar dinal such that E has κ -ﬁlter e d c olimits. We say that the κ -ﬁlter e d c olimits ar e exact in E if the class of short exact se quenc es in E is close d under κ -ﬁlter e d c olimits. This exactness holds in the follo wing basic situation. Prop osition 1.13. L et E b e a lo c al ly κ -c oher ent exact c ate gory. Then the κ -ﬁlter e d c olimits ar e exact in E . Pr o of. Again, the stable case is evident. The general case reduces to the stable case b y Prop osition 1.9: the functor E Ñ Ind κ p St p E κ qq commutes with κ -ﬁltered colimits and reﬂects exactness. □ The following observ ation is sligh tly subtle. Prop osition 1.14. L et E b e an ac c essibly exact c ate gory and let κ b e a r e gular c ar dinal. Supp ose that E has exact κ -ﬁlter e d c olimits. Then the ful l sub c ate gory E κ Ă E is close d under extensions. Pr o of. Let x Ñ y Ñ z b e a short exact sequence in E , and supp ose that x, z P E κ . Let p w i q i P I b e a κ -directed system in E . W e need to prov e that the map lim Ý Ñ i Map E p y , w i q Ñ Map E p y , lim Ý Ñ i w i q is an equiv alence of spaces. By assumption this holds if we replace y with x or z . Using the long exact sequences and the ﬁv e-lemma, we reduce the question to sho wing that the map of ab elian groups (1.2) lim Ý Ñ i Ext 1 E p z , w i q Ñ Ext 1 E p z , lim Ý Ñ i w i q is a monomorphism. T ak e some i 0 P I and an elemen t α P Ext 1 E p z , w i 0 q . It corresp onds to a short exact sequence w i 0 Ñ u Ñ z . F or i ě i 0 consider the pushout u i “ u \ w i 0 w i . Since κ -ﬁltered colimits are exact b y assumption, the image of α in Ext 1 E p z , lim Ý Ñ i w i q corresp onds to the short exact sequence lim Ý Ñ i ě i 0 w i Ñ lim Ý Ñ i ě i 0 u i Ñ z . Supp ose that this extension splits, i.e. w e THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 17 ha v e a section s : z Ñ lim Ý Ñ i ě i 0 u i . Then s factors through u i 1 for some i 1 ě i 0 . This means that the image of α in Ext 1 E p z , w i 1 q v anishes. This pro ves the injectivit y of (1.2) and the prop osition. □ In the situation of Proposition 1.14 we consider E κ as an exact category with the induced exact structure from E . W e mak e further observ ations ab out the subcategories E λ for v arious λ. Prop osition 1.15. L et E b e a lo c al ly κ -c oher ent exact c ate gory and let λ ě κ b e a r e gular c ar dinal. (i) F or any x P E κ the functor Ext 1 E p x, ´q : E Ñ Ab c ommutes with κ -ﬁlter e d c olimits. (ii) The inclusion functor E κ Ñ E λ is left sp e cial. In p articular, the functor St p E κ q Ñ St p E λ q is ful ly faithful. Pr o of. W e pro ve (i). If E is κ -accessible stable with the standard exact structure, then the statemen t is eviden t. The general case reduces to the stable case b y the pro of of Prop osition 1.9: we ha ve a fully faithful κ -accessible functor i : E Ñ Ind κ p St p E κ qq “ C , and Ext 1 E p x, ´q – Ext 1 C p i p x q , i p y qq . This pro ves (i). The second assertion of (ii) follo ws from the ﬁrst one by Theorem 1.7. W e prov e the left sp ecialit y . Consider a short exact sequence x Ñ y Ñ z in E λ with z P E κ . Denote b y α P Ext 1 E p z , x q the corresp onding elemen t. Let x “ lim Ý Ñ i P I x i , where I is λ -small and κ -directed, and x i P E κ for i P I . By (i) there exists some i 0 P I and an element β P Ext 1 E p z , x i 0 q whic h maps to α. W e obtain the asso ciated short exact sequence x i 0 Ñ w Ñ z . B y Prop osition 1.14 we hav e w P E κ . By construction, the pro jection w Ñ z factors through y , whic h pro ves that E κ is left special in E λ . □ In the situation of Proposition 1.15 w e deﬁne the category St p E q to be the directed union of St p E λ q o ver all regular λ ě κ. 1.3. Grothendieck exact categories. Deﬁnition 1.16. L et E b e an ac c essibly exact c ate gory. We say that E is a Gr othendie ck exact c ate gory if E has exact ﬁlter e d c olimits. It is con venien t to use the following conv ention: if E is a lo cally κ -coherent exact category , then Ind p E q is a union of exact categories Ind p E λ q for all regular λ ě κ. Prop osition 1.17. L et E b e a lo c al ly κ -c oher ent exact c ate gory with ﬁlter e d c olimits. Sup- p ose that the c olimit functor colim : Ind p E κ q Ñ E is exact. Then E is a Gr othendie ck exact c ate gory. 18 ALEXANDER I. EFIMOV Pr o of. W e need to show that the functor colim : Ind p E q Ñ E is exact. By Prop osition 1.13 for an y small exact category D the colimit functor Ind p Ind p D qq Ñ Ind p D q is exact. Hence, so is the comp osition Ind p E q » Ind p Ind κ p E κ qq Ñ Ind p Ind p E κ qq colim Ý Ý Ý Ñ Ind p E κ q colim Ý Ý Ý Ñ E . This prov es the prop osition. □ Example 1.18. A Gr othendie ck ab elian c ate gory c an b e c onsider e d as a Gr othendie ck exact c ate gory. Example 1.19. L et E b e a smal l exact c ate gory. Then Ind p E q is a Gr othendie ck exact c ate gory by Pr op osition 1.13. Example 1.20. Any pr esentable stable c ate gory c an b e c onsider e d as a Gr othendie ck exact c ate gory. Mor e gener al ly, any Gr othendie ck pr estable c ate gory ( [Lur18, Deﬁnition C.1.4.2] ) is a Gr othendie ck exact c ate gory. 1.4. Criterion for an exact category to be locally κ -coherent. W e will use the follo wing general characterization of locally κ -coherent exact categories. Prop osition 1.21. L et E b e an ac c essibly exact c ate gory and let κ b e a r e gular c ar dinal. Then E is lo c al ly κ -c oher ent if and only if the fol lowing c onditions hold. (i) E is κ -ac c essible (as an abstr act c ate gory). (ii) The κ -ﬁlter e d c olimits ar e exact in E . (iii) F or x P E κ the functor Ext 1 E p x, ´q : E Ñ Ab c ommutes with κ -ﬁlter e d c olimits. Mor e over, in this c ase the ful l sub c ate gory E κ Ă E is close d under ﬁb ers of pr oje ctions in E . Pr o of. W e ﬁrst prov e the “only if ” direction. (i) holds by deﬁnition, (ii) holds b y Prop osition 1.13 and (iii) holds by Prop osition 1.15. W e pro ve the “moreo ver” assertion. Let x Ñ y Ñ z b e a short exact sequence in E with y , z P E κ . Consider the functor i : E Ñ Ind κ p St p E κ qq from Proposition 1.9. Then the ob jects i p y q , i p z q are κ -compact, hence so is i p x q . Since i is fully faithful and commutes with κ -ﬁltered colimits, it follows that x is κ -compact in E . No w w e pro v e the “if ” direction. By (ii) and by Prop osition 1.14 the full sub category E κ Ă E is closed under extensions. W e consider E κ as a small exact category with the induced exact structure from E . W e equip Ind κ p E κ q with the exact structure from Proposi- tion 1.9. By (ii) the natural functor Φ : Ind κ p E κ q Ñ E is exact. By (i) Φ is an equiv alence (of abstract categories). It remains to pro ve that Φ reﬂects exactness. In other words, THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 19 w e need to pro ve that the class of short exact sequences in E is the smallest class whic h con tains short exact sequences in E κ and is closed under κ -ﬁltered colimits. Let x Ñ y Ñ z be a short exact sequence in E , and denote by α P Ext 1 E p z , x q the corresp onding element. Let z – lim Ý Ñ i P I z i , where I is κ -directed and z i P E κ . Putting y i “ y ˆ z z i , w e see that x Ñ y Ñ z is a κ -directed colimit of short exact sequences x Ñ y i Ñ z i . Hence, we may and will assume that z P E κ . Let x – lim Ý Ñ j P J x j , where J is κ -directed and x j P E κ . By (iii) there exists j 0 P J and an elemen t β P Ext 1 E p z , x j 0 q whic h maps to α. Consider the corresp onding short exact sequence x j 0 Ñ y j 0 Ñ z . F or j ě j 0 put y j “ y j 0 \ x j 0 x. By Prop osition 1.14 w e hav e y j P E κ . W e conclude that x Ñ y Ñ z is the κ -directed colimit of x j Ñ y j Ñ z ov er j ě j 0 , whic h prov es that Φ reﬂects exactness. □ 1.5. Presentable stable env elop es. Let E be a Grothendieck exact category , and let C is presen table stable category . Denote b y F un cont , ex p E , C q the category of contin uous (i.e. ﬁltered colimit-preserving) exact functors E Ñ C . Deﬁnition 1.22. F or a Gr othendie ck exact c ate gory E we denote by ˇ St p E q a pr esentable stable c ate gory with a c ontinuous exact functor j : E Ñ ˇ St p E q , such that for any C P Pr L st the pr e c omp osition with j induc es an e quivalenc e F un L p ˇ St p E q , C q „ Ý Ñ F un cont , ex p E , C q . We c al l ˇ St p E q the pr esentable stable envelop e of E . W e use the notation ˇ St p E q b ecause this is a generalization of the unseparated deriv ed category ˇ D p A q of a Grothendiec k ab elian category A , see Proposition 1.26 b elo w. Prop osition 1.23. F or a Gr othendie ck exact c ate gory E , its pr esentable stable envelop e exists and it is given by (1.3) ˇ St p E q “ t F : E op Ñ Sp | F is exact and c ommutes with c oﬁlter e d limits u . Pr o of. Let us temporarily denote by C the righ t hand side of (1.3). Cho ose a regular cardinal κ suc h that E is lo cally κ -coherent. Then restriction to E κ deﬁnes a fully faithful functor F : C ã Ñ F un ex pp E κ q op , Sp q » Ind p St p E κ qq . Its essential image consists of functors which comm ute with κ -small coﬁltered colimits. It follo ws that C is presen table stable, and F is accessible and commutes with inﬁnite pro ducts. Therefore, F has a left adjoint F L . Consider the composition j 1 : E κ Ñ St p E κ q Ñ Ind p St p E κ qq . 20 ALEXANDER I. EFIMOV Then the kernal of F L is generated (as a lo calizing sub category) by ob jects of the form Cone p lim Ý Ñ i P I j 1 p x i q Ñ j 1 p lim Ý Ñ i x i qq , where I is a κ -small directed poset and I Ñ E κ , i ÞÑ x i , is a functor. No w deﬁne j 2 : E Ñ Ind p St p E κ qq to b e the left Kan extension of j 1 . Then j 2 is exact. W e further deﬁne j “ F L ˝ j 2 : E Ñ C . W e claim that j satisﬁes the required universal prop erty . First, note that j commutes with ﬁltered colimits. T o c hec k this, we only need to see that j | E κ comm utes with κ -small directed colimits, and this follo ws from the ab ov e description of the k ernel of F L . No w for an y D P Pr L st the functor j 2 induces an equiv alence F un L p Ind p St p E κ qq , D q – F un κ - cont ,ex p E , D q , where the sup erscript “ κ - cont ” means the comm utation with κ -ﬁltered colimits. It follo ws formally that j induces an equiv alence F un L p C , D q „ Ý Ñ F un cont , ex p E , D q . □ The case of lo cally coheren t exact categories is studied in [NW25]. Prop osition 1.24. L et E b e a lo c al ly c oher ent exact c ate gory. Then we have an e quivalenc e ˇ St p E q » Ind p St p E ω qq . Pr o of. The description of ˇ St p E q follo ws from the univ ersal prop ert y , see also [NW25, Corol- lary 2.24]. □ The following basic example is familiar from [Lur18]. Prop osition 1.25. L et C b e a Gr othendie ck pr estable c ate gory, which we c onsider as a Gr othendie ck exact c ate gory. Then we have an e quivalenc e ˇ St p C q » Sp p C q , c omp atible with the universal functors fr om C . Pr o of. This follo ws from the universal properties. Namely , for a presen table stable category D a functor F : C Ñ D is con tinuous and exact if and only if F comm utes with colimits. □ Recall that for a Grothendeick abelian category A the so-called unseparated derived category ˇ D p A q can b e describ ed as (the dg nerve of ) the dg category Ch p Inj A q of complexes of injectiv e ob jects of A . It has a natural right complete t -structure, compatible with ﬁltered colimits, and we hav e ˇ D p A q ` » D ` p A q . Moreo ver, ˇ D p A q is generated by the heart as a lo calizing subcategory . W e refer to [Kr05, Kr15] and [Lur18, Section C.5.8] for a detailed account. THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 21 Prop osition 1.26. F or a Gr othendie ck ab elian c ate gory A we have ˇ St p A q » ˇ D p A q . Pr o of. This is explained in [Lur17b]. □ Remark 1.27. Essential ly the same ar gument as in lo c. cit. pr oves a similar r esult for the so- c al le d Gr othendie ck ab elian p n ` 1 q -c ate gories, i.e. c ate gories of the form A “ τ ď n C , wher e C is a Gr othendie ck pr estable c ate gory and n ě 0 [Lur18, Deﬁnition C.5.4.1] . Namely, if we e quip A with the induc e d exact structur e fr om C , then ˇ St p A q is e quivalent to Sp p Φ n p A qq , wher e Φ n : Groth lex n Ñ Groth lex 8 is left adjoint to τ ď n p´q . Her e Groth lex 8 r esp. Groth lex n is the c ate gory of Gr othendie ck pr estable c ate gories r esp. Gr othendie ck ab elian p n ` 1 q - c ate gories, wher e the 1 -morphisms ar e c olimit-pr eserving left exact functors. We wil l crucial ly use such c ate gories ˇ St p τ ď n C q b elow, but only in the r elatively elementary c ase when C is the c onne ctive p art of a c omp actly assemble d t -structur e on a dualizable c ate gory (Deﬁnition 2.8). We wil l not ne e d the ab ove description, inste ad we wil l apply the gener al machinery of c oher ently assemble d exact c ate gories fr om Subse ction 1.7. 1.6. Reminder on compactly assembled and dualizable categories. W e recall some basic notions from [Lur18] and [E24], and also ﬁx the notation and terminology . Recall that w e simply sa y “category” instead of “ 8 -category”, and say “ordinary category” when the spaces of morphisms are discrete. If C is an acc essible category with ﬁltered colimits, then C is called c omp actly assemble d if the colimit functor colim : Ind p C q Ñ C has a left adjoint ˆ Y “ ˆ Y C : C Ñ Ind p C q . This is [Lur18, Deﬁnition 21.1.2.1]. By [E24, Proposition 1.24] an y compactly assembled category C is ω 1 -accessible, hence the functor ˆ Y C tak es v alues in Ind p C ω 1 q Ă Ind p C q (the latter follo ws for example by the same argumen t as in [E24, Corollary 1.22]). If C is a compactly assem bled category , then a morphism f : x Ñ y in C is called c omp act if the map Y p x q Ñ Y p y q in Ind p C q factors through ˆ Y p y q , where Y : C Ñ Ind p C q is the Y oneda embedding. Recall that any compact morphism in C is (homotopic to) a comp osition of tw o compact morphisms, by the same argument as in [E24, Corollary 1.41]. A functor F : C Ñ D b etw een compactly assem bled categories is called c ontinuous if it comm utes with ﬁltered colimits. F urther, F is called str ongly c ontinuous if it is con tin uous and the follo wing (lax commutativ e) square comm utes: C D Ind p C q Ind p D q . F ˆ Y C ˆ Y D Ind p F q 22 ALEXANDER I. EFIMOV Note that if F has a righ t adjoint F R , then the strong contin uity of F exactly means that F R is con tinuous (the same pro of as in [E24, Prop osition 1.14]). In general, the strong con tin uit y of a con tinuous functor F : C Ñ D exactly means that F preserv es compact morphisms (it suﬃces to c heck this for morphisms b etw een the ob jects of C ω 1 ). W e recall that Pr L denotes the category of presentable categories and colimit-preserving functors. It is symmetric monoidal with the Lurie tensor pro duct, and the unit ob ject is giv en b y the category of spaces S . W e denote by Pr L st the full sub category of presentable stable categories, which is also the category of modules ov er the idemp oten t E 8 -algebra Sp . In particular, Pr L st is closed under tensor pro ducts in Pr L , and the unit ob ject is giv en b y Sp . A presen table stable category C is called dualizable if it is a dualizable ob ject in Pr L st . W e will simply say “dualizable category”, assuming that it is presentable and stable. By [Lur18, Prop osition D.7.3.1], C P Pr L st is dualizable if and only if it is compactly assem bled. In particular, the abov e notions apply . In this pap er w e will deal with some functors b etw een dualizable categories, whic h are strongly con tinuous but not exact, suc h as a truncation endofunctor τ ě 0 for a nice t -structure. F or this reason w e will specify if the functor is supp osed to be exact. W e denote by Cat dual st Ă Pr L st the non-full sub category of dualizable categories and strongly con tin uous exact functors. F or C , D P Cat dual st w e denote b y F un LL p C , D q the (idemp otent-complete small stable) category of s trongly con tinuous exact functors. W e recall that again by [Lur18, Proposition D.7.3.1] a presen table stable category C is dualizable if and only if it is a retract in Pr L st of a compactly generated category . W e ha v e a fully faithful embedding Ind p´q : Cat perf Ñ Cat dual st . W e will use the con v enti on from [E25c] for Calkin categories. Namely , for C P Cat dual st w e put Calk ω 1 p C q “ Ind p C ω 1 q{ ˆ Y p C q » ker p colim : Ind p C ω 1 q Ñ C q . This is a compactly generated (presen table stable) category . W e ha ve Calk ω 1 p C q » Ind p Calk cont ω 1 p C qq , where the category Calk cont ω 1 p C q P Cat perf is introduced in [E24, Deﬁnition 1.60]. In this pap er we will simply write Calk ω 1 p C q ω for the latter category . Finally , w e recall the contin uous K -theory and more general (accessible) lo calizing inv ari- an ts for dualizable categories, as deﬁned in [E24]. In this pap er w e consider the Sp -v alued lo calizing inv ariants F : Cat perf Ñ Sp [BGT13]. F or such F and for C P Cat dual st w e put F cont p C q “ Ω F p Calk ω 1 p C q ω q . THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 23 Then F cont : Cat dual st Ñ Sp is a lo calizing inv ariant b y [E24, Prop osition 4.6] and we ha ve a functorial isomorphism F cont p Ind p A qq » F p A q for A P Cat perf b y [E24, Prop osition 4.7]. Moreo v er, F cont is uniquely determined by these prop erties b y [E24, Theorem 4.10]. 1.7. Coherently assem bled exact categories. The follo wing deﬁnition speciﬁes the class of large exact categories for whic h one can deﬁne K -theory and other localizing in v arian ts. Deﬁnition 1.28. L et E b e a Gr othendie ck exact c ate gory. We say that E is c oher ently assemble d if E is c omp actly assemble d and the functor ˆ Y : E Ñ Ind p E q is exact. We denote by CohAss ex the c ate gory of c oher ently assemble d exact c ate gories, wher e the 1 -morphisms ar e str ongly c ontinuous exact functors. Remark 1.29. In the situation of Deﬁnition 1.28 supp ose that E is lo c al ly κ -c oher ent for a r e gular c ar dinal κ. Then the essential image of ˆ Y is c ontaine d in Ind p E κ q . The inclusion Ind p E κ q Ñ Ind p E q r eﬂe cts exactness by Pr op ositions 1.11 and 1.14, henc e the functor ˆ Y : E Ñ Ind p E κ q is exact. Example 1.30. L et E b e a dualizable (pr esentable stable) c ate gory with the standar d exact structur e. Then E is c oher ently assemble d. This gives a ful ly faithful functor Cat dual st ã Ñ CohAss ex . W e will see b elow that a coheren tly assembled exact category is alw ays lo cally ω 1 - coheren t. First w e observ e an equiv alen t c haracterization. Prop osition 1.31. L et E b e a Gr othendie ck exact c ate gory. The fol lowing ar e e quivalent. (i) E is c oher ently assemble d. (ii) Ther e exists a lo c al ly c oher ent exact c ate gory E 1 and a r etr action E F Ý Ñ E 1 G Ý Ñ E , such that b oth F and G ar e exact and c ommute with ﬁlter e d c olimits. In p articular, if E is lo c al ly c oher ent, then it is c oher ently assemble d. Pr o of. Note that the ﬁnal assertion is ob vious: if E is locally coherent, then ˆ Y is obtained b y applying Ind to the exact inclusion E ω Ñ E . (i) ù ñ (ii) . Supp ose that E is locally κ -coheren t. Consider the retraction E ˆ Y Ý Ñ Ind p E κ q colim Ý Ý Ý Ñ E . Both functors comm ute with ﬁltered colimits. By Remark 1.29 the functor ˆ Y is exact, and b y deﬁnition of a Grothendiec k exact category the functor colim is exact. This prov es the implication. (ii) ù ñ (i) . Since F and G are con tinuous, the functor ˆ Y E is isomorphic to the comp osition E F Ý Ñ E 1 ˆ Y E 1 Ý Ý Ñ Ind p E 1 q Ind p G q Ý Ý Ý Ý Ñ Ind p E q . 24 ALEXANDER I. EFIMOV Here the ﬁrst and the third functors are exact by assumption, and the second functor is exact b y the abov e observ ation on the locally coheren t case. This prov es the prop osition. □ W e explain a natural analogue of split exact structures in the con text of large exact categories, see also Remark 1.33 b elo w. Prop osition 1.32. L et E b e a c omp actly assemble d additive c ate gory. Then E has an exact structur e such that a map f : x Ñ y in E is an exact pr oje ction if and only if any c omp act morphism z Ñ y factors thr ough x. With this exact structur e E is a c oher ently assemble d exact c ate gory. Mor e over, short exact se quenc es in E ar e exactly the ﬁlter e d c olimits of split short exact se quenc es. Pr o of. W e ﬁrst consider the case when E is compactly generated, and put A “ E ω . Then E is identiﬁed with the category of ﬂat additiv e functors A op Ñ Sp ě 0 , this is a straightforw ard generalization of [Lur17a, Theorem 7.2.2.15]. The deﬁnition of an exact pro jection in this case gives the class of morphisms which are eﬀectiv e epimorphisms in the am bien t category F un add p A op , Sp ě 0 q . Hence, w e simply obtain the standard exact structure on E » Ind p A q . No w consider the general case. Cho ose a retraction E F Ý Ñ E 1 G Ý Ñ E , where E 1 is addi- tiv e and compactly generated, F and G are additive and con tinuous, and F is moreov er strongly contin uous (e.g. tak e E 1 “ Ind p E ω 1 q ). Denote b y ˇ St add p E q the category of coﬁl- tered limit-preserving functors E op Ñ Sp . As in Prop osition 1.23 we see that w e hav e a univ ersal con tinuous additiv e functor j : E Ñ ˇ St add p E q . Consider also a similar functor j 1 : E 1 Ñ ˇ St add p E 1 q . The latter is simply the ind-completion of the functor E 1 ω Ñ St p E 1 ω q , where E 1 ω is equipp ed with the split exact structure. It follo ws from the retraction that j is fully faithful and its essential image is closed under extensions. Consider the induced exact structure on E . Using the retraction again we see that E is a coheren tly assem bled exact category (by Prop osition 1.31), and any short exact sequence in E is a ﬁltered colimit of split short exact sequences. It remains to obtain the stated description of exact pro jections in E . If f : x Ñ y is an exact pro jection in E , then f is a ﬁltered colimit of split exact pro jections x i Ñ y i . Any compact morphism z Ñ y factors through some y i , hence through x i , hence through x. Con v ersely , left f : x Ñ y b e a morphism in E such that any compact morphism z Ñ y factors through x. Let ˆ Y p y q – “lim Ý Ñ i P I ” y i , where I is directed. Eac h map y i Ñ y is compact, hence so is F p y i q Ñ F p y q b y the strong con tinuit y of F . Moreo v er, for eac h i the map F p y i q Ñ F p y q factors through F p x q . It follo ws from the abov e sp ecial case that F p x q Ñ F p y q is an exact pro jection in E 1 . Applying G : E 1 Ñ E , we conclude that x Ñ y is an exact pro jection in E . □ THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 25 Remark 1.33. The c ate gory ˇ St add p E q fr om the pr o of of Pr op osition 1.32 has a natur al t - structur e such that the c onne ctive p art C “ ˇ St add p E q ě 0 is identiﬁe d with the c ate gory of c oﬁlter e d limit-pr eserving functors E op Ñ Sp ě 0 . Mor e over, C is a dualizable mo dule over Sp ě 0 , i.e. a dualizable obje ct of Pr L Add – the symmetric monoidal c ate gory of additive pr e- sentable 8 -c ate gories [LLS] . Mor e over, any dualizable additive c ate gory C c an b e obtaine d in this way: one r e c overs E as the ful l sub c ate gory of ﬂat obje cts of C . Her e an obje ct x P C is ﬂat if the functor e v p x, ´q : C _ Ñ Sp ě 0 c ommutes with ﬁnite limits. The dualizable additive c ate gories ar e also exactly the c ate gories of c onne ctive almost mo dules in the setting of higher almost ring the ory, studie d by Heb estr eit and Scholze [HS24] . The notion of a ﬂat obje ct is c omp atible with the standar d notion of an almost ﬂat almost mo dule [F a02, GabRam03] . Next we show that the presen table stable env elop e is w ell-b ehav ed. Prop osition 1.34. (i) L et E b e a c oher ently assemble d exact c ate gory. The c ate gory ˇ St p E q is dualizable and the functor j : E Ñ ˇ St p E q is str ongly c ontinuous and ful ly faithful. Mor e over, the essential image of E in ˇ St p E q is close d under extensions, and j r eﬂe cts exactness. (ii) L et F : E Ñ E 1 b e a str ongly c ontinuous exact functor b etwe en c oher ently assemble d exact c ate gories. Then the induc e d functor ˇ St p F q : ˇ St p E q Ñ ˇ St p E 1 q is also str ongly c ontinuous. Ther efor e, we have a wel l-deﬁne d functor ˇ St : CohAss ex Ñ Cat dual st . It is left adjoint to the inclusion. Pr o of. (i) W e ﬁrst consider the case when E is locally coheren t. Then b y Prop osition 1.24 the category ˇ St p E q » Ind p St p E ω qq is compactly generated, hence dualizable. The strong con tin uit y is immediate: the functor j is identiﬁed with Ind p E ω Ñ St p E ω qq . The fully faithfulness follows from Prop osition 1.2. The remaining assertions follo w from Prop osition 1.9. No w consider the general case. By Prop osition 1.31 we ha v e a retraction E F Ý Ñ E 1 G Ý Ñ E , where E 1 is locally coheren t and b oth F and G are exact and contin uous. No w, b oth functors ˇ St p F q and ˇ St p G q are contin uous, and the functor j : E Ñ ˇ St p E q is a retract of the functor j 1 : E 1 Ñ ˇ St p E 1 q . Hence, all the assertions ab out j and ˇ St p E q follow from the ab o ve sp ecial case when E is locally coheren t. (ii) Put Φ “ ˇ St p F q . W e need to sho w that for x P ˇ St p E q we hav e an isomorphism ˆ Y p Φ p x qq „ Ý Ñ Ind p Φ qp ˆ Y p x qq . It suﬃces to consider the case x “ j p y q , where y P E and j : E Ñ ˇ St p E q is the universal contin uous exact functor. Then the isomorphism follows 26 ALEXANDER I. EFIMOV directly from (i): w e ha v e a comm utative square of compactly as sem bled categories E E 1 ˇ St p E q ˇ St p E 1 q , F Φ in which b oth v ertical arro ws and the upp er horizon tal arro w are strongly contin uous. It follo ws formally that the functor ˇ St : CohAss ex Ñ Cat dual st is left adjoin t to the inclusion: the adjunction counit E Ñ ˇ St p E q is well-deﬁned b y (i). □ The ab o v e prop osition allows to deﬁne the con tin uous K -theory and other lo calizing in v arian ts (suc h as K H ) for coherently assem bled exact categories. Deﬁnition 1.35. L et F : Cat perf Ñ Sp b e a lo c alizing invariant. F or a c oher ently assemble d exact c ate gory E we put F cont p E q “ F cont p ˇ St p E qq . This deﬁnes a functor F cont : CohAss ex Ñ Sp . Next, we show the exp ected local ω 1 -coherence. Prop osition 1.36. L et E b e a c oher ently assemble d exact c ate gory. Then E is lo c al ly ω 1 - c oher ent. Pr o of. It suﬃces to c heck that the conditions from Proposition 1.21 are satisﬁed. Since E is compactly assem bled, it is ω 1 -accessible b y [E24, Prop osition 1.24]. Next, all ﬁl- tered colimits are exact in E , in particular, ω 1 -ﬁltered colimits are exact. It remains to pro v e that for x P E ω 1 the functor Ext 1 E p x, ´q : E Ñ Ab comm utes with ω 1 -ﬁltered colimits. By Prop osition 1.34 the functor j : E Ñ ˇ St p E q is strongly contin uous, in par- ticular, it preserv es ω 1 -compact ob jects. Hence, we ha ve j p x q P ˇ St p E q ω 1 . It follows that the functor Ext 1 ˇ St p E q p j p x q , ´q : ˇ St p E q Ñ Ab commutes with ω 1 -ﬁltered colimits by sta- bilit y . Applying Prop osition 1.34 again, w e see that for y P E we hav e an isomorphism Ext 1 E p x, y q – Ext 1 ˇ St p E q p j p x q , j p y qq . Hence, the functor Ext 1 E p x, ´q also comm utes with ω 1 - ﬁltered colimits. □ F rom no w on, for a coheren tly assem bled exact category E we consider the small category E ω 1 as an exact category with the induced exact structure from E , and similarly for any regular κ ě ω 1 . Recall that by Proposition 1.15 for any uncountable regular cardinals κ ď λ the functor St p E κ q Ñ St p E λ q is fully faithful, and w e deﬁned St p E q to b e the directed union of St p E κ q . THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 27 Prop osition 1.37. L et E b e a c oher ently assemble d exact c ate gory. Then the natur al functor St p E q Ñ ˇ St p E q is ful ly faithful. Pr o of. By Proposition 1.31 we ha v e a retraction E F Ý Ñ E 1 G Ý Ñ E , where E 1 is lo cally coheren t and b oth F and G are con tinuous and exact. Hence, the functor St p E q Ñ ˇ St p E q is a retract of the functor St p E 1 q Ñ ˇ St p E 1 q . Th us, w e ma y and will assume that E is locally coheren t. Consider the functor j : E Ñ ˇ St p E q , whic h is fully faithful b y Prop osition 1.34. Let κ b e an uncoun table regular cardinal. By Proposition 1.15 it suﬃces to prov e that the functor St p E κ q Ñ ˇ St p E q is fully faithful. T ake the restriction i “ j | E κ : E κ Ñ ˇ St p E q . By [SW26, Lemmas 4.2 and 4.3] it suﬃces to pro ve that for x P E κ and for n ą 0 the functor Ext n ˇ St p E q p i p´q , i p x qq : p E κ q op Ñ Ab is weakly eﬀaceable. Namely , we need to sho w that for an y y P E κ and for an y element α P Ext n ˇ St p E q p i p y q , i p x qq there exists a pro jection z Ñ y in E κ suc h that the image of α in Ext n ˇ St p E q p i p z q , i p y qq is zero. W e ﬁrst consider the case y P E ω . Let x – lim Ý Ñ s P S x s , where S is directed and x s P E ω . By Prop osition 1.24 i p y q is compact in ˇ St p E q , hence there exists s P S suc h that α is the image of some element β P Ext n ˇ St p E q p i p y q , i p x s qq – Ext n E ω p y , x s q . By [SW26, Lemma 4.3] there exists a pro jection z Ñ y in E ω suc h that the image of β in Ext n ˇ St p E q p i p z q , i p x s qq v anishes. Then the image of α in Ext n ˇ St p E q p i p z q , i p x qq v anishes, as required. Next, if y is a κ -small direct sum of some y k P E ω , then it suﬃces to take z to be the direct sum of z k P E ω , where the pro jections z k Ñ y k are as ab o ve. Now consider the general case, i.e. y “ lim Ý Ñ t P T y t , where T is a κ -small directed p oset, and y t P E ω . It suﬃces to prov e that the map (1.4) à t P T y t Ñ lim Ý Ñ t P T y t is a pro jection in E κ . By Proposition 1.21 it suﬃces to prov e that (1.4) is a pro jection in E . W e ma y and will assume that the exact structure on E ω is split, i.e. we ha ve only split short exact sequences. Then E is iden tiﬁed with the category of ﬂat additiv e functors p E ω q op Ñ Sp ě 0 , with the induced exact structure from the Grothendiec k prestable category C “ F un add pp E ω q op , Sp ě 0 q . The class of ﬂat ob jects in C is closed under ﬁbers of eﬀective epimorphisms, which implies that (1.4) is a pro jection in E , as required. □ The following corollary is immediate. Corollary 1.38. L et E b e a c oher ently assemble d exact c ate gory, and let C b e a pr esentable stable c ate gory. Supp ose that we have a c ontinuous exact functor Φ : E Ñ C which is ful ly faithful, r eﬂe cts exactness and its essential image is close d under extensions. Consider the induc e d (c ontinuous exact) functor Ψ : ˇ St p E q Ñ C , and denote by j : E Ñ ˇ St p E q the 28 ALEXANDER I. EFIMOV universal c ontinuous exact functor, so that Ψ ˝ j – Φ . Then for x, y P E the map (1.5) Ext n ˇ St p E q p j p x q , j p y qq Ñ Ext n C p Φ p x q , Φ p y qq is an isomorphism for n ď 1 , and a monomorphism for n “ 2 . Pr o of. Let κ b e an uncoun table regular cardinal such that x, y P E κ . By Prop ositions 1.37 and 1.15 the source of (1.5) is iden tiﬁed with Ext n E κ p x, y q . F urther, b y Prop osition 1.14 the essen tial image Φ p E κ q in C is also closed under extensions. Hence, the assertion follo ws from Prop osition 1.5. □ Finally , we prov e a v ersion of [SW26, Theorem 4.5], whic h is deduced from loc. cit. Prop osition 1.39. L et Φ : U Ñ E b e a functor b etwe en c oher ently assemble d exact c ate gories, which is ful ly faithful, str ongly c ontinuous, r eﬂe cts exactness, and such that the essential image of Φ is close d under extensions. Supp ose that the inclusion Φ p U ω 1 q ã Ñ E ω 1 is left sp e cial (Deﬁnition 1.6). Then the induc e d functor Ψ : ˇ St p U q Ñ ˇ St p E q is ful ly faithful. Pr o of. The functor Ψ is a retract (in Pr L st ) of the functor Ind p St p U ω 1 qq Ñ Ind p St p E ω 1 qq , whic h is fully faithful b y Theorem 1.7. □ Remark 1.40. In Pr op osition 1.39 one c an e quivalently assume that Φ p U q is left sp e cial in E . However, we pr efer to formulate the c ondition in terms of smal l c ate gories. 1.8. Coherently assem bled ab elian categories. Deﬁnition 1.41. A Gr othendie ck ab elian c ate gory A is c oher ently assemble d if it is such as an exact c ate gory. This me ans that if A is c omp actly assemble d and the functor ˆ Y : A Ñ Ind p A q is exact. We denote by CohAss ab the c ate gory of c oher ently assemble d ab elian c ate gories, wher e the 1 -morphisms ar e str ongly c ontinuous exact functors. Remark 1.42. Note that a Gr othendie ck ab elian c ate gory A is c omp actly assemble d if and only if A satisﬁes (AB6) (the same pr o of as in [E24, Prop osition 1.53] ). W e recall the following statemen t from [E24]. Prop osition 1.43. [E24, Prop osition E.8] L et A b e a Gr othendie ck ab elian c ate gory which is c omp actly gener ate d. Then A is c oher ently assemble d if and only if it is lo c al ly c oher ent. F or completeness we give the follo wing non-standard characterization of lo cally no ether- ian ab elian categories. Prop osition 1.44. L et A b e a Gr othendie ck ab elian c ate gory. The fol lowing ar e e quivalent. THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 29 (i) A is lo c al ly no etherian. (ii) A is c omp actly assemble d and for any obje ct x P A the map ˆ Y p x q Ñ Y p x q is a monomorphism in A . Pr o of. (i) ù ñ (ii) . An y ob ject x P A is a directed union of no etherian sub ob jects x i Ă x, hence the map ˆ Y p x q – lim Ý Ñ i Y p x i q Ñ Y p x q is a monomorphism. (ii) ù ñ (i) . W e ﬁrst observe that A is coherently assembled. Indeed, the functor ˆ Y : A Ñ Ind p A q is automatically righ t exact, so w e only need to c heck that it preserves monomorphisms. If x Ñ y is a monomorphism in A , then in the commutativ e square ˆ Y p x q ˆ Y p y q Y p x q Y p y q b oth v ertical arrows and the lo wer horizon tal arrow are monomorphisms, hence so is the upp er horizontal arrow, as required. Next, we see that the functor Φ : A Ñ Ind p A q , Φ p x q “ cok er p ˆ Y p x q Ñ Y p x qq , is exact. Moreo v er, an ob ject x P A is compact if and only if Φ p x q “ 0 . No w take some x P A , and let ˆ Y p x q – lim Ý Ñ i P I Y p x i q , where I is directed. Since the map ˆ Y p x q Ñ Y p x q is a monomorphism, w e may and will assume that eac h map x i Ñ x is a monomorphism in A . F or each i P I the map Φ p x i q Ñ Φ p x q is zero, hence Φ p x i q “ 0 b y the left exactness of Φ . Therefore, each x i is compact and A is compactly generated. By Proposition 1.43 A is lo cally coherent. It remains to show that the class of compact ob jects in A is closed under sub ob jects. This again follo ws from the left exactness of Φ : if x P A is compact and y Ă x, then Φ p x q “ 0 , hence Φ p y q “ 0 , i.e. y is compact. □ W e record the sp ecial cases of general results on coheren tly assem bled exact categories. Corollary 1.45. L et A b e a c oher ently assemble d ab elian c ate gory. (i) A is lo c al ly ω 1 -c oher ent. (ii) The unsep ar ate d derive d c ate gory ˇ D p A q is dualizable and the functor A Ñ ˇ D p A q is str ongly c ontinuous. Mor e over, we have a wel l-deﬁne d functor ˇ D p´q : CohAss ab Ñ Cat dual st . Pr o of. (i) is a sp ecial case of Proposition 1.36, and (ii) follo ws from Propositions 1.26 and 1.34. □ 30 ALEXANDER I. EFIMOV The localizing in v ariants of coheren tly assem bled ab elian categories are obtained by ap- plying Deﬁnition 1.35. Prop osition 1.46. L et A b e a Gr othendie ck ab elian c ate gory and let B b e a c oher ently assemble d ab elian c ate gory. Supp ose that we have a r etr action A F Ý Ñ B G Ý Ñ A , such that b oth F and G ar e left exact and c ommute with ﬁlter e d c olimits. Then A is c oher ently assemble d. Pr o of. The retraction implies that A is compactly assem bled. The functor ˆ Y A : A Ñ Ind p A q is automatically right exact (it is a left adjoin t). Hence, w e only need to c heck the left exactness. This follows from the retraction since both F and G are left exact. □ Some non-trivial examples of (not lo cally coherent) coherently assem bled ab elian cate- gories are giv en in Section 8. W e recall the follo wing notation. W e denote b y Groth lex ab the category of Grothendieck ab elian categories, where the 1 -morphisms are colimit-preserving left exact functors. W e denote b y Groth c, lex ab Ă Groth lex the non-full sub category with the same ob jects, where 1 -morphisms are functors F : C Ñ D suc h that the righ t adjoint F R : D Ñ C comm utes with ﬁltered colimits. The follo wing statement is certainly kno wn to experts, but w e could not ﬁnd a reference. Prop osition 1.47. The c ate gories Groth c, lex ab and Groth lex ab have ﬁlter e d c olimits, and the (non-ful l sub c ate gory inclusion) functors Groth c, lex ab Ñ Groth lex ab Ñ Pr L c ommute with ﬁl- ter e d c olimits. Pr o of. Consider an ind-system p A i q i P I . By [Lur18, Prop osition C.5.5.20] we ha ve a fully faithful functor Groth lex ab Ñ Groth lex 8 , A ÞÑ ˇ D p A q ě 0 , left adjoin t to the functor C ÞÑ τ ď 0 C . Putting C i “ ˇ D p A i q ě 0 , we obtain an ind-system p C i q i P I in Groth lex 8 suc h that A i » τ ď 0 C i (functorially in i P I ). By [Lur18, Prop osition C.3.3.5] the category Groth lex 8 has ﬁltered colimits, which are preserved by the functor Groth lex 8 Ñ Pr L . Recall that a colimit in Pr L can be identiﬁed with the limit of the associated diagram in Pr R (with righ t adjoin t transition functors). The right adjoin ts preserv e discrete ob jects, which implies that lim Ý Ñ i A i – τ ď 0 p lim Ý Ñ i ˇ D p A i q ě 0 q is a Grothendieck ab elian category (the former colimit is also computed in Pr L ). Using again the fully faithfulness of the functor A ÞÑ ˇ D p A q ě 0 , w e conclude that the functor Groth lex ab Ñ Pr L creates ﬁltered colimits, as stated. It follo ws formally that the functor Groth c, lex ab Ñ Groth lex ab also creates ﬁltered colimits, b y applying the argumen t from the pro of of [Lur18, Prop osition 3.5.3] or [E24, Proposition 1.7]. □ THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 31 Note that CohAss ab Ă Groth c, lex is a (strictly) full sub category . The follo wing result is an (almost) straightforw ard generalization of the follo wing fact: if p R i q i P I is an ind-system of (ordinary) associative righ t coherent rings with left ﬂat transition maps, then the ring lim Ý Ñ i R i is right coherent. Prop osition 1.48. The ful l sub c ate gory CohAss ab Ă Groth c, lex ab is close d under ﬁlter e d c ol- imits. Pr o of. Let I b e a directed p oset, and let I Ñ CohAss ab , i ÞÑ A i , be a functor. First suppose that each A i is locally coheren t. Then w e ha v e an I -indexed diagram of small abelian categories, i ÞÑ A ω i . The transition functors are exact, hence the colimit is ab elian and w e ha ve A » Ind p lim Ý Ñ i A ω i q . The general case follo ws since the diagram i ÞÑ A i in Groth lex ab is a retract of the diagram i ÞÑ Ind p A ω 1 i q . □ Finally , we p oint out that the weak (AB5) axiom holds. Prop osition 1.49. The c ate gories Groth c, lex ab and CohAss ab satisfy the we ak (AB5) axiom: the class of ful ly faithful functors is close d under ﬁlter e d c olimits. Pr o of. By Prop osition 1.48 it suﬃces to prov e this for Groth c, lex ab . In this case the pro of is the same as in [E24, Prop osition 1.67]. □ Finally , we mention an exp ected statemen t about localizations (Serre quotients). Prop osition 1.50. L et A b e a c oher ently assemble d ab elian c ate gory, and let B Ă A b e a c oher ently assemble d lo c alizing sub c ate gory such that the inclusion functor i : B Ñ A is str ongly c ontinuous. Then the c ate gory A { B is c oher ently assemble d and the quotient functor q : A Ñ A { B is str ongly c ontinuous. If mor e over A and B ar e lo c al ly c oher ent, then we have an e quivalenc e Ind p A ω { B ω q „ Ý Ñ A { B . Pr o of. The equiv alence in the lo cally coheren t case follows from the univ ersal properties. Denoting b y i R and q R the righ t adjoin ts, for x P A w e ha ve a functorial short exact sequence 0 Ñ i p i R p x qq Ñ x Ñ q R p q p x qq Ñ 0 . By assumption, i R is con tinuous, hence so is q R . The retraction A { B q R Ý Ñ A q Ý Ñ A { B implies that A { B is coheren tly assem bled b y Prop osition 1.46. The strong con tinuit y of q is already established. □ 32 ALEXANDER I. EFIMOV 1.9. The situation of d ´ evissage. The conditions of Quillen’s D ´ evissage theorem [Qui73, The- orem 4] naturally generalize to coherently assem bled abelian categories. Ho wev er, this is not immediately straigh tforward and requires some clariﬁcation. W e ﬁrst treat the case of small categories. Prop osition 1.51. L et A b e a smal l ab elian c ate gory, and let B Ă A b e a strictly ful l ab elian sub c ate gory such that the inclusion functor i : B Ñ A is exact. Supp ose that every obje ct of A has a ﬁnite ﬁltr ation with sub quotients in B . Then B is close d under taking sub obje cts and quotients in A . Pr o of. It suﬃces to prov e that for an y morphism f : x Ñ y in A , if x P B then k er p f q P B . Let F ‚ y b e a ﬁnite increasing ﬁltration suc h that F 0 y “ 0 and F n y “ y . If n “ 0 , then there is nothing to prov e. If n ą 0 , then put y 1 “ F n ´ 1 y , x 1 “ k er p x Ñ y { y 1 q and f 1 “ f | x 1 : x 1 Ñ y 1 . Then k er p f q “ ker p f 1 q , and applying induction on n w e obtain k er p f q P B . □ Next we consider the lo cally coherent case. Prop osition 1.52. L et A b e a smal l ab elian c ate gory and B Ă A a strictly ful l ab elian sub c ate gory such that the inclusion functor i : B Ñ A is exact. We identify Ind p B q with its essential image in Ind p A q . The fol lowing ar e e quivalent. (i) Every obje ct of A has a ﬁnite ﬁltr ation with sub quotients in B . (ii) Ind p B q Ă Ind p A q is close d under sub obje cts and quotients, and every obje ct of Ind p A q has an N -indexe d incr e asing exhaustive ﬁltr ation with sub quotients in Ind p B q . (iii) Ind p B q gener ates Ind p A q via extensions and ﬁlter e d c olimits. Pr o of. (i) ù ñ (ii) . By Prop osition 1.51, B Ă A is closed under sub ob jects and quotien ts, hence the same holds for Ind p B q Ă Ind p A q . Abusing the notation w e write i for Ind p i q : Ind p B q Ñ Ind p A q . Then the righ t adjoin t i R : Ind p A q Ñ Ind p B q comm utes with ﬁltered colimits and for an y x P Ind p A q the map i p i R p x qq Ñ x is a monomorphism. More precisely , i p i R p x qq is the largest sub ob ject of x which is contained in Ind p B q . Deﬁne the functorial non-negativ e ﬁltration F ‚ x inductiv ely: F 0 x “ 0 and F n ` 1 x Ă x is the preimage of i p i R p x { F n x qq Ă x { F n x. By induction eac h of the endofunctors F n : Ind p A q Ñ Ind p A q comm utes with ﬁltered colimits. It follows from (i) that for any x P A w e ha v e F n x “ x for large n. It follows that for any x P Ind p A q the ﬁltration F ‚ x is exhaustive. By construction, its subquotients are in Ind p B q , whic h prov es the implication. The implication (ii) ù ñ (iii) is evident. (iii) ù ñ (i) . W e identify Ind p A q with the heart of the standard t -structure on Ind p D b p A qq » ˇ D p Ind p A qq . Then (iii) implies that B generates Ind p D b p A qq as a lo calizing THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 33 sub category . Hence, B generates D b p A q as an idemp otent-complete stable subcategory . Let A 1 Ă A b e the smallest weak Serre sub category containing B , i.e. A 1 is generated by B via extensions. Denote by D b A 1 p A q Ă D b p A q the (idemp otent-complete stable) sub category consisting of complexes with homology in A 1 . Then B is con tained in D b A 1 p A q , hence the latter category coincides with D b p A q and A 1 “ A . This pro ves the implication. □ Finally , we consider the coheren tly assembled case. Prop osition 1.53. L et A b e a c oher ently assemble d ab elian c ate gory, and let B Ă A b e a strictly ful l c oher ently assemble d ab elian sub c ate gory such that the inclusion functor i : B Ñ A is str ongly c ontinuous and exact. The fol lowing ar e e quivalent. (i) B Ă A is close d under sub obje cts and quotients, and any obje ct of A has an N - indexe d exhaustive ﬁltr ation with sub quotients in B . (ii) B gener ates A via extensions and ﬁlter e d c olimits. Pr o of. Again, the implication (i) ù ñ (ii) is evident. (ii) ù ñ (i) . Denote b y E Ă A ω 1 the w eak Serre subcategory generated b y B ω 1 (note that E is not necessarily closed under coun table copro ducts in A ω 1 ). By Prop osition 1.11 the full subcategory Ind p E q Ă Ind p A ω 1 q is closed under extensions. By assumption the functor ˆ Y A : A Ñ Ind p A ω 1 q sends B to Ind p B ω 1 q Ă Ind p E q . Since ˆ Y A is exact, the condition (ii) implies that w e ha ve ˆ Y A p A q Ă Ind p E q . W e show that B Ă A is closed under sub ob jects, and the assertion ab out quotients follo ws. Let x P B and y Ă x, a priori y P A . Then ˆ Y A p y q Ă ˆ Y A p x q P Ind p B ω 1 q , hence b y Prop osition 1.52 (applied to B ω 1 Ă E ) we hav e ˆ Y A p y q P Ind p B ω 1 q . Applying the colimit functor, we obtain y P B , as stated. The same argument shows the existence of a ﬁltration with required properties. Namely , let x P A , then by Prop osition 1.52 the ob ject ˆ Y A p x q P Ind p E q has an N -indexed ﬁltration with sub quotien ts in Ind p B ω 1 q . Applying the colimit functor Ind p E q Ñ A , we obtain the required ﬁltration on x. This pro ves the implication. □ The following is a basic example when the d ´ evissage condition holds. Prop osition 1.54. L et B b e a c oher enly assemble d c ate gory. L et A b e the c ate gory of p airs p x, N q , wher e x P B and N : x Ñ x is a lo c al ly nilp otent endomorphism, i.e. colim p x N Ý Ñ x N Ý Ñ . . . q “ 0 . Then A is c oher ently assemble d, and the inclusion B Ñ A , x ÞÑ p x, 0 q , satisﬁes the c onditions of Pr op osition 1.53. Pr o of. The only statement requiring a pro of is that A is coheren tly assembled. This can b e deduced from Prop osition 2.25 b elo w. F or comp eteness w e giv e a more direct pro of. 34 ALEXANDER I. EFIMOV F or an y n P N consider the full subcategory A n Ă A consisting of pairs p x, N q such that N n “ 0 . Then we hav e a direct sequence p A n q n ě 0 in Groth c, lex ab . By Prop osition 1.49 the functor lim Ý Ñ n A n Ñ A is fully faithful. It is an equiv alence since A is generated by all A n via ﬁltered colimits. By Prop osition 1.48 it suﬃces to pro ve that A n is coheren tly assem bled for eac h n ě 0 . W e ﬁrst note that A n is compactly assem bled: the forgetful functor Φ n : A n Ñ B , Φ n p x, N q “ x, commutes with colimits, and the essential image of its left adjoint Φ L n generates the target via colimits. Next, the righ t adjoin t Φ R n is con tinuous, and Φ n is exact and conserv ativ e, hence Ind p Φ n q : Ind p A n q Ñ Ind p B q is also exact and conserv ative. This formally implies that A n is coherently assembled since B is. □ W e mention an example showing that one has to b e cautious when dealing with the d´ evissage condition for coherently assem bled categories. Supp ose that we ha ve a Grothendiec k ab elian category A and a strictly full Grothendieck ab elian sub category B Ă A , suc h that the inclusion functor i : B Ñ A is exact and colimit-preserving, the righ t adjoin t i R : A Ñ B is con tinuous, B is closed under sub ob jects and quotien ts in A , and ev ery ob ject of A has an N -indexed exhaustiv e ﬁltration with subquotients in B . It is tempting to think that if B is coherently assem bled, then so is A . Ho wev er, the follo wing example shows that this is not the case in general, ev en if B is lo cally coheren t. Example 1.55. L et k b e a ﬁeld, and c onsider the ring R “ k r x, y 1 , y 2 , . . . s{p xy n ; n ě 1 q . L et A b e the c ate gory of R -mo dules M such that x acts lo c al ly nilp otently on M . Then A is a Gr othendie ck c ate gory. L et B “ Mod - R { x Ă A . Then B is lo c al ly c oher ent sinc e R { x – k r y 1 , y 2 , . . . s is a se quential c olimit of no etherian rings with ﬂat tr ansition maps. F urther, B Ă A is close d under sub obje cts and quotients, and any obje ct M P A has a ﬁltr ation F ‚ M , given by F n M “ k er p x n : M Ñ M q , with Ť n F n M “ M and gr F n M P B . The right adjoint to the inclusion is given by M ÞÑ ker p x : M Ñ M q , henc e it is c ontinuous. However, A is not c oher ently assemble d. Assume the c ontr ary. Then by Cor ol lary 1.45 for any c omp act obje ct M P A ω the functor Ext 1 A p M , ´q : A Ñ Ab c ommutes with ﬁlter e d c olimits. Cle arly, R { x is c omp act in A . Consider the dir e ct se quenc e p N n q n ě 0 , given by N n “ R {p x, y 1 , . . . , y n q . Put N “ lim Ý Ñ n N n – k . T ake the element α P Ext 1 A p R { x, N q c orr esp onding to the short exact se quenc e 0 Ñ N Ñ R { x 2 Ñ R { x Ñ 0 . Then α do es not factor thr ough N n for any n, sinc e Ext 1 A p R { x, N n q is a quotient of Hom R p Rx, N n q “ 0 . This gives a c ontr adiction, henc e A is not c oher ently assemble d. THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 35 2. Comp a ctl y assembled t -structures on dualizable ca tegories 2.1. Reminder on t -structures. W e start by recalling the basic notions related to t - structures, also ﬁxing the notation. The basic reference is [BBD82] in the con text of trian- gulated categories, and [Lur17a, Section 1.2], [Lur18, Appendix C] in the context of stable categories. Recall from [BBD82] that a t -structure on a triangulated category T is a pair of full sub categories p T ě 0 , T ď 0 q , suc h that T ě 0 r 1 s Ă T ě 0 , T ď 0 r´ 1 s Ă T ď 0 , w e hav e Hom T p y , z q “ 0 for y P T ě 0 , z P T ď 0 r´ 1 s , and for an y x P T there exists an exact triangle τ ě 0 x Ñ x Ñ τ ď´ 1 x, τ ě 0 x P T ě 0 , τ ď´ 1 x P T ď 0 r´ 1 s . A t -structure on a stable category C is a pair of full sub categories p C ě 0 , C ď 0 q suc h that p h C ě 0 , h C ď 0 q is a t -structure on h C . W e will only deal with t -structures on stable cate- gories. If D is another stable category with a t -structure p D ě 0 , D ď 0 q , then an exact functor F : C Ñ D is called left t -exact if F p C ď 0 q Ă D ď 0 , and right t -exact if F p C ě 0 q Ă D ě 0 . F urther, F is called t -exact if it is both left and righ t t -exact. Giv en a t -structure p C ě 0 , C ď 0 q on a stable category C , for an y a P Z we put C ě a “ C ě 0 r a s , C ď a “ C ď 0 r a s . W e denote b y τ ě a : C Ñ C ě a the right adjoin t to the inclusion, and b y τ ď a : C Ñ C ď a the left adjoint to the inclusion. W e also denote by the same symbols the compositions C τ ě a Ý Ý Ñ C ě a ã Ñ C , C τ ď a Ý Ý Ñ C ď a ã Ñ C . This should not lead to confusion. F or in tegers a ď b w e put C r a,b s “ C ě a X C ď b . In particular, C r 0 , 0 s “ C ♡ is the heart of the t -structure, whic h is an ab elian category . W e denote b y τ r a,b s : C Ñ C r a,b s the functor τ ě a ˝ τ ď b – τ ď b ˝ τ ě a . Since w e are working with not necessarily Z -linear categories, we use the notation π a p x q “ τ r a,a s p x qr´ a s P C ♡ , a P Z , x P C . W e recall the notation C ` “ ď a P Z C ď a , C ´ “ ď a P Z C ě a , C b “ C ` X C ´ “ ď n ě 0 C r´ n,n s . The t -structure p C ě 0 , C ď 0 q is called left bounded resp. righ t b ounded resp. b ounded if C “ C ` resp. C “ C ´ resp. C “ C b . Recall that for a t -structure p C ě 0 , C ď 0 q on C , the left completion is giv en b y the limit p C l “ lim Ð Ý n P N C ď n . This is a stable category , and it naturally has a t -structure p p C l ě 0 , p C l ď 0 q , where p C l ě 0 “ lim Ð Ý n P N C r 0 ,n s , p C l ď 0 “ C ď 0 . W e ha ve a natural exact and t -exact functor C Ñ p C l . If it is 36 ALEXANDER I. EFIMOV an equiv alence, then the t -structure on C is called left c omplete . Dually , we ha ve a notion of the righ t completion p C r “ lim Ð Ý n P N C ě´ n , and similarly for right completeness. Note that in general the right completion of p C l is equiv alent to the left completion of p C r , and both are giv en by lim Ð Ý n P N C r´ n,n s . It is con venien t to use the follo wing terminology for brevity . Deﬁnition 2.1. A smal l t -c ate gory is a smal l stable c ate gory C with a b ounde d t -structur e p C ě 0 , C ď 0 q . W e recall that if A is a small ab elian category (considered also as an exact category), then w e hav e D b p A q » St p A q . In particular, if C is a small t -category , then we ha ve a realization functor D b p C ♡ q Ñ C , corresp onding to the inclusion C ♡ Ñ C . 2.2. t -structures compatible with ﬁltered colimits. In this subsection we consider t - structures on presen table stable categories, before specializing to dualizable categories. Recall that an accessible t -structure p C ě 0 , C ď 0 q on C P Pr L st is said to be compatible with ﬁltered colimits if the sub category C ď 0 is closed under ﬁltered colimits (note that C ě 0 is automatically closed under all colimits). Equiv alen tly , this means that the functor τ ě 0 : C Ñ C ě 0 comm utes with ﬁltered colimits. In this case C ě 0 is a Grothendiec k prestable category [Lur18, Proposition C.1.4.1]. In particular, the heart C ♡ is a Grothendiec k ab elian category . It is con venien t to introduce the follo wing notation. Deﬁnition 2.2. L et C b e a pr esentable stable c ate gory, and let p C ě 0 , C ď 0 q b e an ac c essible t - structur e c omp atible with ﬁlter e d c olimits. We denote by x C ´ y Ă C the lo c alizing sub c ate gory gener ate d by C ´ , and similarly for x C ` y and x C b y . We say that the t -structur e is c ontinuously b ounde d if x C b y “ C . Note that equiv alently x C ´ y is generated b y C ě 0 , x C ` y is generated b y C ď 0 , and x C b y is generated b y C ♡ . Since the t -structure is accessible, each of the categories x C ´ y , x C ` y and x C b y is presentable. W e ﬁrst mak e some almost tautological observ ations. Prop osition 2.3. L et C and p C ě 0 , C ď 0 q b e as in Deﬁnition 2.2. Then the t -structur e on C induc es the t -structur es on x C ´ y , x C ` y and x C b y , with the same he art. Mor e over, we have e qualities xx C ´ y ` y “ xx C ` y ´ y “ x C b y (of strictly ful l sub c ate gories of C ). Pr o of. The ﬁrst statemen t is eviden t: the (contin uous) comp osition C τ ě 0 Ý Ý Ñ C ě 0 ã Ñ C preserv es each of the sub categories x C ´ y , x C ` y and x C b y . Next, we hav e an inclusion C b “ C ´ X C ` Ă x C ´ y ` , hence x C b y Ă xx C ´ y ` y . On the other hand, x C ´ y ď 0 is generated THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 37 via extensions and ﬁltered colimits b y C ´ ď 0 Ă C b , hence we ha ve x C b y “ xx C ´ y ` y . The same argumen t prov es the equality x C b y “ xx C ` y ´ y . □ W e recall the following relation with the right completion. Prop osition 2.4. L et C and p C ě 0 , C ď 0 q b e as in Deﬁnition 2.2. (i) We have x C ´ y » Sp p C ě 0 q . (ii) The right adjoint to the inclusion x C ´ y Ñ C is given by x ÞÑ lim Ý Ñ n P N τ ě´ n x. In p artic- ular, it c ommutes with c olimits. (iii) We have C “ x C ´ y if and only if the t -structur e p C ě 0 , C ď 0 q is right c omplete. Pr o of. All the assertions are w ell-known. Namely , the righ t completion of C is iden tiﬁed with Sp p C ě 0 q , see for example [AN21, Lemma A.8]. F or con venience, w e write the ob jects of Sp p C ě 0 q as p x n q n ě 0 , where x n P C ě 0 and w e assume the isomorphisms x n – Ω x n ` 1 . The natural functor Φ : C Ñ Sp p C ě 0 q is giv en b y x ÞÑ p τ ě 0 p Σ n x qq n ě 0 . Its left adjoint is giv en by Φ L : Sp p C ě 0 q Ñ C , Φ L pp x n q n ě 0 q “ lim Ý Ñ n Ω n C x n Since the t -structure is compatible with ﬁltered colimits, the adjunction counit Id Ñ Φ ˝ Φ L is an isomorphism, i.e. Φ L is fully faithful. It follows that the essential image of Φ L is iden tiﬁed with x C ´ y . This prov es all the assertions. □ W e deduce the following statement, whic h will ha ve an imp ortant K -theoretic applica- tion. Prop osition 2.5. L et C and p C ě 0 , C ď 0 q b e as in Deﬁnition 2.2. (i) The fol lowing (lax c ommutative) squar e c ommutes: (2.1) x C ` y C x C b y x C ´ y . Her e the horizontal functors ar e the inclusions and the vertic al functors ar e right adjoints to the inclusions. (ii) We have a semi-ortho gonal de c omp osition in Pr L st : (2.2) C {x C b y “ xx C ` y{x C b y , x C ´ y{x C b yy In p articular, we have an e quivalenc e (2.3) x C ´ y{x C b y „ Ý Ñ C {x C ` y . 38 ALEXANDER I. EFIMOV Pr o of. (i) follo ws directly from Propositions 2.3 and 2.4: both v ertical functors in (2.1) are giv en by x ÞÑ lim Ý Ñ n P N τ ě´ n x. Next, (ii) follo ws directly from (i). Namely , the inclusion x C ´ y{x C b y Ñ C {x C b y is colimit- preserving, hence it has a right adjoin t. The right orthogonal is identiﬁed with x C ´ y K Ă C , or equiv alently with t x P C | lim Ý Ñ n P N τ ě´ n x “ 0 u . It follo ws from (i) that the image of x C ` y{x C b y in C {x C b y is con tained in the righ t orthogonal to x C ´ y{x C b y . Since x C ´ y and x C ` y generate C , w e obtain the semi-orthogonal decomp osition (2.2). The equiv alence (2.3) follo ws. □ 2.3. Compactly assem bled t -structures. W e ﬁrst recall the notion of a compactly generated t -structure. Prop osition 2.6. L et T b e a smal l stable c ate gory with a t -structur e p T ě 0 , T ď 0 q . Then the c ate gory Ind p T q has a t -structur e p Ind p T q ě 0 , Ind p T q ď 0 q , wher e Ind p T q ě 0 is the essential image of the functor Ind p T ě 0 q ã Ñ Ind p T q , and similarly for Ind p T q ď 0 . Mor e over, this t -structur e on Ind p T q is c ontinuously b ounde d if and only if the t - structur e p T ě 0 , T ď 0 q is b ounde d. Pr o of. The ﬁrst assertion is [Lur18, Lemma C.2.4.3]. F or the second assertion, note that the t -structure p T ě 0 , T ď 0 q is bounded iﬀ T is generated b y T ♡ as an idemp oten t-complete sta- ble sub category , whic h equiv alen tly means that Ind p T q is generated b y Ind p T ♡ q » Ind p T q ♡ as a lo calizing sub category , i.e. the t -structure p Ind p T q ě 0 , Ind p T q ď 0 q is con tinuously b ounded. □ The following is immediate. Corollary 2.7. In the situation of Pr op osition 2.6, for any inte gers a ď b the ful l sub c ate gory Ind p T q r a,b s Ă Ind p T q is the essential image of the functor Ind p T r a,b s q ã Ñ Ind p T q . Pr o of. Clearly , the essential image of Ind p T r a,b s q is con tained in Ind p T q r a,b s . On the other hand, for an ob ject x “ “lim Ý Ñ i ” x i P Ind p T q r a,b s the maps “lim Ý Ñ i ” τ ě a x i Ñ “lim Ý Ñ i ” x i and “lim Ý Ñ i ” τ ě a x i Ñ “lim Ý Ñ i ” τ r a,b s x i are isomorphisms. This prov es the inv erse inclusion. □ It is technically conv enient to consider the natural t -structure on the “t wice large” category Ind p C q when C is a presen table stable category with an accessible t -structure. Namely , Ind p C q is a union of Ind p C κ q , for suﬃcien tly large regular κ the t -structure on C induces a t -structure on C κ , and the inclusion functors Ind p C κ q Ñ Ind p C λ q are t - exact. The compatibilit y of p C ě 0 , C ď 0 q with ﬁltered colimits exactly means that the functor colim : Ind p C q Ñ C is t -exact (it is automatically righ t t -exact). W e in tro duce the following natural generalization of a compactly generated t -structure. THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 39 Deﬁnition 2.8. L et C b e a dualizable c ate gory, and let p C ě 0 , C ď 0 q b e an ac c essible t - structur e c omp atible with ﬁlter e d c olimits. We say that this t -structur e is c omp actly as- semble d if the functor ˆ Y : C Ñ Ind p C q is t -exact. We wil l say that C is a dualizable t -c ate gory if the t -structur e is c omp actly assemble d and c ontinuously b ounde d. Remark 2.9. (i) In the situation of the Deﬁnition 2.8 the functor ˆ Y is automatic al ly right t -exact, sinc e its right adjoint is t -exact. (ii) A c omp actly gener ate d t -structur e is c omp actly assemble d. (iii) L et p C ě 0 , C ď 0 q b e a c omp actly assemble d t -structur e on a dualizable c ate gory C . It fol lows fr om Pr op osition 2.5 that the c ondition C “ x C b y is e quivalent to the e qual- ities C “ x C ´ y “ x C ` y . By Pr op osition 2.4 the c ondition C “ x C ´ y is e quivalent to the right c ompleteness. One c an show that C “ x C ` y if and only if the Gr othendie ck pr estable c ate gory C ě 0 is antic omplete in the sense of [Lur18, Deﬁnition C.5.5.4] T o av oid confusion we sp ell out the following sp ecial case. Prop osition 2.10. L et C b e a c omp actly gener ate d pr esentable stable c ate gory, and let p C ě 0 , C ď 0 q b e an ac c essible t -structur e c omp atible with ﬁlter e d c olimits. Then this t - structur e is c omp actly assemble d if and only if it induc es a t -structur e on C ω , which e quiv- alently me ans that we ar e in the situation of Pr op osition 2.6. Pr o of. The “if ” direction follows from Prop osition 2.6. F or the “only if ” direction w e need to sho w that for x P C ω w e ha v e τ ě 0 x P C ω . This follo ws from the isomorphisms in Ind p C q : ˆ Y p τ ě 0 x q – τ ě 0 ˆ Y p x q – τ ě 0 Y p x q – Y p τ ě 0 x q . □ As usual with dualizable categories, w e obtain the ω 1 -accessibilit y statement. Prop osition 2.11. L et p C ě 0 , C ď 0 q b e a c omp actly assemble d t -structur e on a dualizable c at- e gory C . Then this t -structur e is ω 1 -ac c essible. Pr o of. Let x P C ω 1 , then ˆ Y p x q – “lim Ý Ñ n P N ” x n . W e obtain ˆ Y p τ ě 0 x q – “lim Ý Ñ n ” τ ě 0 x n , hence τ ě 0 x P C ω 1 . This prov es that the t -structure on C induces a t -structure on C ω 1 . Recall that C is ω 1 -presen table b y [E24, Corollary 1.21]. Since the t -structure on C commutes with ﬁltered colimits, it follo ws that this t -structure is ω 1 -accessible. □ Next, we hav e an in terpretation via retracts. Prop osition 2.12. L et C b e a dualizable c ate gory with a t -structur e p C ě 0 , C ď 0 q c omp atible with ﬁlter e d c olimits. The fol lowing ar e e quivalent. (i) The t -structur e is c omp actly assemble d. 40 ALEXANDER I. EFIMOV (ii) Ther e exists a smal l stable c ate gory T with a t -structur e p T ě 0 , T ď 0 q and a r e- tr action C F Ý Ñ Ind p T q G Ý Ñ C , such that b oth F and G ar e c olimit-pr eserving and t -exact. (iii) Ther e exists a dualizable c ate gory C 1 with a c omp actly assemble d t -structur e p C 1 ě 0 , C 1 ď 0 q and a r etr action C F Ý Ñ C 1 G Ý Ñ C , such that b oth F and G ar e c olimit- pr eserving and left t -exact. Mor e over, similar assertions ar e e quivalent for c omp actly assemble d c ontinuously b ounde d t -structur es, wher e in (ii) the t -structur e p T ě 0 , T ď 0 q is r e quir e d to b e b ounde d. Pr o of. (i) ù ñ (ii) . By Proposition 2.11 it suﬃces to tak e T “ C ω 1 , F “ ˆ Y , G “ colim . If the t -structure p C ě 0 , C ď 0 q is con tinuously b ounded, then w e instead take T “ p C ω 1 q b . Note that the functor ˆ Y indeed sends C to Ind pp C ω 1 q b q since it sends C ♡ to Ind pp C ω 1 q ♡ q . (ii) ù ñ (iii) is trivial. (iii) ù ñ (i) . By Remark 2.9 the functor ˆ Y C is automatically righ t t -exact, so we only need to sho w the left t -exactness. This follo ws from the retraction, since ˆ Y C is iden tiﬁed with the composition of left t -exact functors C F Ý Ñ C 1 ˆ Y C ” Ý Ý Ñ Ind p C 1 q Ind p G q Ý Ý Ý Ý Ñ Ind p C q . □ Supp ose that the t -structure on C 1 is con tin uously b ounded. W e need to show that the t -structure on C is also con tinuously b ounded. By Remark 2.9 it suﬃces to sho w the equalities C “ x C ´ y “ x C ` y . First, C is generated b y G p C 1 ď 0 q Ă C ď 0 , hence C “ x C ` y . It remains to show that C “ x C ´ y . T ake x P C , and put y “ Cone p lim Ý Ñ n P N τ ě´ n x Ñ x q . Then y P Ş n C ď´ n , hence F p y q P Ş n C 1 ď´ n “ 0 . W e get y – G p F p y qq “ 0 , so x P x C ´ y . This pro v es the proposition. Corollary 2.13. L et C b e a dualizable c ate gory with a c omp actly assemble d t -structur e p C ě 0 , C ď 0 q . L et D Ă C b e a dualizable sub c ate gory such that the inclusion functor i : D Ñ C is str ongly c ontinuous. Supp ose that the t -structur e on C induc es a t -structur e p D ě 0 , D ď 0 q on D . Then the latter t -structur e is also c omp actly assemble d. If mor e over the t -structur e p C ě 0 , C ď 0 q is c ontinuously b ounde d, then so is the t -structur e p D ě 0 , D ď 0 q . Pr o of. Denote by i R : C Ñ D the (contin uous) right adjoint to i. Then i R is left t - exact and i is t -exact. Hence, the retraction D i Ý Ñ C i R Ý Ñ D implies that p D ě 0 , D ď 0 q is compactly assem bled b y Proposition 2.12. The “moreo ver” assertion also follows from loc. cit. □ It is useful to k eep in mind a reform ulation in terms of compact morphisms (although w e will not use it). THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 41 Prop osition 2.14. L et C b e a dualizable c ate gory, and let p C ě 0 , C ď 0 q b e an ac c essible t - structur e c omp atible with ﬁlter e d c olimits. The fol lowing ar e e quivalent. (i) The t -structur e p C ě 0 , C ď 0 q is c omp actly assemble d. (ii) F or any c omp act morphism f : x Ñ y in C , the morphism τ ě 0 f : τ ě 0 x Ñ τ ě 0 y is also c omp act in C . Pr o of. This is formal: the t -structure is compactly assem bled if and only if the endofunctor τ ě 0 : C Ñ C is strongly con tin uous. F or completeness we giv e the details. (i) ù ñ (ii) . Compactness of f : x Ñ y means that Y p f q : Y p x q Ñ Y p y q factors through ˆ Y p y q in Ind p C q . Since ˆ Y p τ ě 0 y q – τ ě 0 ˆ Y p y q , w e conclude that τ ě 0 f is also a compact morphism. (ii) ù ñ (i) . Consider τ ě 0 as a con tin uous endofunctor of C , and similarly for Ind p C q . W e need to sho w that the morphism ˆ Y ˝ τ ě 0 Ñ τ ě 0 ˝ ˆ Y is an isomorphism of functors from C to Ind p C q . Let x P C and ˆ Y p x q “ “lim Ý Ñ i P I ” x i , where I is directed. Then for each i P I there exists j ě i suc h that the transition morphism x i Ñ x j is compact, hence so is τ ě 0 x i Ñ τ ě 0 x j . This pro v es that the map ˆ Y p τ ě 0 x q Ñ “lim Ý Ñ i ” τ ě 0 x i is an isomorphism, as required. □ The following basic properties of the categories C r a,b s are almost immediate. Prop osition 2.15. L et C b e a dualizable c ate gory with a c omp actly assemble d t -structur e p C ě 0 , C ď 0 q . (i) L et a ď b, wher e a P Z Y t´8u , b P Z Y t`8u . The exact c ate gory C r a,b s (with the induc e d exact structur e fr om C ) is c oher ently assemble d and the inclusion functor C r a,b s Ñ C is str ongly c ontinuous. In p articular the ab elian c ate gory C ♡ is c oher ently assemble d. (ii) The c ate gories x C ´ y , x C ` y and x C b y ar e dualizable, and their inclusion functors into C ar e str ongly c ontinuous. The induc e d t -structur es ar e c omp actly assemble d. Pr o of. (i) By Prop osition 2.11 we hav e the induced t -structure on C ω 1 . It suﬃces to prov e that the functor ˆ Y : C Ñ Ind p C ω 1 q takes C r a,b s to (the essen tial image of ) Ind pp C ω 1 q r a,b s q . Let x P C r a,b s and let ˆ Y p x q “ “lim Ý Ñ i ” x i P Ind p C ω 1 q . By Corollary 2.7 w e ha ve an isomorphism “lim Ý Ñ i ” x i – “lim Ý Ñ i ” τ r a,b s x i , as required. (ii) By (i) the functor ˆ Y C tak es C ě 0 to Ind p C ě 0 q , hence it tak es x C ´ y to Ind p C ´ q Ă Ind px C ´ yq . This prov es that x C ´ y is dualizable and its inclusion functor into C is strongly con tin uous. Similar argumen ts show the same for x C ` y and x C b y (replace C ě 0 with C ď 0 resp. C ♡ ). □ 42 ALEXANDER I. EFIMOV F rom no w on, in the situation of Prop osition 2.15 (i) we write C ω 1 r a,b s for p C ω 1 q r a,b s » p C r a,b s q ω 1 . Also, for a morphism f : x Ñ y in C r a,b s the compactness of f in C is equ iv alen t to its compactness in C r a,b s , whic h w e will tacitly use in the pro ofs. Note that for a 1 ď a ď b ď b 1 the inclusion functor C r a,b s Ñ C r a 1 ,b 1 s is exact and strongly con tinuous. W e form ulate the follo wing statemen t only for dualizable t -categories, but it holds in a m uc h more general setting, see Remark 2.17 b elo w. Prop osition 2.16. L et C b e a dualizable t -c ate gory (Deﬁnition 2.8). Then we have an e quivalenc e (2.4) lim Ý Ñ n ˇ St p C r 0 ,n s q „ Ý Ñ C . Pr o of. By Proposition 2.12 we hav e a retraction C F Ý Ñ C 1 G Ý Ñ C , where C 1 » Ind p T q , T is equipp ed with a bounded t -structure, and the functors F and G are con tinuous, exact and t -exact. Then the functor lim Ý Ñ n ˇ St p C r 0 ,n s q Ñ C is a retract of the functor lim Ý Ñ n ˇ St p C 1 r 0 ,n s q Ñ C 1 (in Pr L st ), hence w e ma y and will assume that C “ C 1 . In this case b y Prop osition 1.24 w e ha ve ˇ St p C r 0 ,n s q » Ind p St p T r 0 ,n s qq . Hence, it suﬃces to consider the corresponding colimit in Cat perf . W e hav e lim Ý Ñ n St p T r 0 ,n s q „ Ý Ñ St p lim Ý Ñ n T r 0 ,n s q » St p T ě 0 q » T . T aking the ind-completions, we obtain the equiv alence (2.4). □ Remark 2.17. The statement of Pr op osition 2.16 holds mor e gener al ly when C is pr esentable stable, the ac c essible t -structur e p C ě 0 , C ě 0 q is right c omplete and c omp atible with ﬁlter e d c olimits, and the Gr othendie ck pr estable c ate gory C ě 0 is antic omplete. The following tw o important statements are sp ecial cases of m uc h more general results on Grothendieck ab elian p m ` 1 q -categories, but our pro ofs use only the general machinery of presentable stable en velopes of coheren tly assembled exact categories. Prop osition 2.18. L et C b e a dualizable t -c ate gory, and let m ě 0 b e an inte ger. Denote by j : C r 0 ,m s Ñ ˇ St p C r 0 ,m s q the universal functor. Then for x, y P C ♡ the map Ext n ˇ St p C r 0 ,m s q p j p x q , j p y qq Ñ Ext n C p x, y q is an isomorphism for n ď m ` 1 , and a monomorphism for n “ m ` 2 . Pr o of. W e hav e isomorphisms Ext n ˇ St p C r 0 ,m s q p j p x q , j p y qq – Ext n ´ m ˇ St p C r 0 ,m s q p j p x q , j p y r m sqq , n P Z . THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 43 Applying Corollary 1.38, we see that the map Ext n ´ m ˇ St p C r 0 ,m s q p j p x q , j p y r m sqq Ñ Ext n C p x, y q is an isomorphism for n ď m ` 1 , and a monomorphism for n “ m ` 2 , as required. □ Recall that in the situation of Prop osition 2.16 we hav e ˇ St p C r 0 , 0 s q » ˇ D p C ♡ q – the unsep- arated derived category of the heart. Prop osition 2.19. L et C b e a dualizable t -c ate gory and let m ě 0 b e an inte ger. Consider the (str ongly c ontinuous exact) functor Φ : ˇ D p C ♡ q Ñ ˇ St p C r 0 ,m s q induc e d by the inclusion C ♡ ã Ñ C r 0 ,m s . The fol lowing ar e e quivalent. (i) Φ is an e quivalenc e. (ii) F or x, y P C ♡ , the map Ext n C ♡ p x, y q Ñ Ext n C p x, y q is an isomorphism for n ď m ` 1 , and a monomorphism for n “ m ` 2 . (iii) F or x, y P C ω 1 , ♡ , the map Ext n C ♡ p x, y q Ñ Ext n C p x, y q is an isomorphism for n ď m ` 1 . Pr o of. The implication (i) ù ñ (ii) follo ws directly from Proposition 2.18. The implication (ii) ù ñ (iii) is trivial. (iii) ù ñ (i) . Clearly , the essential image of Φ generates ˇ St p C r 0 ,m s q as a lo calizing sub category . Hence, w e only need to sho w that Φ is fully faithful. By Prop osition 1.39, it suﬃces to show that for x P C ω 1 r 0 ,m s there exists y P C ω 1 , ♡ and a map f : y Ñ x such that the map π 0 p f q : y Ñ π 0 p x q is an epimorphism. If m “ 0 , then there is nothing to prov e. If m ą 0 , consider the map g : π 0 p x q Ñ τ ě 1 x r 1 s . It follo ws from (iii) (applied to Ext 2 ) that w e can ﬁnd an epimorphism f 1 : y 1 Ñ π 0 p x q in C ω 1 , ♡ suc h that the comp osition y 1 f 1 Ý Ñ π 0 p x q g Ý Ñ τ ě 1 x r 1 s Ñ π 1 p x qr 2 s is zero. Then g ˝ f 1 factors through τ ě 2 x r 1 s , so w e obtain a map g 1 : y 1 Ñ τ ě 2 x r 1 s . Con tinuing the pro cess, we ﬁnd an epimorphism f m : y m Ñ π 0 p x q in C ω 1 , ♡ suc h that the comp osition y m f m Ý Ý Ñ π 0 p x q g Ý Ñ τ ě 1 x r 1 s is zero. This means that f m factors through x, whic h pro ves the implication. □ W e conclude this subsection with a K -theoretic application of the abov e results. Corollary 2.20. L et C b e a dualizable c ate gory with a c omp actly assemble d t -structur e p C ě 0 , C ď 0 q . Then the fol lowing squar e of sp e ctr a is c artesian: (2.5) K cont px C b yq K cont px C ´ yq K cont px C ` yq K cont p C q . 44 ALEXANDER I. EFIMOV Pr o of. By Proposition 2.15 this square is w ell-deﬁned: the four categories are dualizable and the inclusion functors are strongly contin uous. It suﬃces to pro ve that the map b et w een the coﬁb ers of horizon tal arro ws is an equiv alence. This follows directly from Prop osition 2.5: we ha v e K cont px C ´ y{x C b yq „ Ý Ñ K cont p C {x C ` yq . □ 2.4. General constructions of t -structures. W e will need a general eﬃcient metho d for constructing a t -structure. First w e deal with small categories, for whic h the construction seems to b e a folklore kno wledge. In the following prop osition w e do not a priori assume the idemp otent-completeness, but it holds a posteriori. Prop osition 2.21. L et C b e a smal l stable c ate gory, and let A b e a smal l ab elian c ate gory. Supp ose that F : A Ñ C is a ful ly faithful exact functor such that F p A q gener ates C as a stable sub c ate gory. Then C has a unique b ounde d t -structur e such that F p A q Ă C ♡ . Each obje ct of C ♡ has a ﬁnite ﬁltr ation with sub quotients in F p A q . Mor e over, if F induc es isomorphisms Ext 1 A p x, y q „ Ý Ñ Ext 1 C p F p x q , F p y qq for x, y P A , then we have C ♡ » A . Pr o of. W e will construct the t -structure with required prop erties, and its uniqueness follows from the proof. W e deﬁne C ě 0 Ă C resp. C ď 0 Ă C to be the full sub category generated via extensions b y F p A qr n s , where n ě 0 resp. n ď 0 . Clearly , C ě 0 r 1 s Ă C ě 0 C ď 0 r´ 1 s Ă C ď 0 and F p A q Ă C ě 0 X C ď 0 . Also, the orthogonality holds since Ext ă 0 C p F p x q , F p y qq “ 0 for x, y P A b y assumption. As usual w e denote b y r m s the p oset t 0 , 1 . . . , m u with the usual order. W e only need to pro v e that for an y x P C there exists some m ě 0 and a functor r m s ÞÑ C , i ÞÑ x i , suc h that x 0 “ 0 , x m – x and we hav e Cone p x i ´ 1 Ñ x i q P F p A qr k i s for 1 ď i ď m, where k 1 ě k 2 ě ¨ ¨ ¨ ě k m . If this is the case for some ob ject x, we sa y that x has a go o d ﬁltration. Since F p A q generates C as a stable sub category , w e only need to show that for an y ob ject x P C with a goo d ﬁltration, for an y y P A , for any l P Z and for any morphism f : F p y qr l s Ñ x, the ob ject Cone p f q has a goo d ﬁltration. Since the class of ob jects with a goo d ﬁltration is closed under shifts, we ma y and will assume that l “ 0 . Let r m s Ñ C , x i and k i b e as abov e, with x m – x etc. If m “ 0 , then x “ 0 and there is nothing to prov e. So w e assume that m ą 0 . If k 1 ă 0 , then f “ 0 and Cone p f q – x ‘ F p y qr 1 s has a go o d ﬁltration.. Hence, w e assume that k 1 ě 0 . W e take the largest p suc h that k p ě 0 , then we get a unique factorization f : F p y q g Ý Ñ x p Ñ x. It suﬃces to pro v e that Cone p g q has a go o d ﬁltration with sub quotients in F p A qrě 0 s . W e pro v e this b y induction on s “ |t i : k i “ 0 u| . THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 45 If s “ 0 , then k p ě 1 , hence the coﬁb er sequence x p Ñ Cone p g q Ñ F p y qr 1 s shows that Cone p g q has a go o d ﬁltration with subquotients in F p A qrě 1 s . No w supp ose that s ą 0 and the assertion holds for smaller v alues of s. W e hav e k p “ 0 , so there is some z P A and an isomorphism Cone p x p ´ 1 Ñ x p q – F p z q . Denote b y φ : y Ñ z the morphism in A such that the comp osition F p y q g Ý Ñ x p Ñ F p z q is homotopic to F p φ q . First suppose that φ is not an epimorphism. Consider the comp osition ψ : x p Ñ F p z q Ñ F p cok er p φ qq , and put x 1 p “ Fiber p ψ q . Then we hav e a unique factorization g : F p y q g 1 Ý Ñ x 1 p Ñ x p , and it suﬃces to pro ve that Cone p g 1 q has a go o d ﬁltration with sub quotien ts in F p A qrě 0 s . Hence, w e ma y and will assume that φ : y Ñ z is an epimorphism. Put w “ ker p φ q , then the composition F p w q Ñ F p y q g Ý Ñ x p factors uniquely through x p ´ 1 . Moreov er, w e ha v e Cone p g q – Cone p F p w q Ñ x p ´ 1 q . By the induction h yp othesis, this ob ject has a go o d ﬁltration with subquotients in F p A qrě 0 s . This prov es the induction step. Finally , the “moreo v er” assertion is immediate: the assumption implies that F p A q is closed under extensions in C ♡ , hence the functor A Ñ C ♡ is essentially surjective. □ W e deduce an analogue for dualizable categories. Prop osition 2.22. L et C b e a dualizable c ate gory, and let A b e a c oher ently assemble d ab elian c ate gory. Supp ose that F : A Ñ C is a str ongly c ontinuous ful ly faithful exact functor, such that F p A q gener ates C as a lo c alizing sub c ate gory. (i) C has a unique c omp actly assemble d c ontinuously b ounde d t -structur e such that F p A q Ă C ♡ . Mor e over, F p A q gener ates C ♡ via extensions and ﬁlter e d c olimits. (ii) If F induc es isomorphisms Ext 1 A p x, y q „ Ý Ñ Ext 1 C p F p x q , F p y qq for x, y P A ω 1 , then we have C ♡ » A . (iii) L et D b e another dualizable t -c ate gory and let G : C Ñ D b e a str ongly c ontinuous exact functor. Then G is t -exact if and only if we have G p F p A ω 1 qq Ă D ♡ . Pr o of. (i) Again, w e construct the t -structure with required prop erties, and its uniqueness follo ws automatically . Denote by T Ă C ω 1 the (small) stable subcategory generated b y F p A ω 1 q . By Prop o- sition 2.21, T has a unique bounded t -structure such that F p A ω 1 q Ă T ♡ . The strong con tin uit y of F implies that ˆ Y C p F p A qq Ă Ind p T q . Since F p A q generates C as a lo calizing sub category , it follo ws that ˆ Y C p C q Ă Ind p T q . W e claim that the t -structure on Ind p T q induces a t -structure on ˆ Y C p C q . T o see this, it suﬃces to pro v e that the composition Φ : Ind p T q colim Ý Ý Ý Ñ C ˆ Y C Ý Ý Ñ Ind p T q is t -exact. Since the t -structure on T is bounded, we only need to sho w the inclusion Φ p T ♡ q Ă Ind p T ♡ q . By Prop osition 2.21, each ob ject of T ♡ has a ﬁnite ﬁltration with subquotients in F p A ω 1 q . 46 ALEXANDER I. EFIMOV Hence, it suﬃces to show that Φ p F p A ω 1 qq Ă Ind p F p A ω 1 qq . The latter inclusion follows from the strong con tinuit y of F . W e claim that the constructed t -structure on C satisﬁes the desired properties. Since the t -structure on Ind p T q is compactly assem bled and con tinuously b ounded, so is the t -structure on C by Corollary 2.13. Since F p A ω 1 q generates T ♡ via extensions (by Propo- sition 2.21), the retraction C ♡ Ñ Ind p T ♡ q Ñ C ♡ implies that C ♡ is generated b y F p A q via extensions and ﬁltered colimits. (ii) follo ws from the pro of of (i): we hav e T ♡ » A ω 1 b y Proposition 2.21. Hence, C ♡ is generated b y F p A ω 1 q via ﬁltered colimits, which shows that the functor A Ñ C ♡ is essen tially surjective. (iii) The “only if ” direction is clear. F or the “if ” direction w e ﬁrst note that G p C ♡ q Ă D ♡ since C ♡ is generated b y F p A ω 1 q via extensions and ﬁltered colimits. Next, we ha ve F p C ě 0 q Ă D ě 0 since C ♡ generates C ě 0 via extensions and colimits. Finally , w e hav e F p C ď 0 q Ă D ď 0 since C ď 0 is generated by C ♡ via ﬁnite limits, extensions and ﬁltered colimits (by the righ t completeness). □ Corollary 2.23. L et C b e dualizable t -c ate gory and let m ě 0 b e an inte ger. Ther e is a unique c omp actly assemble d c ontinuously b ounde d t -structur e on ˇ St p C r 0 ,m s q such that the functor ˇ D p C ♡ q Ñ ˇ St p C r 0 ,m s q is t -exact. Mor e over, we have ˇ St p C r 0 ,m s q ♡ » C ♡ . Pr o of. This follows directly from Prop osition 2.22, applied to the functor C ♡ Ñ C r 0 ,m s Ñ ˇ St p C r 0 ,m s q . □ Let C b e a dualizable category and let A be a contin uous exact monad on C , i.e. A P Alg E 1 p F un L p C , C qq . Here we consider the category F un L p C , C q as a presentable stable E 1 -monoidal category (i.e. an E 1 -algebra in Pr L st ). Then w e can consider the dualizable category Mo d A p C q of A -mo dules in C . The forgetful functor Mo d A p C q Ñ C has a left adjoin t x ÞÑ A p x q , whic h is automatically strongly con tinuous. By [E24, Prop osition C.1] this gives a fully faithful functor (2.6) Alg E 1 p F un L p C , C qq ã Ñ p Cat dual st q C { . Its essential image consists of pairs p D , Φ : C Ñ D q such that Φ p C q generates D as a lo calizing sub category . F or such p D , Φ q the functor Φ R is monadic, so w e hav e an equiv alence D » Mo d Φ R ˝ Φ p C q . More generally , the right adjoin t to (2.6) is giv en b y p D , Φ q ÞÑ Φ R ˝ Φ . W e now apply this p oin t of view to dualizable t -categories. THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 47 Corollary 2.24. L et C b e a dualizable t -c ate gory. L et A P Alg E 1 p F un L p C , C qq b e a c on- tinuous exact monad. Supp ose that the functor Cone p Id C Ñ A qr 1 s is left t -exact. Then the c ate gory Mo d A p C q has a unique c omp actly assemble d c ontinuously b ounde d t -structur e such that the functor C Ñ Mo d A p C q is t -exact. Pr o of. By adjunction, the left t -exactness of the functor Cone p Id C Ñ A qr 1 s implies that the comp osition C ♡ Ñ C Ñ Mo d A p C q is fully faithful. Since its image generates the target as a lo calizing subcategory , the assertion of the corollary is a sp ecial case of Prop osition 2.22. □ W e also deduce the follo wing basic result. Prop osition 2.25. L et C b e a dualizable t -c ate gory. L et y b e a formal variable of de gr e e 0 , and c onsider the (c omp actly gener ate d) c ate gory Mo d y - tors - S r y s of S r y s -mo dules such that y acts lo c al ly nilp otently on the homotopy gr oups. Denote by i : Sp Ñ Mo d y - tors - S r y s the r estriction of sc alars functor for S r y s Ñ S , y ÞÑ 0 . (i) The c ate gory p Mo d y - tors - S r y sq b C has a unique c omp actly assemble d c ontinuously b ounde d t -structur e such that the functor i b C : C Ñ p Mod y - tors - S r y sq b C is t - exact. Mor e over, the ab elian c ate gory pp Mo d y - tors - S r y sq b C q ♡ is identiﬁe d with the c ate gory of p airs p X , f q , wher e X P C ♡ and f : X Ñ X is a lo c al ly nilp otent endomorphism. (ii) If we have an e quivalenc e ˇ D p C ♡ q „ Ý Ñ C , then we also have an e quivalenc e ˇ D ppp Mo d y - tors - S r y sq b C q ♡ q „ Ý Ñ p Mo d y - tors - S r y sq b C . Pr o of. (i) It is w ell-known that the category Mo d y - tors - S r y s is generated b y the single com- pact ob ject i p S q – Cone p S r y s y Ý Ñ S r y sq , in particular the functor i is strongly con tin uous. Th us, the functor i b C : C Ñ p Mo d y - tors S r y sq b C is also strongly contin uous and its image generates the target as a lo calizing sub category . W e hav e Cone p S Ñ i R p i p S qqq – S r´ 1 s , in particular the functor Cone p Id C Ñ p i b C q R ˝ p i b C qqr 1 s – Id C is left t -exact. Applying Corollary 2.24, we obtain the stated t -structure and its uniqueness. By construction the forgetful functor p Mod y - tors S r y sq b C Ñ C is t -exact and conserv a- tiv e. This giv es the description of the heart and prov es (i). (ii) By Prop osition 1.31 and its pro of, w e may and will assume that C ♡ is lo cally coherent, i.e. C is compactly generated. Put A “ p C ♡ q ω , and let Nil p A q b e the category of pairs p X , f q , where X P A and f : X Ñ X is a nilpotent endomorphism. W e need to pro ve 48 ALEXANDER I. EFIMOV that the realization functor (2.7) D b p Nil p A qq Ñ Perf y - tors p S r y sq b D b p A q “ D is an equiv alence. By [Nee21, Lemma 2.1] it suﬃces to show that for M P D ě 0 there exists N P D ♡ and a map f : N Ñ M such that Cone p f q P D ě 1 . Denote by Φ : D Ñ D b p A q the forgetful functor. Then there exists some N 1 P A and a map g : N 1 Ñ Φ p M q suc h that Cone p g q P D b p A q ě 1 . By adjunction g corresponds to a map g 1 : N 1 r y s Ñ M , where we consider N 1 r y s as an ob ject of P erf p S r y sq b D b p A q . Choose n ą 0 such that y n is acting b y zero on M . Then g 1 factors through N “ Cone p N 1 r y s y n Ý Ñ N 1 r y sq (non-uniquely), whic h gives a map f : N Ñ M such that π 0 p f q : N Ñ π 0 p M q is an epimorphism, i.e. Cone p f q P D ě 1 . Since N P D ♡ , this shows that (2.7) is an equiv alence and prov es (ii). □ 2.5. Suﬃcient conditions for left t -exactness. In this subsection we consider dualizable t - categories, an d we record some semi-trivial observ ations on ho w to c heck the left t -exactness of a functor b etw een them. The statements of course hold in more general situations. Prop osition 2.26. L et F : C Ñ D b e a c ontinuous exact functor b etwe en dualizable t - c ate gories. The fol lowing ar e e quivalent. (i) F is left t -exact. (ii) We have F p C ω 1 , ♡ q Ă D ď 0 . Pr o of. The t -structure on C is righ t complete and ω 1 -accessible, hence C ď 0 is generated b y C ω 1 , ♡ via ﬁnite limits, extensions and ﬁltered colimits. This implies the equiv alence. □ Prop osition 2.27. L et C and D b e dualizable t -c ate gories, and let F : C Ñ D b e a str ongly c ontinuous, exact and t -exact functor, such that F p C q gener ates D as a lo c alizing sub c ate gory and mor e over the r estriction F | C ♡ : C ♡ Ñ D ♡ is ful ly faithful. F or a c ontinuous exact endofunctor G : D Ñ D the fol lowing ar e e quivalent. (i) G is left t -exact. (ii) The c omp osition F R ˝ G ˝ F is left t -exact. Pr o of. The implication (i) ù ñ (ii) is trivial since b oth F and F R are left t -exact. (ii) ù ñ (i) . W e ﬁrst sho w that F R ˝ G : D Ñ C is left t -exact. By Prop osition 2.22, F p C ♡ q generates D ♡ via extensions and ﬁltered colimits. Hence, F R p G p D ♡ qq Ă C ď 0 . Th us, by Prop osition 2.26 the functor F R ˝ G is left t -exact. No w w e show that G is left t -exact. Let x P D ď 0 . By adjunction for y P C ♡ w e ha ve Hom D p F p y q , G p x qq P Sp ď 0 . Again, since F p C ♡ q generates D ♡ via extensions and ﬁltered colimits, for any z P D ♡ w e hav e Hom D p z , G p x qq P Sp ď 0 . But D ♡ generates D ě 0 via colimits and extensions, since ˆ Y p D ě 0 q Ă Ind p D b ě 0 q . W e conclude that G p x q P D ď 0 , as required. □ THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 49 2.6. Filtered colimits of dualizable t -categories. The following result is elementary , and again it is a special case of a more general fact. Prop osition 2.28. L et p C i q i P I b e a dir e cte d system of dualizable t -c ate gories, wher e we assume the tr ansition functors F ij : C i Ñ C j to b e str ongly c ontinuous, exact and t -exact. Then the c olimit C “ lim Ý Ñ i C i is natur al ly a dualizable t -c ate gory and al l the functors C i Ñ C ar e t -exact. Pr o of. If all C i are compactly generated, then the assertion follo ws directly from Prop osition 2.10: w e hav e a diagram of small t -categories p C ω i q i P I , and the colimit lim Ý Ñ i C ω i (tak en in Cat perf ) has a natural b ounded t -structure suc h that all the functors C ω i Ñ lim Ý Ñ i C ω i are t -exact. W e ha ve C “ Ind p lim Ý Ñ i C ω i q , whic h pro ves the prop osition in this case. The general case follows: take the colimit B “ lim Ý Ñ i C ω 1 ,b i in Cat perf , and let F : C Ñ Ind p B q b e the colimit (in Cat dual st ) of functors ˆ Y : C i Ñ Ind p C ω 1 ,b i q . Then F is fully faithful and the comp osition F ˝ F R : Ind p B q Ñ Ind p B q is t -exact, hence by Corollary 2.13 w e obtain a compactly assem bled con tin uously b ounded t -structure on C . The functors C i Ñ C are t -exact since they are compositions of t -exact functors C i Ñ Ind p C ω 1 ,b i q Ñ Ind p B q F R Ý Ý Ñ C . □ W e will need the follo wing elemen tary application. Prop osition 2.29. L et C b e a dualizable t -c ate gory. L et S b e a p oset, and c onsider the t -structur e on the (dualizable) c ate gory D “ F un p S op , C q , given by D ě 0 “ F un p S op , C ě 0 q , D ď 0 “ F un p S op , C ď 0 q . Then this t -structur e is ac c essible, c omp atible with ﬁlter e d c olimits and is c ontinuously b ounde d. Mor e over, this t -structur e is c omp actly assemble d in the fol lowing two c ases: (i) S is ﬁnite. (ii) S is a me et-semi-lattic e, i.e. S has the smal lest element and for any i, j P S ther e exists i ^ j “ inf p i, j q P S (in other wor ds, S c onsider e d as a c ate gory has ﬁnite pr o ducts). Pr o of. It is clear that the t -structure is w ell-deﬁned, accessible and compatible with ﬁltered colimits. Denote by Φ i : C Ñ D the left adjoint to F ÞÑ F p i q , i P I . Then the functors Φ R i , i P I , form a conserv ative family , hence D is generated b y Φ i p C ♡ q Ă D ♡ , so the t -structure is con tin uously b ounded. No w consider the cases. 50 ALEXANDER I. EFIMOV (i) By Proposition 2.12 we ma y and will assume that C is compactly generated. Then D is also compactly generated and we hav e D ω “ F un p S op , C ω q since S is ﬁnite. It follo ws that the t -structure on D induces a t -structure on D ω , whic h pro ves (i). (ii) Let I be the p oset of (full) ﬁnite subp osets T Ă S closed under ﬁnite meets (in particular, we require that T con tains the smallest element of S ). Then I is directed, and w e hav e D » lim Ý Ñ T P I F un p T op , C q , where the colimit is tak en in Cat dual st (equiv alen tly , in Pr L st ). Here the transition functors are left Kan extensions. No w, each inclusion of p osets T Ă T 1 in I has a left adjoin t by assumption. Hence, the functor F un p T op , C q Ñ F un p T 1 op , C q is t -exact. It remains to apply (i) and Prop osition 2.28. □ 2.7. Dual t -structures. This subsection can be skipp ed on the ﬁrst reading, w e include it for completeness. Recall that if T is a small stable category with a t -structure p T ě 0 , T ď 0 q , then the category T op has a natural opp osite t -structure giv en b y p T op q ě 0 “ p T ď 0 q op , p T op q ď 0 “ p T ě 0 q op . By taking ind-ob jects we obtain the notion of a dual of a compactly generated t -structure. W e explain a natural generalization for compactly assem bled t - structures. Prop osition 2.30. L et C b e a dualizable c ate gory with a c omp actly assemble d t - structur e p C ě 0 , C ď 0 q . Then the c ate gory C _ has a natur al c omp actly assemble d t -structur e pp C _ q ě 0 , p C _ q ď 0 q which is describ e d as fol lows. Consider the (str ongly c ontinuous ful ly faithful) functor Φ “ p colim q _ : C _ Ñ Ind pp C ω 1 q op q . Then p C _ q ě 0 “ t x P C _ | Φ p x q P Ind pp C ω 1 ď 0 q op q Ă Ind pp C ω 1 q op qu , p C _ q ď 0 “ t x P C _ | @ y P C ω 1 ď 0 we have e v C p y , x q P Sp ď 0 u . Pr o of. Consider the compactly generated t -structure on Ind pp C ω 1 q op q , coming from the opp osite t -structure on p C ω 1 q op . It suﬃces to prov e that it induces the t -structure on C _ via Φ , which w ould b e automatically compactly assembled b y Corollary 2.13. Indeed, b y deﬁnition p C _ q ě 0 is the preimage of Ind pp C ω 1 q op q ě 0 , and the condition x P p C _ q ď 0 means that the mapping space Map p y , Φ p x qq is discrete for y P Ind pp C ω 1 q op q ě 0 , or equiv alently Φ p x q P Ind pp C ω 1 q op q ď 0 . Let x P C _ . It suﬃces to pro ve that the ob ject τ ě 0 Φ p x q is contained in the essen tial image of Φ . If we identify Ind pp C ω 1 q op q with F un ex p C ω 1 , Sp q , then Φ p x q “ e v C p´ , x q . Consider the functor (2.8) F : C ω 1 ď 0 Ñ S , F p y q “ Ω 8 e v C p y , x q . Then F comm utes with ﬁnite limits (since ﬁnite limits in C ω 1 ď 0 are the same as in C ω 1 ). Hence, F is pro-represen table, i.e. there exists a directed p oset I and a functor I op Ñ C ω 1 ď 0 , THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 51 i ÞÑ y i , suc h that F – lim Ý Ñ i Map p y i , ´q . Then τ ě 0 Φ p x q – lim Ý Ñ i Hom C p y i , ´q P F un ex p C ω 1 , Sp q . The essential image of Φ in F un ex p C ω 1 , Sp q is identiﬁed with the full sub category F un ω 1 - rex p C ω 1 , Sp q . Hence, we need to prov e that the functor lim Ý Ñ i Hom C p y i , ´q : C ω 1 Ñ Sp comm utes with coun table colimits. It suﬃces to sho w that for an y i P I there exists j ě i suc h that the map y j Ñ y i is compact in C ď 0 , hence also compact in C . This follo ws directly from the fact that the functor (2.8) commutes with ﬁltered colimits. Namely , it suﬃces to sho w that for z P C ω 1 ď 0 and for α P π 0 F p z q there exists z 1 P C ω 1 ď 0 , a compact map f : z 1 Ñ z and an element β P π 0 F p z 1 q suc h that π 0 F p f qp β q “ α. It suﬃces to tak e z 1 “ z n for large n, where ˆ Y p z q – “lim Ý Ñ n P N ” z n . This pro ves that τ ě 0 Φ p x q P Φ p C _ q , as required. □ The ab ov e construction is functorial in the follo wing sense. Prop osition 2.31. L et C and D b e dualizable c ate gories with c omp actly assemble d c ontin- uously b ounde d t -structur es p C ě 0 , C ď 0 q r esp. p D ě 0 , D ď 0 q . L et F : C Ñ D b e a str ongly c ontinuous, exact and t -exact functor. We e quip C _ and D _ with t -structur es as in Pr op osition 2.30. Then the functor p F R q _ : C _ Ñ D _ is t -exact. Pr o of. W e hav e a comm utativ e square C _ Ind pp C ω 1 q op q D _ Ind pp D ω 1 q op q , p colim q _ p F R q _ Ind pp F ω 1 q op q p colim q _ in which the functors are strongly con tin uous and the horizon tal arro ws are fully faithful. Clearly , the righ t vertical arrow is t -exact. By the pro of of Prop osition 2.30 the horizon tal arro ws are t -exact. Hence, the left vertical arro w is also t -exact. □ Remark 2.32. It is e asy to show that in the situation of Pr op osition 2.30 the double dual t -structur e on C __ » C is identiﬁe d with the original t -structur e p C ě 0 , C ď 0 q . F urthermor e, the t -structur e p C ě 0 , C ď 0 q is c ontinuously b ounde d if and only if so is pp C _ q ě 0 , p C _ q ď 0 q . This is also e quivalent to the right c ompleteness of b oth t -structur es. 3. Refined K -theoretic theorem of the hear t for dualizable ca tegories In this section we pro v e the follo wing result. 52 ALEXANDER I. EFIMOV Theorem 3.1. L et C b e a dualizable t -c ate gory. L et m ě 0 b e an inte ger. Then the map K cont j p C r 0 ,m s q Ñ K cont j p C q is an isomorphism for j ě ´ m ´ 2 , and a monomorphism for j “ ´ m ´ 3 . W e ﬁrst men tion an immediate corollary . Corollary 3.2. L et C b e a dualizable t -c ate gory. Supp ose that for some n ě 1 we have isomorphisms Ext i C ♡ p x, y q Ñ Ext i C p x, y q for i ď n and x, y P C ω 1 , ♡ (this assumption always holds for n “ 1 ). Then the map K cont j p C ♡ q Ñ K cont j p C q is an isomorphism for j ě ´ n ´ 1 , and a monomorphism for j “ ´ n ´ 2 . Pr o of. By Prop osition 2.19 our assumption implies that we ha ve an equiv alence ˇ D p C ♡ q „ Ý Ñ ˇ St p C r 0 ,n ´ 1 s q . Hence, the assertion follo ws from Theorem 3.1. □ Our arguments are completely diﬀerent from [Bar15]. Even if we are only interested in K cont ě 0 , still the pro of of [Bar15, Theorem 6.1] do es not generalize to dualizable (not compactly generated) categories. W e also note that an example b elow (Corollary 8.2) sho ws that the group K cont ´ 1 migh t b e non-zero for a dualizable t -category . In other w ords, the statemen t of [AGH19, Theorem 2.35] do es not hold for dualizable categories. Ho wev er, a diﬀeren t version of a construction from loc. cit. will be very useful for us, see Propositions 3.4 and 3.5 b elow. One of the key statemen ts for our pro of is the following sp ecial case of Theorem 3.1. Lemma 3.3. L et C b e a dualizable t -c ate gory. Then for any m ě 0 the map K cont ´ m ´ 2 p C r 0 ,m s q Ñ K cont ´ m ´ 2 p C q is surje ctive. The pro of of this lemma will require some preparations. The general idea is to ﬁnd “suﬃcien tly b ounded” represen tatives of classes in the negative contin uous K -theory of C . The k ey ingredient is the follo wing construction. W e consider the set Q ě 0 as a totally ordered set with the usual order. Prop osition 3.4. L et C b e a dualizable t -c ate gory. Consider the smal l stable c ate gory C ω 1 ,b with the b ounde d t -structur e induc e d fr om C , and let D b e its simultane ous left and right c ompletion, i.e. D “ lim Ð Ý n C ω 1 r´ n,n s . Equivalently, D is the left c ompletion of the c ate gory C ω 1 , ` , i.e. D » lim Ð Ý n C ω 1 ď n . Consider the lo c alizing sub c ate gory E Ă Ind p D q gener ate d by obje cts of the form “ lim Ý Ñ a P Q ě 0 ” F p a q , wher e F : Q ě 0 Ñ D is a functor such that for any r ational 0 ď a ă b and for any n P N the morphism τ r´ n,n s F p a q Ñ τ r´ n,n s F p b q is c omp act in C . Her e we identify D r´ n,n s with C ω 1 r´ n,n s Ă C . THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 53 (i) The c ate gory E is dualizable and the inclusion functor E Ñ Ind p D q is str ongly c ontinuous. Mor e over, the ab ove gener ating c ol le ction of obje cts of E is exactly the c ol le ction of al l ω 1 -c omp act obje cts of E , up to isomorphism. (ii) The (c omp actly gener ate d) t -structur e on Ind p D q induc es a t -structur e on E . The latter t -structur e is c omp actly assemble d. The c ate gory E ω 1 , ♡ c onsists of obje cts isomorphic to “ lim Ý Ñ a P Q ě 0 ” F p a q , wher e F : Q ě 0 Ñ D ♡ » C ω 1 , ♡ is a functor such that for any r ational 0 ď a ă b the map F p a q Ñ F p b q is c omp act in C (e quivalently, c omp act in C ♡ ). (iii) We have a t -exact e quivalenc e C „ Ý Ñ x E b y , so that the fol lowing squar e c ommutes: C x E b y Ind p C ω 1 ,b q Ind p D q . „ ˆ Y Her e the lower horizontal functor is induc e d by the inclusion C ω 1 ,b » D b Ñ D . Pr o of. (i) The argument is the same as for similar constructions with dualizable categories, but we sp ell it out for completeness. Denote by T Ă E the full sub category formed b y ob jects isomorphic to an ob ject in the generating collection describ ed ab ov e. T ak e some x “ “ lim Ý Ñ a P Q ě 0 ” F p a q P T , where F : Q ě 0 Ñ D is a functor as ab ov e. Then for any n P N ą 0 the ob ject x n : “ “ lim Ý Ñ 0 ď a ă n ” F p a q is also in T , and w e ha v e x – lim Ý Ñ n x n . Moreov er, the map x n Ñ x n ` 1 in Ind p D q factors through the constant ind-ob ject F p n q , hence this map is compact in Ind p D q . This implies that E is dualizable and the inclusion functor E Ñ Ind p D q is strongly con tinuous, see also [E24, Lemma1.83 and its proof ]. T o pro ve the equalit y E ω 1 “ T (of strictly full sub categories of E ), we only need to sho w that T is closed under coun table colimits and shifts. Clearly , for x P T we ha ve x r n s P T for n P Z . Next, let p F n : Q ě 0 Ñ D q n P N b e a collection of functors as ab ov e. F or n ě 0 deﬁne G n : Q ě 0 Ñ D to be the left Kan extension of p F n q | Q ě n , i.e. G n p a q “ $ & % F n p a q for a ě n ; 0 for 0 ď a ă n. Then for any a P Q ě 0 there are only ﬁnitely man y n P N suc h that G n p a q ‰ 0 , hence the direct sum G “ À n G n : Q ě 0 Ñ D is well-deﬁned. Moreov er, for an y rational 0 ď a ă b and for k P N the map τ r´ k,k s G p a q Ñ τ r´ k,k s G p b q is compact in C . W e hav e “ lim Ý Ñ a P Q ě 0 ” G p a q – à n P N “ lim Ý Ñ a P Q ě 0 ” F n p a q , 54 ALEXANDER I. EFIMOV whic h shows that T Ă E is closed under coun table copro ducts. It remains to show that T Ă E is closed under cones (coﬁb ers). Let φ : x Ñ x 1 b e a morphism in T , and x – “ lim Ý Ñ a P Q ě 0 ” F p a q , x 1 – “ lim Ý Ñ a P Q ě 0 ” F 1 p a q , where the functors F , F 1 : Q ě 0 Ñ D satisfy the ab ov e conditions. W e ma y assume that φ is induced b y a morphism of functors F Ñ F 1 . Indeed, iden tifying x resp. x 1 with “lim Ý Ñ n P N ” F p n q resp. “lim Ý Ñ n P N ” F 1 p n q , we ﬁrst see that there exists a strictly increasing function f : N Ñ N such that φ is induced by a morphism of functors F | N Ñ p F 1 | N q ˝ f . T ake any strictly increasing function g : Q ě 0 Ñ Q ě 0 suc h that g p n q “ f p n ` 1 q for n P N . W e obtain a morphism of functors F Ñ F 1 ˝ g (which factors through the right Kan extension of p F 1 | N q ˝ f ), which induces φ : x Ñ x 1 . Replacing F 1 b y F 1 ˝ g , w e ma y assume that w e ha ve a morphism u : F Ñ F 1 whic h induces φ. No w w e ha v e Cone p φ : x Ñ x 1 q – “ lim Ý Ñ a P Q ě 0 ” Cone p u a : F p a q Ñ F 1 p a qq . It suﬃces to chec k that for 0 ď a ă b and for k P N the map τ r´ k,k s Cone p u a q Ñ τ r´ k,k s Cone p u b q is compact in C . By assumptions on F and F 1 the maps τ r´ k ´ 1 ,k ` 1 s F p a q Ñ τ r´ k ´ 1 ,k ` 1 s F ˆ a ` b 2 ˙ , τ r´ k ´ 1 ,k ` 1 s F 1 ˆ a ` b 2 ˙ Ñ τ r´ k ´ 1 ,k ` 1 s F 1 p b q are compact in C . It follo ws that the map (3.1) Cone p τ r´ k ´ 1 ,k ` 1 s F p a q Ñ τ r´ k ´ 1 ,k ` 1 s F 1 p a qq Ñ Cone p τ r´ k ´ 1 ,k ` 1 s F p b q Ñ τ r´ k ´ 1 ,k ` 1 s F 1 p b qq is also compact in C . Applying τ r´ k,k s to (3.1) we deduce that the map τ r´ k,k s Cone p u a q Ñ τ r´ k,k s Cone p u b q is compact in C . This ﬁnishes the pro of of (i). No w (ii) follo ws directly from (i) and Corollary 2.13. Indeed, it suﬃces to show that for x P E ω 1 w e hav e τ ě 0 x P E , where τ ě 0 is the truncation endofunctor of Ind p D q . By (i) w e hav e x – “ lim Ý Ñ a P Q ě 0 ” F p a q , where F : Q ě 0 Ñ D satisﬁes the ab o v e conditions. Then the functor τ ě 0 F : Q ě 0 Ñ D also satisﬁes the abov e conditions, hence τ ě 0 x – “ lim Ý Ñ a P Q ě 0 ” τ ě 0 F p a q P E . W e also ha ve π 0 x – “ lim Ý Ñ a P Q ě 0 ” π 0 F p a q , whic h gives the stated description of E ω 1 , ♡ . This prov es (ii). Finally , w e prov e (iii). Consider the comp osition Φ : C ˆ Y Ý Ñ Ind p C ω 1 ,b q Ñ Ind p D q . Clearly , Φ is strongly con tin uous and fully faithful. It suﬃces to show that the essential image of Φ | C ω 1 , ♡ coincides with E ω 1 , ♡ . T ak e some ob ject x P C ω 1 , ♡ , and let ˆ Y p x q – “ lim Ý Ñ n P N ” x n , THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 55 where x n P C ω 1 , ♡ and each map x n Ñ x n ` 1 is compact in C ♡ . Arguing as in [E24, Pro of of Proposition 1.82] we can extend the functor N Ñ C ω 1 , ♡ , n ÞÑ x n , to a functor F : Q ě 0 Ñ C ω 1 , ♡ suc h that for an y rational 0 ď a ă b the map F p a q Ñ F p b q is compact in C ♡ . Then w e hav e Φ p x q – “ lim Ý Ñ a P Q ě 0 ” F p a q , where we iden tify C ω 1 , ♡ with D ♡ . In particular, w e ha ve Φ p x q P E ω 1 , ♡ and b y (ii) an y isomorphism class in E ω 1 , ♡ can be obtained in this w a y . This pro ves (iii). □ The following is a sligh tly more sophisticated version of Eilenberg swindle. Prop osition 3.5. L et C , D , E b e as in Pr op osition 3.4. Then any lo c alizing invariant vanishes on x E ´ y and x E ` y . In p articular, we have K cont px E ´ yq “ K cont px E ` yq “ 0 . We obtain an isomorphism (3.2) K cont p C q » Ω K cont p E q . Pr o of. Assuming the v anishing of K cont px E ´ yq and K cont px E ` yq , the isomorphism (3.2) follo ws from Corollary 2.20. It suﬃces to show that we hav e the v anishings (3.3) r Id x E ´ y s “ 0 P K 0 p F un LL px E ´ y , x E ´ yqq , r Id x E ` y s “ 0 P K 0 p F un LL px E ` y , x E ` yqq . Note that x E ´ y is contained in Ind p D ´ q as a sub category of Ind p D q , and the inclusion functor is strongly contin uous. F or an ob ject x P D ´ the copro duct À n ě 0 x r 2 n s exists in D ´ . Hence, we ha ve a strongly contin uous endofunctor Φ “ Ind p à n P N Σ 2 n q : Ind p D ´ q Ñ Ind p D ´ q . W e claim that Φ preserv es the sub category x E ´ y . It suﬃces to prov e that for x P E ω 1 ě 0 w e ha v e Φ p x q P x E ´ y . By Prop osition 3.4 we ha ve x – “ lim Ý Ñ a P Q ě 0 ” F p a q , where F : Q ě 0 Ñ D ě 0 is a functor such that for any rational 0 ď a ă b and for any k P N the morphism τ r 0 ,k s F p a q Ñ τ r 0 ,k s F p b q is compact in C . Deﬁne G : Q ě 0 Ñ D by the formula G p a q “ à n P N F p a qr 2 n s . Then for a P Q ě 0 and k P N w e hav e τ r´ k,k s G p a q – à 0 ď n ď t k 2 u τ r 0 ,k ´ 2 n s p F p a qqr 2 n s . Hence, for rational 0 ď a ă b the map τ r´ k,k s G p a q Ñ τ r´ k,k s G p b q is compact. Therefore, Φ p x q – “ lim Ý Ñ a P Q ě 0 ” G p a q P x E ´ y , as required. 56 ALEXANDER I. EFIMOV Denote by Ψ : x E ´ y Ñ x E ´ y the restriction of Φ . Then Ψ is strongly con tinuous and b y construction w e hav e Ψ – Ψ r 2 s ‘ Id x E ´ y . This pro v es the ﬁrst v anishing in (3.3). The other v anishing is prov ed by a similar argument. □ W e will need to consider the natural t -structures on the iterated Calkin categories. Recall from Subsection 1.6 that for a dualizable category C we denote b y Calk ω 1 p C q the compactly generated category Ind p C ω 1 q{ ˆ Y p C q . W e will denote by Calk n ω 1 p C q the n -th iteration of this construction, n ě 1 . It is conv enien t to deal with them inductiv ely , using the following observ ation. Prop osition 3.6. L et C b e a dualizable c ate gory with an ω 1 -ac c essible t -structur e p C ď 0 , C ě 0 q , c omp atible with ﬁlter e d c olimits. Supp ose that for some n ě 0 the functor ˆ Y C r´ n s : C Ñ Ind p C ω 1 q is left t -exact. Consider the functor Φ : C » Ind ω 1 p C ω 1 q ã Ñ Ind p C ω 1 q Ñ Calk ω 1 p C q . (i) The t -structur e on Ind p C ω 1 q induc es an ω 1 -ac c essible t -structur e on Calk ω 1 p C q via the inclusion Calk ω 1 p C q » ker p colim : Ind p C ω 1 q Ñ C q Ă Ind p C ω 1 q . Mor e over, this t -structur e is c omp atible with ﬁlter e d c olimits. (ii) The functor Φ : C Ñ Calk ω 1 p C q is right t -exact, and the functor Φ r´ n ´ 1 s is left t -exact. (iii) F or D “ Calk ω 1 p C q , the functor ˆ Y D r´ n ´ 1 s : D Ñ Ind p D ω 1 q is left t -exact. Pr o of. (i) is straightforw ard since the colimit functor Ind p C ω 1 q Ñ C is t -exact. Note that the ω 1 -accessibilit y of the induced t -structure on Calk ω 1 p C q follows from the identiﬁcation of Calk ω 1 p C q ω 1 with the k ernel of the t -exact functor colim : Ind p C ω 1 q ω 1 Ñ C ω 1 . T o pro ve (ii), it suﬃces to consider the restriction of Φ to C ω 1 . Considering Calk ω 1 p C q as a full sub category of Ind p C ω 1 q , w e ha ve Φ p x qr´ n ´ 1 s – Fib er p ˆ Y C p x qr´ n s Ñ Y C p x qr´ n sq P Calk ω 1 p C q ď 0 , x P C ω 1 ď 0 , since the functor ˆ Y C r´ n s is left t -exact b y assumption. This prov es that the functor Φ r´ n ´ 1 s is left t -exact, as stated. It is also clear that Φ is righ t t -exact. W e pro ve (iii). Consider again D “ Calk ω 1 p C q as a full subcategory of Ind p C ω 1 q , and recall that for x P C ω 1 the ob ject Cone p ˆ Y C p x q Ñ Y C p x qq P D is compact. Now take an y ob ject “lim Ý Ñ i ” x i P D ď 0 , where x i P C ω 1 ď 0 and lim Ý Ñ i x i “ 0 P C . Consider the category Ind p D ω 1 q THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 57 as a full sub category of Ind p Ind p C ω 1 q ω 1 q . Then we ha ve ˆ Y D p “lim Ý Ñ i ” x i qr´ n ´ 1 s “ ˆ Y D p lim Ý Ñ i Cone p ˆ Y C p x i q Ñ Y C p x i qqqr´ n ´ 1 s – Fib er p “lim Ý Ñ i ” ˆ Y C p x i qr´ n s Ñ “lim Ý Ñ i ” Y C p x i qr´ n sq P Ind p Ind p C ω 1 q ω 1 q ď 0 , b ecause again b y assumption we ha ve ˆ Y C p x i qr´ n s P Ind p C ω 1 q ω 1 ď 0 . □ In the following pro of w e use the notation Ind ω 1 p A q for Ind p A q ω 1 , where A is a small category . W e write p Ind ω 1 q n p A q for the n -th iteration of this construction, and for con ve- nience we put p Ind ω 1 q 0 p A q “ A . Pr o of of L emma 3.3. Le t D and E b e as in Prop osition 3.4. By lo c. cit. w e can identify C with x E b y , and C r 0 ,m s with E r 0 ,m s . By Prop osition 3.5 we ha ve the follo wing isomorphisms: (3.4) φ : K cont ´ m ´ 1 p E q „ Ý Ñ K cont ´ m ´ 1 p E {x E ` yq „ Ý Ñ K cont ´ m ´ 1 px E ´ y{x E b yq , ψ : K cont ´ m ´ 1 px E ´ y{x E b yq „ Ý Ñ K cont ´ m ´ 2 px E b yq . It suﬃces to pro ve that the image of (the isomorphism) ψ ˝ φ is con tained in the image of the map K cont ´ m ´ 2 p E r 0 ,m s q Ñ K cont ´ m ´ 2 px E b yq . W e iden tify K cont ´ m ´ 1 p E q w ith K 0 p Calk m ` 1 ω 1 p E q ω q . Below for any n ě 1 w e iden tify the category Calk n ω 1 p E q ω 1 with its essen tial image in p Ind ω 1 q n p E ω 1 q , and similarly for ˇ St p E r 0 ,m s q and x E b y . More precisely , Calk n ω 1 p E q ω 1 Ă p Ind ω 1 q n p E ω 1 q is the intersection of k ernels of the functors p Ind ω 1 q k p colim q : p Ind ω 1 q n p E ω 1 q Ñ p Ind ω 1 q n ´ 1 p E ω 1 q , 0 ď k ď n ´ 1 . T ak e some class α P K cont ´ m ´ 1 p E q , and choose an ob ject x P Calk m ` 1 ω 1 p E q ω suc h that r x s “ α. Consider the natural t -structure on Calk m ` 1 ω 1 p E q ω 1 , whic h is obtained b y an iterated application of Prop osition 3.6. Then the ob ject τ ě 0 x P Calk m ` 1 ω 1 px E ´ yq ω 1 is a lift of the image of x in Calk m ` 1 ω 1 p E {x E ` yq ω » Calk m ` 1 ω 1 px E ´ y{x E b yq ω . The latter ob ject represen ts the elemen t φ p α q P K cont ´ m ´ 1 px E ´ y{x E b yq . W e can now describ e a representativ e for ψ p φ p α qq P K ´ m ´ 2 px E b yq . Namely , consider the comp osition Φ : Calk m ` 1 ω 1 p E q ω 1 Ñ p Calk m ` 1 ω 1 p E q ω 1 { Calk m ` 1 ω 1 p E q ω q Kar » Calk m ` 2 ω 1 p E q ω ã Ñ Calk m ` 2 ω 1 p E q ω 1 . Then the ob ject y “ Φ p τ ě 0 x q is con tained in the full sub category Calk m ` 2 ω 1 px E b yq ω Ă Calk m ` 2 ω 1 p E q . W e then ha ve r y s “ ψ p φ p α qq . No w, by (an iterated application of ) Proposition 3.6 the functor Φ is righ t t -exact and the functor Φ r´ m ´ 2 s is left t -exact. In particular, w e ha v e y P Calk m ` 2 ω 1 px E b yq ω 1 ě 0 . On 58 ALEXANDER I. EFIMOV the other hand, since Φ p x q “ 0 , we obtain y “ Φ p τ ě 0 x q – Φ p τ ď´ 1 x qr´ 1 s P Calk m ` 2 ω 1 px E b yq ω 1 ď m . Therefore, we hav e y P Calk m ` 2 ω 1 px E b yq ω 1 r 0 ,m s . It follo ws formally that the image of y in p Ind ω 1 q m ` 2 px E b y ω 1 q is contained in p Ind ω 1 q m ` 2 p E ω 1 r 0 ,m s q . Denote by z P p Ind ω 1 q m ` 2 p ˇ St p E r 0 ,m s q ω 1 q the image of y under the functor p Ind ω 1 q m ` 2 p E ω 1 r 0 ,m s q Ñ p Ind ω 1 q m ` 2 p ˇ St p E r 0 ,m s q ω 1 q . By construction, w e ha ve z P Calk m ` 2 ω 1 p ˇ St p E r 0 ,m s qq ω 1 . W e claim that z is compact. Equiv alently , this means that z is annihilated by the functor Ψ : Calk m ` 2 ω 1 p ˇ St p E r 0 ,m s qq ω 1 Ñ Calk m ` 3 ω 1 p ˇ St p E r 0 ,m s qq ω 1 T o see this, consider the similar functor Θ : Calk m ` 2 ω 1 px E b yq ω 1 Ñ Calk m ` 3 ω 1 px E b yq ω 1 . By Prop osition 3.6, b oth Ψ and Θ preserve bounded ob jects. W e obtain a comm utative square Calk m ` 2 ω 1 p ˇ St p E r 0 ,m s qq ω 1 ,b Calk m ` 2 ω 1 px E b yq ω 1 ,b Calk m ` 3 ω 1 p ˇ St p E r 0 ,m s qq ω 1 ,b Calk m ` 3 ω 1 px E b yq ω 1 ,b . Ψ F Θ G Note that G is t -exact and it induces an equiv alence b etw een the hearts. Indeed, the hearts of b oth the source and the target are iden tiﬁed with the full subcategory of p Ind ω 1 q m ` 3 p E ω 1 , ♡ q , whic h is the in tersection of the kernels of the functors p Ind ω 1 q k p colim q : p Ind ω 1 q m ` 3 p E ω 1 , ♡ q Ñ p Ind ω 1 q m ` 2 p E ω 1 , ♡ q , 0 ď k ď m ` 2 . It follows that G is conserv ative. Now w e ha v e F p z q – y and Θ p y q “ 0 , hence G p Ψ p z qq – Θ p F p z qq “ 0 . Therefore, Ψ p z q “ 0 , i.e. z P Calk m ` 2 ω 1 p ˇ St p E r 0 ,m s qq ω , as stated. W e conclude that the class r y s “ ψ p φ p α qq P K cont ´ m ´ 2 px E b yq is the image of r z s P K cont ´ m ´ 2 p E r 0 ,m s q . Since α was arbitrary , this pro ves the lemma. □ T o deduce Theorem 3.1 from Lemma 3.3 we need to analyze the coherently assembled exact categories C r 0 ,m s . THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 59 Prop osition 3.7. L et C b e a dualizable t -c ate gory and let m ě 1 b e an inte ger. Consider the exact c ate gory r C r 0 ,m s whose underlying c ate gory is C r 0 ,m s and the exact structur e is the fol lowing. A morphism f : x Ñ y is an inclusion in r C r 0 ,m s if and only if b oth morphisms π m ´ 1 p f q : π m ´ 1 p x q Ñ π m ´ 1 p y q and π m p f q : π m p x q Ñ π m p y q ar e monomorphisms in C ♡ . F urther, f : x Ñ y is a pr oje ction in r C r 0 ,m s if and only if b oth morphisms π 0 p f q : π 0 p x q Ñ π 0 p y q and π m p f q : π m p x q Ñ π m p y q ar e epimorphisms in C ♡ . (i) r C r 0 ,m s is a c oher ently assemble d exact c ate gory. We denote by i : r C r 0 ,m s Ñ ˇ St p r C r 0 ,m s q the universal c ontinuous exact functor. (ii) We have str ongly c ontinuous ful ly faithful exact functors j 1 : C r 0 ,m ´ 1 s Ñ r C r 0 ,m s , j 2 : C ♡ Ñ r C r 0 ,m s , given by j 1 p x q “ x, j 2 p y q “ y r m s . They induc e ful ly faithful functors on pr esentable stable envelop es, which we denote by the same symb ols. They give a semi-ortho gonal de c omp osition in Cat dual st : (3.5) ˇ St p r C r 0 ,m s q “ x j 1 p ˇ St p C r 0 ,m ´ 1 s qq , j 2 p ˇ D p C ♡ qqy . (iii) We have a short exact se quenc e in Cat dual st : 0 Ñ T m Ñ ˇ St p r C r 0 ,m s q q m Ý Ý Ñ ˇ St p C r 0 ,m s q Ñ 0 . Her e q m is induc e d by the identity functor r C r 0 ,m s Ñ C r 0 ,m s , and T m is the kernel of q m . The functor Φ m : C ♡ Ñ T m , Φ m p x q “ Cone p i p x r m ´ 1 sqr 1 s Ñ i p x r m sqq , is str ongly c ontinuous and ful ly faithful, and its essential image is the he art of a c omp actly assemble d c ontinuously b ounde d t -structur e. Mor e over, the maps Ext s C ♡ p x, y q „ Ý Ñ Ext s T m p Φ m p x q , Φ m p y qq ar e isomorphisms for s ď m ` 1 , wher e x, y P C ♡ . Pr o of. The proof of (i) is straightforw ard: it is clear that the exact structure is w ell-deﬁned, it induces an exact structure on r C ω 1 r 0 ,m s , and the functors colim : Ind p r C ω 1 r 0 ,m s q Ñ r C r 0 ,m s and ˆ Y : r C r 0 ,m s Ñ Ind p r C ω 1 r 0 ,m s q are exact. W e pro ve (ii). Both functors j 1 : C r 0 ,m ´ 1 s Ñ r C r 0 ,m s and j 2 : C ♡ Ñ r C r 0 ,m s are exact. The left adjoin t to j 1 is giv en by j L 1 p x q “ τ ď m ´ 1 x, hence it is exact. The right adjoint to j 2 is given by j R 2 p x q “ π m p x q , hence it is also e xact. W e hav e the isomorphisms j L 1 j 1 „ Ý Ñ Id C r 0 ,m ´ 1 s , Id C ♡ „ Ý Ñ j R 2 j 2 , and the v anishing j R 2 j 1 “ 0 . Moreo v er, w e hav e a functorial short exact sequence j 2 j R 2 p x q Ñ x Ñ j 1 j L 1 p x q , x P r C r 0 ,m s . 60 ALEXANDER I. EFIMOV P assing to presen table stable env elop es w e deduce the fully faithfulness of j 1 and j 2 , and obtain the semi-orthogonal decomp osition (3.5). This prov es (ii). Finally w e prov e (iii). It is clear that q m is a strongly contin uous quotien t functor. W e ﬁrst show that T m (i.e. the kernel of q m ) is generated b y the essen tial image of Φ m as a lo calizing sub category . Denote by C ω 1 , add r 0 ,m s the additiv e category C ω 1 r 0 ,m s with a split exact structure. Clearly , T m is generated by the images of ob jects of the form E p f q P St p C ω 1 , add r 0 ,m s q , where f : x Ñ y is an exact inclusion in C ω 1 r 0 ,m s , which means that π m p f q is a monomorphism in C ♡ (the notation E p f q is introduced in (1.1)). Denote by E p f q the image of E p f q in ˇ St p r C r 0 ,m s q . It suﬃces to show that w e hav e an isomorphism E p f q – Φ m p z q , where z “ k er p π m ´ 1 p f qq P C ω 1 , ♡ . First we observ e that τ ď m ´ 1 p f q is also an exact inclusion in C ω 1 r 0 ,m s , and E p f q – E p τ ď m ´ 1 p f qq . Hence, we ma y and will assume that π m p x q “ π m p y q “ 0 . Put x 1 “ Cone p z r m ´ 1 s Ñ x q , then f factors uniquely through x 1 , and the induced morphism f 1 : x 1 Ñ y is an exact inclusion in r C r 0 ,m s . Hence E p f q – E p z r m ´ 1 s Ñ 0 q – Φ m p z q , as stated. It is clear that Φ m is a strongly con tinuous exact functor (b y deﬁnition of the exact category r C r 0 ,m s ). Hence, b y Propositions 1.26 and 1.34 Φ m induces a strongly contin uous exact functor ˇ D p C ♡ q Ñ T m . By Prop osition 2.22 it remains to pro ve that the induced maps Ext s C ♡ p x, y q Ñ Ext s T m p Φ m p x q , Φ m p y qq are isomorphisms for s ď m ` 1 and x, y P C ♡ (where s can b e strictly negativ e). W e tak e such x, y and denote b y i 1 : C r 0 ,m ´ 1 s Ñ ˇ St p C r 0 ,m ´ 1 s q the universal functor. By (ii) we ha v e a cartesian square of sp ectra: (3.6) Hom T m p Φ m p x q , Φ m p y qq Hom ˇ St p C r 0 ,m ´ 1 s q p i 1 p x r m ´ 1 sq , i 1 p y r m ´ 1 sqq Hom ˇ D p C ♡ q p x, y q Hom ˇ St p r C r 0 ,m s q p i p x r m ´ 1 sqr 1 s , i p y r m sqq . W e ﬁrst deal with the relev ant homotop y groups of the bottom right spectrum. The fully faithfulness of the functor i giv es the isomorphisms π ´ s Hom ˇ St p r C r 0 ,m s q p i p x r m ´ 1 sqr 1 s , i p y r m sqq – Ext s C p x, y q for s ď 1 . Next, for 2 ď s ď m ` 1 we compute: π ´ s Hom ˇ St p r C r 0 ,m s q p i p x r m ´ 1 sqr 1 s , i p y r m sqq – π ´ s Hom ˇ St p r C r 0 ,m s q p i p x r m ´ s ` 1 sqr s ´ 1 s , i p y r m sqq – Ext 1 r C r 0 ,m s p x r m ´ s ` 1 s , y r m sq – Ext s C p x, y q . Therefore, b y Prop osition 2.18 the righ t v ertical arro w in (3.6) induces isomorphisms on π ´ s for s ď m. and a monomorphism for s “ m ` 1 . Hence, the same holds for the left v ertical arrow in (3.6). But it has a right inv erse, induced by the functor Φ m . W e conclude THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 61 that the map Ext s C ♡ p x, y q Ñ Ext s T m p Φ m p x q , Φ m p y qq is an isomorphism for s ď m ` 1 , as required. This pro ves (iii). □ Pr o of of The or em 3.1. W e start with the k ey coﬁb er sequence of sp ectra, whic h will be used rep eatedly . Let m ě 1 and consider the exact category r C r 0 ,m s and the dualizable category T m from Prop osition 3.7. By lo c. cit. we ha v e a coﬁber sequence (3.7) K cont p T m q p α,β q Ý Ý Ý Ñ K cont p C r 0 ,m ´ 1 s q ‘ K cont p C ♡ q Ñ K cont p C r 0 ,m s q . Next, we pro ve that for m ě 1 the map K cont j p C r 0 ,m ´ 1 s q Ñ K cont j p C r 0 ,m s q is an isomor- phism for eac h j P r´ m ´ 1 , ´ 2 s , and a monomorphism for j “ ´ m ´ 2 . By Proposition 3.7 the category T m from (3.7) has a compactly assem bled con tin uously b ounded t -structure, and we hav e an equiv alence C ♡ » T ♡ m . By the assertion on Ext ď m ` 1 from lo c. cit. and b y Prop osition 2.19 w e ha ve an equiv alence ˇ D p C ♡ q „ Ý Ñ ˇ St pp T m q r 0 ,l s q for 0 ď l ď m. Hence, b y Lemma 3.3 we hav e a surjection K cont j p C ♡ q ↠ K cont j p T m q for j P r´ m ´ 2 , ´ 2 s . On the other hand, the composition T m Ñ ˇ St p r C r 0 ,m s q j R 2 Ý Ñ ˇ D p C ♡ q is left in v erse to the realization functor. Therefore the map β from (3.7) induces an isomorphism K cont j p T m q „ Ý Ñ K cont j p C ♡ q for j P r´ m ´ 2 , ´ 2 s The long exact sequence of homotop y groups for (3.7) shows that the map K cont j p C r 0 ,m ´ 1 s q Ñ K cont j p C r 0 ,m s q is an isomorphism for j P r´ m ´ 1 , ´ 2 s , and a monomorphism for j “ ´ m ´ 2 , as stated. P assing to the colimit ov er m and using Prop osition 2.16 w e deduce that for m ě 0 the map K cont j p C r 0 ,m s q Ñ K cont j p C q is an isomorphism for j P r´ m ´ 2 , 2 s , and a monomorphism for j “ ´ m ´ 3 . It remains to prov e that w e also ha ve isomorphisms for j ě ´ 1 . F or this w e ma y and will assume that m “ 0 , i.e. w e will prov e that the map K cont j p C ♡ q Ñ K cont j p C q is an isomorphism for j ě ´ 1 . W e will use the already established case j “ ´ 2 as a base of induction on j. Supp ose that the statemen t is pro ven for j P r´ 2 , n s , where n P Z ě´ 2 . W e need to pro v e that the map K cont n ` 1 p C ♡ q Ñ K cont n ` 1 p C q is an isomorphism. T ak e some m ě 1 . It follo ws from the induction hypothesis that in (3.7) the map π n β : K cont n p T m q Ñ K cont n p C ♡ q is an isomorphism (it has a righ t in verse, whic h is an isomorphism). It follo ws from the long exact sequence of homotopy groups that the images of the maps K cont n ` 1 p C r 0 ,m ´ 1 s q Ñ K cont n ` 1 p C r 0 ,m s q and K cont n ` 1 p C ♡ q Ñ K cont n ` 1 p C r 0 ,m s q generate the target. But the image of the latter map is con tained in the image of the former map, hence the former map is surjective. Passing to the colimit ov er m and using Prop osition 2.16 again, we conclude that the map K cont n ` 1 p C ♡ q Ñ K cont n ` 1 p C q is surjective. Now let again m ě 1 be a p ositiv e integer. Applying the abov e surjectivit y to T m , we see that the realization functor induces a surjection K cont n ` 1 p C ♡ q ↠ K cont n ` 1 p T m q . Therefore, its left in verse π n ` 1 β is an isomorphism. Applying again the long 62 ALEXANDER I. EFIMOV exact sequence for (3.7) we deduce that the map K cont n ` 1 p C r 0 ,m ´ 1 s q Ñ K cont n ` 1 p C r 0 ,m s q is an isomorphism. P assing to the colimit o v er m we conclude that the map K cont n ` 1 p C ♡ q Ñ K cont n ` 1 p C q is an isomorphism. This prov es the induction step and the theorem. □ 4. Coconnectivity estima tes 4.1. Estimates for K -groups. Using Theorem 3.1 w e pro ve the following key result, which is a dual analogue of W aldhausen’s connectivity estimates. Theorem 4.1. L et C and D b e dualizable t -c ate gories, and let n ě 1 b e an inte ger. Supp ose that F : C Ñ D is a functor satisfying the fol lowing c onditions: ‚ F is str ongly c ontinuous, exact and t -exact; ‚ F p C q gener ates D as a lo c alizing sub c ate gory; ‚ for x, y P C ω 1 , ♡ the map Ext i C p x, y q Ñ Ext i D p F p x q , F p y qq is an isomorphism for i ď n ´ 1 , and a monomorphism for i “ n. Then the induc e d map K cont j p C q Ñ K cont j p D q is an isomorphism for j ě ´ n, and a monomorphism for j “ ´ n ´ 1 . It is con venien t to introduce the follo wing notion. Deﬁnition 4.2. L et C , D and n ě 1 b e as in The or em 4.1. We say that a functor F : C Ñ D is p´ n q -c o c onne ctive if it satisﬁes the c onditions of The or em 4.1. Remark 4.3. It is e asy to se e that for a p´ n q -c o c onne ctive functor F : C Ñ D the c ondition on maps b etwe en Ext gr oups automatic al ly holds for al l x, y P C ♡ . On the other hand, if C is c omp actly gener ate d, then it suﬃc es to che ck this c ondition for x, y P C ω , ♡ . These assertions fol low for example fr om Pr op osition 4.4 b elow. W e ﬁrst in terpret the p´ n q -co connectivit y condition via monads. Prop osition 4.4. L et C b e a dualizable t -c ate gory. L et A P Alg E 1 p F un L p C , C qq b e a c on- tinuous exact monad. Supp ose that the functor Cone p Id C Ñ A qr 1 s is left t -exact, so by Pr op osition 2.24 ther e is a unique c omp actly assemble d c ontinuously b ounde d t -structur e on Mo d A p C q such that the natur al functor C Ñ Mo d A p C q is t -exact. F or n ě 1 the fol lowing ar e e quivalent. (i) The functor C Ñ Mo d A p C q is p´ n q -c o c onne ctive. (ii) The functor Cone p Id C Ñ A qr n s : C Ñ C is left t -exact. Pr o of. The implication (ii) ù ñ (i) is immediate b y adjunction. (i) ù ñ (ii) . Again, by adjunction the functor Cone p Id C Ñ A qr n s tak es C ω 1 , ♡ to C ď 0 . Here w e of course used that C ě 0 is generated by C ω 1 , ♡ via extensions and colimits. By Prop osition 2.26 w e conclude that this functor is left t -exact. □ THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 63 Corollary 4.5. L et F : C Ñ D b e a p´ n q -c o c onne ctive functor b etwe en dualizable t - c ate gories, wher e n ě 2 . Then F induc es an e quivalenc e of exact c ate gories C r 0 ,n ´ 2 s „ Ý Ñ D r 0 ,n ´ 2 s . Pr o of. By Prop osition 2.22 (ii) the functor F induces an equiv alence C ♡ „ Ý Ñ D ♡ . By Prop osition 4.4 the functor Cone p Id C Ñ F R ˝ F q takes C ď 0 to C ď´ n , hence it takes C r 0 ,n ´ 2 s to C ď´ 2 . It follows b y adjunction that the functor C r 0 ,n ´ 2 s Ñ D r 0 ,n ´ 2 s is fully faithful, and it induces isomorphisms on Ext 1 . T o see that it is essen tially surjective, note that its essential image is closed under extensions, and it contains D ♡ r i s for 0 ď i ď n ´ 2 . Therefore, this essen tial image coincides with D r 0 ,n ´ 2 s and w e ha ve an equiv alence of categories C r 0 ,n ´ 2 s „ Ý Ñ D r 0 ,n ´ 2 s . It remains to observ e that this equiv alence automatically reﬂects exactness: in b oth categories the short exact sequences are precisely all ﬁb er-coﬁb er sequences. □ W e now recall the mac hinery of deformed tensor algebras, which will b e crucial for our argumen ts. W e refer to [E25c, Section 1.9] for this notion in an abstract setting. Here we need the follo wing special case. Let C b e a dualizable category . W e consider the functor T def p´q : F un L p C , C q { Σ C Ñ Alg E 1 p F un L p C , C qq , whic h is left adjoint to the comp osition Alg E 1 p F un L p C , C qq Ñ Alg E 0 p F un L p C , C qq » F un L p C , C q Id C { » F un L p C , C q { Σ C . W e will typically write T def p F q , assuming that F : C Ñ C is a con tinuous exact functor with a c hosen morphism F Ñ Σ C . Note that b y adjunction we hav e a natural morphism of functors Fib er p F Ñ Σ C q Ñ T def p F q . W e consider T def p F q as a contin uous exact monad on C . W e will use the natural non-negative multiplicativ e ﬁltration Fil ‚ on T def p F q as ex- plained in [E25c, Section 1.9]. Its associated graded is the (non-deformed) tensor algebra T p F q . Note that w e hav e Fil 0 T def p F q – Id C and Fil 1 T def p F q – Fib er p F Ñ Σ C q . T o deal with monads whic h are deformed tensor algebras w e will use the key short exact sequence given by the follo wing prop osition. W e recall the following notation: for an accessible exact functor F : D Ñ C betw een presen table stable categories w e denote b y C i F D the category of triples p x, y , φ q , where x P C , y P D and φ : x Ñ F p y q . By [E24, Proposition 1.2] this category is also presen table stable. Moreov er, if C and D are dualizable and F is contin uous, then the category C i F D is also dualizable b y [E24, Prop osition 1.78]. 64 ALEXANDER I. EFIMOV Prop osition 4.6. L et C b e a dualizable c ate gory, and let F P F un L p C , C q b e a c ontinuous exact functor with a morphism f : Id C Ñ F . (i) We have a short exact se quenc e in Cat dual st : 0 Ñ D Ψ Ý Ñ C i F C Φ Ý Ñ Mo d T def p Cone p f qq p C q Ñ 0 . Her e D Ă C i F C is gener ate d as a lo c alizing sub c ate gory by the image of the str ongly c ontinuous functor (4.1) Θ : C Ñ C i F C , x ÞÑ p x, x, f x q . We c onsider Θ as a (str ongly c ontinuous) functor fr om C to D . (ii) We denote by i 1 , i 2 : C Ñ C i F C the (str ongly c ontinuous) inclusion functors i 1 p x q “ p x r´ 1 s , 0 , 0 q , i 2 p x q “ p 0 , x, 0 q , and denote by i L 1 r esp. i R 2 the (str ongly c ontinuous) left adjoint to i 1 r esp. right adjoint to i 2 Then the functors Φ ˝ i 1 , Φ ˝ i 2 : C Ñ Mo d T def p Cone p f qq p C q ar e isomor- phic, and they ar e left adjoint to the for getful functor. The functors i L 1 r´ 1 s ˝ Ψ and i R 2 ˝ Ψ ar e left inverses to the functor Θ : C Ñ D . (iii) The adjunction c ounit Id C Ñ Θ R ˝ Θ is a split monomorphism, its left inverse is given by (4.2) Θ R ˝ Θ Θ R ˝ η ˝ Θ Ý Ý Ý Ý Ý Ñ Θ R ˝ p i R 2 ˝ Ψ q R ˝ i R 2 ˝ Ψ ˝ Θ – Id C . This gives a dir e ct sum de c omp osition in F un L p C , C q : Θ R ˝ Θ – Id C ‘ Fib er p f q . Pr o of. W e prov e (i) using the universal prop erty of the monad T def p Cone p f qq . First, b y [E24, Prop osition 1.78] the functors i 1 and i 2 giv e a semi-orthogonal decomposition in Cat dual st , i.e. C i F C “ x i 1 p C q , i 2 p C qy . By [E24, Proposition 1.79] the functor Θ from (4.1) is strongly con tinuous. Hence, by [E24, Corollary 1.57] the category D (as deﬁned in (i)) is dualizable and the inclusion functor Ψ : D Ñ C i F C is strongly con tinuous. W e put E “ p C i F C q{ Ψ p D q , and consider the (strongly contin uous) composition G : C i 2 Ý Ñ C i F C Ñ E . By construction, E is dualizable, the functor G is strongly con tinuous and G p C q generates E as a lo calizing sub category (since Ψ p D q and i 2 p C q generate C i F C ). Hence, p E , G q P THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 65 p Cat dual st q C { is con tained in the essential image of (2.6). F or any other pair p E 1 , G 1 q P p Cat dual st q C { using [E24, Proposition 1.79] we obtain a natural equiv alence of spaces: Map p Cat dual st q C { pp E , G q , p E 1 , G 1 qq » Map F un L p C , C q Id C { p F , G 1 R ˝ G 1 q » Map Alg E 0 p F un L p C , C qq p F , G 1 R ˝ G 1 q . Hence, the fully faithfulness of (2.6) implies an equiv alence E » Mo d T def p Cone p f qq p C q . This pro v es (i) and also the ﬁrst assertion of (ii). Namely , we see that Φ ˝ i 2 is left adjoin t to the forgetful functor, and the comp ositions Φ ˝ i 1 and Φ ˝ i 2 are isomorphic b y the construction of D Ă C i F C . The second assertion of (ii) follo ws from the deﬁnitions. It remains to prov e (iii). Since we hav e an isomorphism i R 2 ˝ Ψ ˝ Θ – Id C , the map (4.2) is indeed a left inv erse to the adjunction counit. The direct sum decomposition is obtained b y a direct computation: Θ R ˝ Θ – Fib er p Id C ‘ Id C p f ,f q Ý Ý Ý Ñ F q – Id C ‘ Fib er p f q . □ W e apply this general construction to dualizable t -categories. Prop osition 4.7. L et C , F and f : Id C Ñ F b e as in Pr op osition 4.6, and we ke ep the notation fr om ther ein. Supp ose that p C ě 0 , C ď 0 q is a c omp actly assemble d c ontinuously b ounde d t -structur e on C . Supp ose that for some n ě 1 the functor Cone p f qr n s is left t -exact. Then the c ate gories Mo d T def p Cone p f qq p C q and D have natur al c omp actly assemble d c ontinuously b ounde d t -structur es such that the functor C Ñ Mo d T def p Cone p f qq p C q is p´ n q - c o c onne ctive and the functor Θ : C Ñ D is p´ n ´ 1 q -c o c onne ctive. Pr o of. This follo ws almost directly from Prop ositions 4.4 and 4.6. Namely , w e need to see that the corresp onding monads on C satisfy the required assumptions. This is clear for Θ , since we ha ve Cone p Id C Ñ Θ R ˝ Θ qr n ` 1 s – Fib er p f qr n ` 1 s – Cone p f qr n s . Next, the endofunctor Cone p Id C Ñ T def p Cone p f qqq has an exhaustive increasing ﬁltration with sub quotien ts Cone p f q ˝ m , m ě 1 . By assumption, Cone p f q ˝ m r mn s is left t -exact, hence Cone p f q ˝ m r n s is left t -exact for m ě 1 . W e conclude that Cone p Id C Ñ T def p Cone p f qqqr n s is left t -exact. This prov es the prop osition. □ The following general construction gives a w ay to reduce certain questions ab out (con- tin uous exact) monads to the case of a deformed tensor algebra. Prop osition 4.8. L et Φ : C Ñ D b e a str ongly c ontinuous exact functor b etwe en dualizable c ate gories, and supp ose that Φ p C q gener ates D as a lo c alizing sub c ate gory. We deﬁne a 66 ALEXANDER I. EFIMOV functor N Ñ p Cat dual st q { D , n ÞÑ p C n , Φ n q , inductively as fol lows. We put p C 0 , Φ 0 q “ p C , Φ q , and for n ě 0 we deﬁne C n ` 1 “ Mo d T def p Cone p Id C n Ñ Φ R n ˝ Φ n qq p C n q . The functor Φ n ` 1 : C n ` 1 Ñ C is induc e d by the natur al map of monads on C n , namely T def p Cone p Id C n Ñ Φ R n ˝ Φ n qq Ñ Φ R n ˝ Φ n . Then we have an e quivalenc e lim Ý Ñ n C n » D . Pr o of. This is in fact the same construction as in [E25c, Pro of of Prop osition 3.15] if we sp ell it out in terms of monads on C . W e giv e the details for completeness. Denote the transition functors b y F nm : C n Ñ C m , n ď m. By construction, F 0 n p C 0 q generates C n for all n ě 0 . Next, for n ě 0 the morphism of functors Id C n Ñ F R n,n ` 1 ˝ F n,n ` 1 factors through Φ R n ˝ Φ n . Indeed, by construction the monad F R n,n ` 1 ˝ F n,n ` 1 is the deformed tensor algebra of Cone p Id C n Ñ Φ R n ˝ Φ n q . The term Fil 1 of the standard ﬁltration is iden tiﬁed with Φ R n ˝ Φ n . Comp osing with F R 0 n and precompos ing with F 0 n w e get a factorization in the category F un L p C 0 , C 0 q : F R 0 n ˝ F 0 n Ñ Φ R 0 ˝ Φ 0 Ñ F R 0 ,n ` 1 ˝ F 0 ,n ` 1 . Moreo v er, the composition Φ R 0 ˝ Φ 0 Ñ F R 0 ,n ` 1 ˝ F 0 ,n ` 1 Ñ Φ R 0 ˝ Φ 0 is homotopic to the iden tity . Hence, we get an isomorphism of monads lim Ý Ñ n F R 0 n ˝ F 0 n „ Ý Ñ Φ R 0 ˝ Φ 0 . By assumption, Φ 0 p C 0 q generates D , hence we obtain an equiv alence lim Ý Ñ n C n » D . This pro v es the proposition. □ Again, we apply this abstract result to dualizable t -categories. Prop osition 4.9. We ke ep the notation and assumption fr om Pr op osition 4.8. Supp ose that C and D ar e mor e over dualizable t -c ate gories and the functor Φ : C Ñ D is p´ n q - c o c onne ctive for some n ě 1 . Ther e ar e c omp actly assemble d c ontinuously b ounde d t - structur es on the c ate gories C k , k ě 1 , which ar e uniquely determine d by the r e quir e- ment that al l the tr ansition functors ar e t -exact. Mor e over, for e ach k ě 0 the functor Cone p Id C k Ñ Φ R k ˝ Φ k qr 2 k p n ´ 1 q ` 1 s : C k Ñ C k is left t -exact. THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 67 Pr o of. W e construct the t -structures inductiv ely , sim ultaneously proving the left t - exactness assertion. W e already ha ve the t -structure on C 0 “ C . By assumption on Φ 0 “ Φ and by Prop osition 4.4, the functor Cone p Id C 0 Ñ Φ R 0 ˝ Φ 0 qr n s is left t -exact. Supp ose that for some l ě 0 we hav e the claimed t -structures on C k for 0 ď k ď l, suc h that for these k the functor Cone p Id C k Ñ Φ R k ˝ Φ k qr 2 k p n ´ 1 q ` 1 s is left t -exact. Applying Prop osition 4.7, w e obtain a unique compactly assem bled con tin uously b ounded t -structure on C l ` 1 suc h that the transition functor F l,l ` 1 : C l Ñ C l ` 1 is t -exact. W e need to prov e that the functor Cone p Id C l ` 1 Ñ Φ R l ` 1 ˝ Φ l ` 1 qr 2 l ` 1 p n ´ 1 q ` 1 s is left t -exact. By Prop osition 2.27 this is equiv alent to the left t -exactness of the functor Cone p F R l,l ` 1 ˝ F l,l ` 1 Ñ Φ R l ˝ Φ l qr 2 l ` 1 p n ´ 1 q ` 1 s . By construction, the monad F R l,l ` 1 ˝ F l,l ` 1 has a non-negativ e increasing ﬁltration Fil ‚ , such that Fil 1 – Φ R l ˝ Φ l . The inclusion of Fil 1 is a right inv erse to the morphism F R l,l ` 1 ˝ F l,l ` 1 Ñ Φ R l ˝ Φ l . Hence, it suﬃces to show that for m ě 2 the functor gr m Fil ‚ r 2 l ` 1 p n ´ 1 q ` 2 s is left t -exact. This follo ws from the induction hypothesis: w e hav e gr m Fil ‚ r 2 l ` 1 p n ´ 1 q ` 2 s – Cone p Id C l Ñ Φ R l ˝ Φ l q ˝ m r 2 l ` 1 p n ´ 1 q ` 2 s – p Cone p Id C l Ñ Φ R l ˝ Φ l qr 2 l p n ´ 1 q ` 1 sq ˝ m rp 2 ´ m qp 2 l p n ´ 1 q ` 1 qs , and the latter functor is left t -exact since Cone p Id C l Ñ Φ R l ˝ Φ l qr 2 l p n ´ 1 q ` 1 s is left t -exact and m ě 2 . This prov es the induction step and the proposition. □ Pr o of of The or em 4.1. W e ﬁrst consider the follo wing special case. Let F : C Ñ D be as in the theorem, and supp ose that moreo v er the monad F R ˝ F is isomorphic to T def p G q , where G P F un L p C , C q is equipp ed with a morphism G Ñ Σ C and the functor G r n s is left t -exact. Put H “ Fiber p G Ñ Σ C q . Applying the construction from Prop osition 4.6 (and c hanging the notation for the categories), w e obtain a short exact sequence in Cat dual st (4.3) 0 Ñ E Ψ Ý Ñ C i H C Φ Ý Ñ D Ñ 0 . As ab o v e, w e denote b y i 1 , i 2 : C Ñ C i H C . the natural inclusions, and consider the adjoin ts i L 1 and i R 2 . Applying K cont p´q to (4.3), w e obtain the coﬁb er sequence K cont p E q Ñ K cont p C q ‘ K cont p C q Ñ K cont p D q . This gives an equiv alence of sp ectra (4.4) Cone p K cont p C q K cont p F q Ý Ý Ý Ý Ý Ý Ñ K cont p D qq – Cone p K cont p E q K cont p i R 2 ˝ Ψ q Ý Ý Ý Ý Ý Ý Ý Ý Ñ K cont p C qq . No w, recall that by Prop osition 4.6 w e ha ve a functor Θ : C Ñ E , righ t in verse to i R 2 ˝ Ψ . Moreo v er, by Proposition 4.7 E is naturally a t -category and the functor Θ is p´ n ´ 68 ALEXANDER I. EFIMOV 1 q -co connective. By Corollary 4.5 Θ induces an equiv alence C r 0 ,n ´ 1 s „ Ý Ñ E r 0 ,n ´ 1 s . By Theorem 3.1 the latter equiv alence of exact categories implies that Θ induces an equiv alence K cont ě´ n ´ 1 p C q „ Ý Ñ K cont ě´ n ´ 1 p E q . Hence the left in verse functor i R 2 ˝ Ψ also induces an equiv alence on K cont ě´ n ´ 1 . W e conclude from (4.4) that Cone p K cont p C q Ñ K cont p D qq P Sp ď´ n ´ 1 . No w consider the general situation of the theorem. By Prop osition 4.9 w e ha ve a sequence C “ C 0 Ñ C 1 Ñ . . . of dualizable t -categories suc h that D » lim Ý Ñ k C k , eac h transition functor F k : C k Ñ C k ` 1 is t -exact, C k ` 1 is generated b y F k p C k q and the monad F R k ˝ F k is the deformed tensor algebra of a functor G k : C k Ñ C k suc h that G k r n s is left t -exact. By the ab ov e sp ecial case, we kno w that the sp ectrum Cone p K cont p F k qq is p´ n ´ 1 q - co connectiv e for k ě 0 . Hence, also the sp ectra Cone p K cont p C 0 q Ñ K cont p C k qq are p´ n ´ 1 q -co connective. T aking the colimit ov er k , w e conclude that Cone p K cont p C q Ñ K cont p D qq is p´ n ´ 1 q -co connective. This prov es the theorem. □ 4.2. Estimates for higher nil groups. F or the proof of the theorem of the heart for K H it will b e important to hav e a v ersion of Theorem 4.1 for higher nil groups N s K cont j . W e will need the follo wing immediate application of Prop osition 2.25. Prop osition 4.10. L et C and D b e a dualizable t -c ate gories, and let F : C Ñ D b e a p´ n q -c o c onne ctive functor for some n ě 1 . L et y b e a formal variable of de gr e e 0 . Then the functor (4.5) Id b F : p Mo d y - tors - S r y sq b C Ñ p Mo d y - tors - S r y sq b D is also p´ n q -c o c onne ctive, wher e the sour c e and the tar get ar e e quipp e d with t -structur es fr om Pr op osition 2.25. Pr o of. By Prop osition 4.4, the p´ n q -co connectivit y of F means that the functor Cone p Id C Ñ F R ˝ F qr n s is left t -exact, and we need to pro ve that the same holds for the adjunction counit of the functor (4.5). As in Prop osition 2.25 w e consider the functor i : Sp Ñ Mo d y - tors S r y s (restriction of scalars for y ÞÑ 0 ). Consider the comp osition Φ : C i b C Ý Ý Ñ p Mo d y - tors S r y sq b C Id b F Ý Ý Ý Ñ p Mo d y - tors S r y sq b D . By Prop osition 2.27 it suﬃces to sho w that the functor Cone pp i b C q R ˝ p i b C q Ñ Φ R ˝ Φ qr n s is left t -exact. This follows from the isomorphism Cone pp i b C q R ˝ p i b C q Ñ Φ R ˝ Φ qr n s – Cone p Id C Ñ F R ˝ F qr n s ‘ Cone p Id C Ñ F R ˝ F qr n ´ 1 s . □ W e recall that for C P Cat perf and for s ě 1 the higher nil sp ectrum N s K p C q is deﬁned as K p C b Perf p S r x 1 , . . . , x s sqq red , where w e tak e the reduced part in the simplicial sense THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 69 and we consider the assignment r s s ÞÑ S r x 1 , . . . , x s s as a simplicial E 8 -ring as explained in [E25c, Section 8]. On the level of homotop y groups w e ha ve N s K j p C q – coker p à 1 ď i ď s K j p C b P erf p S r x 1 , . . . , x i ´ 1 , x i ` 1 , . . . , x s sqq Ñ K j p C b P erf p S r x 1 , . . . , x s sqqq , see [W ei89] for the Z -linear v ersion. As usual, we denote b y N s K cont the asso ciated lo calizing inv ariant of dualizable cate- gories. Corollary 4.11. L et F : C Ñ D b e a p´ n q -c o c onne ctive functor b etwe en dualizable t - c ate gories, wher e n ě 1 . L et s ě 1 b e an inte ger. Then the induc e d map N s K cont j p C q Ñ N s K cont j p D q is an isomorphism for j ě ´ n ` s, and a monomorphism for j “ ´ n ` s ´ 1 . Pr o of. Recall that w e hav e the identiﬁcation of reduced lo calizing motiv es r U loc p S r x sq – Σ r U loc p P erf 8 p P 1 , 5 S qq whic h foll ows from the Beilinson’s full exceptional collection x O p´ 1 q , O y on P 1 , 5 S [Bei78], see for example [E25c, Pro of of Theorem 9.1]. This isomorphism implies that N s K cont p C q is naturally a direct summand of K cont pp Mo d x ´ 1 - tors - S r x ´ 1 sq b s b C qr s s , and similarly for D . Hence, the sp ectrum Cone p N s K cont p F qq is a direct summand of the sp ectrum (4.6) Cone p K cont pp Mo d x ´ 1 - tors - S r x ´ 1 sq b s b C q Ñ K cont pp Mo d x ´ 1 - tors - S r x ´ 1 sq b s b D qqr s s . By the inductive application of Prop ositions 2.25 and 4.10, we see that the tensor products p Mo d x ´ 1 - tors - S r x ´ 1 sq b s b C and p Mo d x ´ 1 - tors - S r x ´ 1 sq b s b D are naturally t -categories, and the functor p Mo d x ´ 1 - tors - S r x ´ 1 sq b s b C Ñ p Mo d x ´ 1 - tors - S r x ´ 1 sq b s b D is p´ n q -co connective. Applying Theorem 4.1 again, w e see that the sp ectrum (4.6) is p´ n ` s ´ 1 q -co connective. □ W e record another immediate application of the ab o ve results. Corollary 4.12. L et C b e a dualizable t -c ate gory. Then we have N s K cont j p C q “ 0 for s ě 1 , j ě s ´ 1 . Pr o of. Put D “ p Mo d x ´ 1 - tors - S r x ´ 1 sq b C . As in Prop osition 2.25 consider the strongly con tin uous functor i : Sp Ñ Mo d x ´ 1 - tors - S r x ´ 1 s and put F “ i b C : C Ñ D . By lo c. cit. D is naturally a dualizable t -category and the functor F is p´ 1 q -co connectiv e. Applying 70 ALEXANDER I. EFIMOV Corollary 4.11 and using again the isomorphism r U loc p S r x sq – Σ r U loc p P erf 8 p P 1 , 5 S qq , we obtain that for s ě 1 the sp ectrum N s K cont p C q – Σ Cone p N s ´ 1 K cont p F qq is p s ´ 2 q -co connective. This prov es the corollary . □ 5. Theorem of the hear t f or K H W e prov e the following result. Theorem 5.1. L et C b e a dualizable t -c ate gory. Then the r e alization functor induc es an e quivalenc e of sp e ctr a K H cont p C ♡ q „ Ý Ñ K H cont p C q . T o pro v e Theorem 5.1 we will deal with the natural ﬁltration on K H . F or k ě 0 we consider the localizing in v ariant U k “ Fil k K H : Cat perf Ñ Sp , i.e. (5.1) U k p C q “ lim Ý Ñ r n sP ∆ op ď k K p C b Perf p S r x 1 , . . . , x n sqq . These functors form a direct sequence U 0 Ñ U 1 Ñ . . . . W e ha ve U 0 p C q – K p C q , lim Ý Ñ k U k p C q – K H p C q , and Cone p U k p C q Ñ U k ` 1 p C qq – N k ` 1 K p C qr k ` 1 s , k ě 0 , C P Cat perf . F or conv enience w e put N 0 K “ K . Throughout this section we will systematically use the notion of a p´ n q -co connective functor from Deﬁnition 4.2. Our ﬁrst step is to deduce the coconnectivity estimates for Cone p U cont k p F qq , where F is p´ n q -co connectiv e for some n ě 1 . Corollary 5.2. L et F : C Ñ D b e a p´ n q -c o c onne ctive functor b etwe en dualizable t - c ate gories, wher e n ě 1 . Then for k ě 0 the sp e ctrum Cone p U cont k p F qq is p´ n ` 2 k ´ 1 q - c o c onne ctive. Pr o of. This follows from Theorem 4.1 and Corollary 4.11 since the sp ectrum Cone p U cont k p F qq has a ﬁnite ﬁltration with sub quotien ts Cone p N s K cont p F qqr s s , 0 ď s ď k . □ T o pro v e Theorem 5.1 we will consider certain statements dep ending on four integer parameters (w e do not claim that they hold for all v alues of the parameters, this is certainly not the case). Namely , for n ě 1 , c P Z and 0 ď k ď l consider the statements THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 71 ‚ A p n, c, k , l q . If F : C Ñ D is a p´ n q -connective functor b etw een dualizable t - categories, then the map of sp ectra τ ě c Cone p U cont k p F qq Ñ τ ě c Cone p U cont l p F qq is null-homotopic. ‚ B p n, c, k , l q . Let C b e a dualizable t -category , and supp ose that G : C Ñ C is a con tin uous exact endofunctor with a map G Ñ Σ C , such that G r n s is left t -exact. Consider the functor F : C Ñ Mo d T def p G q p C q . Then the map of sp ectra τ ě c Cone p U cont k p F qq Ñ τ ě c Cone p U cont l p F qq is null-homotopic. Note that B p n, c, k , l q is a sp ecial case of A p n, c, k , l q by Proposition 4.7. The follo wing lemma is crucial for the pro of of Theorem 5.1. Lemma 5.3. L et n ě 1 , c P Z and 0 ď k ă l. Then the statement A p 2 n ` 1 , c ´ 1 , k , l ´ 1 q implies the statement B p n, c, k , l q . T o pro ve this implication we use a general observ ation on monads which are (non- deformed) tensor algebras. Prop osition 5.4. L et C b e a dualizable c ate gory and let G : C Ñ C b e a c ontinuous exact endofunctor. Consider the monad T p G q , put D “ Mo d T p G q p C q , and denote by F : C Ñ D the left adjoint to the for getful functor. Then for any k ě 0 the map Cone p U cont k p F qq Ñ Cone p U cont k ` 1 p F qq is nul l-homotopic. Pr o of. It is conv enien t to prov e a stronger statement on lo calizing motiv es. Namely , for k ě 0 we put M k “ lim Ý Ñ r n sP ∆ op ď k U loc p ∆ n q P Mot loc . It suﬃces to pro v e that the map Cone p U cont loc p F qq b M k Ñ Cone p U cont loc p F qq b M k ` 1 is n ull-homotopic in Mot loc . T o see this, we ﬁrst recall that by [E25c, Proofs of Theorem 8.1 and Proposition 8.2] the map r U loc p S r x sq b M k Ñ r U loc p S r x sq b M k ` 1 is n ull-homotopic. Thus, it suﬃces to sho w that the ob ject Cone p U cont loc p F qq is a retract of the ob ject r U loc p S r x sq b Cone p U cont loc p F qq . This follo ws from the standard argument: infor- mally sp eaking, the non-negative grading on the tensor algebra T p G q corresp onds to the m ultiplicativ e A 1 , 5 S -action. More precisely , consider the map of monads on C , (5.2) T p G q Ñ S r x s b T p G q in Alg E 1 p F un L p C , C qq , 72 ALEXANDER I. EFIMOV whic h by adjunction corresponds to the composition G x b G Ý Ý Ý Ñ S r x s b G Ñ S r x s b T p G q . The map (5.2) has a left in verse induced b y the map of E 1 -rings S r x s Ñ S , x ÞÑ 1 . P assing to the categories of modules o ver the monads, w e get the retraction Mo d T p G q p C q Ñ p Mo d - S r x sq b Mo d T p G q p C q Ñ Mo d T p G q p C q . P assing to motiv es, w e get a retraction U cont loc p Mo d T p G q p C qq Ñ U loc p S r x sq b U cont loc p Mo d T p G q p C qq Ñ U cont loc p Mo d T p G q p C qq . No w, the ob ject Cone p U cont loc p F qq is naturally a direct summand of U cont loc p Mo d T p G q p C qq . P assing to reduced motives, w e obtain the desired retraction Cone p U cont loc p F qq Ñ r U loc p S r x sq b Cone p U cont loc p F qq Ñ Cone p U cont loc p F qq . This prov es the prop osition. □ Pr o of of L emma 5.3. Supp ose that A p 2 n ` 1 , c ´ 1 , k , l ´ 1 q holds. Let G : C Ñ C and F : C Ñ D b e as in B p n, c, k , l q . Put H “ Fib er p G Ñ Σ C q . Consider the short exact sequence from Proposition 4.6, with modiﬁed notation: 0 Ñ E Ψ Ý Ñ C i H C Ñ D Ñ 0 . W e denote b y i 1 .i 2 : C Ñ C i H C the standard inclusions as in lo c. cit., and denote b y i R 2 the righ t adjoint to i 2 . Arguing as in the pro of of Theorem 4.1, w e obtain the isomorphisms Cone p U cont j p F qq – Cone p U cont j p i R 2 ˝ Ψ qq , j ě 0 , compatible with the transition maps. As in Prop osition 4.6, we consider the functor Θ : C Ñ E , righ t inv erse to i R 2 ˝ Ψ . Recall that Θ p C q generates E by construction, and b y part (iii) of lo c.cit., the adjunction counit Id C Ñ Θ R ˝ Θ is a split monomorphism, giving a direct sum decomp osition Θ R ˝ Θ – Id C ‘ G r´ 1 s . Consider the morphism of monads T p G r´ 1 sq Ñ Θ R ˝ Θ , corresp onding b y adjunction to the inclusion of G r´ 1 s as a direct summand. Put B “ Mo d T p G r´ 1 sq p C q . Then the maps of monads Id C Ñ T p G r´ 1 sq Ñ Θ R ˝ Θ Ñ Id C corresp ond to the chain of (strongly con tinuous exact) functors C Λ Ý Ñ B Φ Ý Ñ E i R 2 ˝ Ψ Ý Ý Ý Ñ C . By Prop osition 4.7 and Prop osition 2.22 (iii), B and E are naturally dualizable t -categories and the functors Λ , Φ and i R 2 ˝ Ψ are t -exact. W e claim that the functor Φ : B Ñ E is p´ 2 n ´ 1 q -co connective. By Prop osition 4.4 w e need to c heck that the functor Cone p Id B Ñ Φ R ˝ Φ qr 2 n ` 1 s is left t -exact. By Proposition THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 73 2.27 it suﬃces to show that the functor Cone p Λ R ˝ Λ Ñ Λ R ˝ Φ R ˝ Φ ˝ Λ qr 2 n ` 1 s is left t -exact. This is clear b y construction: the latter functor is isomorphic to p à m ě 2 G r´ 1 s ˝ m qr 2 n ` 2 s – à m ě 2 G r n s ˝ m rp 2 ´ m qp n ` 1 qs , and G r n s is left t -exact b y assumption. F or j ě 0 consider the coﬁber sequence Cone p U cont j p i R 2 ˝ Ψ ˝ Φ qq Ñ Cone p U cont j p i R 2 ˝ Ψ qq Ñ Cone p U cont j p Φ qqr 1 s . Applying A p 2 n ` 1 , c ´ 1 , k , l ´ 1 q to Φ , w e see that the map τ ě c p Cone p U cont k p Φ qqr 1 sq Ñ τ ě c p Cone p U cont l ´ 1 p Φ qqr 1 sq is null-homotopic. Hence, the map τ ě c Cone p U cont k p i R 2 ˝ Ψ qq Ñ τ ě c Cone p U cont l ´ 1 p i R 2 ˝ Ψ qq factors through τ ě c Cone p U cont l ´ 1 p i R 2 ˝ Ψ ˝ Φ qq . No w the functor i R 2 ˝ Ψ ˝ Φ is left inv erse to Λ . Applying Prop osition 5.4 w e see that the map Cone p U cont l ´ 1 p i R 2 ˝ Ψ ˝ Φ qq Ñ Cone p U cont l p i R 2 ˝ Ψ ˝ Φ qq is null-homotopic. W e conclude that the map τ ě c Cone p U cont k p i R 2 ˝ Ψ qq Ñ τ ě c Cone p U cont l p i R 2 ˝ Ψ qq is null-homotopic. This pro ves the implication A p 2 n ` 1 , c ´ 1 , k , l ´ 1 q ñ B p n, c, k , l q . □ Before pro ving Theorem 5.1 we spell out a trivial suﬃcient condition for v anishing of a comp osition of maps betw een sp ectra equipp ed with ﬁnite ﬁltrations. As usual, w e denote b y r m s the totally ordered set t 0 , 1 , . . . , m u for m ě 0 . Prop osition 5.5. L et m ě 1 and c b e inte gers, and let Φ : r m s ˆ r m s Ñ Sp b e a functor. Supp ose that the fol lowing c onditions hold. (i) Φ p 0 , i q “ 0 for 0 ď i ď m. (ii) The map τ ě c Cone p Φ p m ´ i ´ 1 , i q Ñ Φ p m ´ i, i qq Ñ τ ě c Cone p Φ p m ´ i ´ 1 , i ` 1 q Ñ Φ p m ´ i, i ` 1 qq is nul l-homotopic for 0 ď i ď m ´ 1 . Then the map τ ě c Φ p m, 0 q Ñ τ ě c Φ p m, m q is nul l-homotopic. 74 ALEXANDER I. EFIMOV Pr o of. W e apply induction on m. The base m “ 1 is tautological. Let m ě 2 and supp ose that the statemen t holds for m ´ 1 . Let Φ : r m s ˆ r m s Ñ Sp b e a functor satisfying the conditions (i) and (ii). Applying (ii) to i “ 0 we see that the map τ ě c Φ p m, 0 q Ñ τ ě c Φ p m, 1 q factors through τ ě c Φ p m ´ 1 , 1 q . Hence, it suﬃces to sho w that the map τ ě c Φ p m ´ 1 , 1 q Ñ τ ě c Φ p m ´ 1 , m q is n ull-homotopic. This follo ws from the induction hypothesis applied to the functor Ψ : r m ´ 1 s ˆ r m ´ 1 s Ñ Sp , Ψ p i, j q “ Φ p i, j ` 1 q . □ W e now prov e a v anishing result whic h is in fact stronger than Theorem 5.1. Prop osition 5.6. F or m ě 0 the fol lowing statements hold. ‚ p S m q : the statement A p n, c, k , l q holds if n ě 1 , k ě 0 , l ´ k ě 2 m ´ 1 , c ě ´ 2 m p n ´ 1 q ` 2 k ´ 1 . ‚ p T m q : the statement B p n, c, k , l q holds if n ě 1 , k ě 0 , l ´ k ě 2 m , c ě ´ 2 m ` 1 n ` 2 k . Pr o of. W e apply induction on m. W e already know that p S 0 q holds: this is exactly the assertion of Corollary 5.2. Next, for m ě 0 we hav e the implication p S m q ñ p T m q . Indeed, supp ose that p S m q holds and let p n, c, k , l q b e as in p T m q . Then the quadruple p 2 n ` 1 , c ´ 1 , k , l ´ 1 q satisﬁes the assumptions of p S m q , so A p 2 n ` 1 , c ´ 1 , k , l ´ 1 q holds. The latter statement implies B p n, c, k , l q b y Lemma 5.3. This pro v es that p S m q implies p T m q . It remains to prov e p S m ` 1 q assuming p S 0 q and p T i q for 0 ď i ď m. Let p n, c, k , l q b e as in p S m ` 1 q . W e need to prov e that A p n, c, k , l q holds. W e ma y and will assume that l “ k ` 2 m ` 1 ´ 1 and c “ ´ 2 m ` 1 p n ´ 1 q ` 2 k ´ 1 . Let F : C Ñ D be a p´ n q -co connective functor b etw een dualizable t -categories. W e apply the construction from Prop osition 4.8. By Propositions 4.9 and 4.7, it giv es a direct sequence of dualizable t -categories C “ C 0 Ñ C 1 Ñ . . . , such that lim Ý Ñ i C i » D and each functor F i : C i Ñ D is p´ 2 i p n ´ 1 q ´ 1 q -co connectiv e. Moreov er, if F ij : C i Ñ C j are the transition functors for 0 ď i ď j, then F 0 ,i p C 0 q generates C i , and eac h monad F R i,i ` 1 ˝ F i,i ` 1 is isomorphic to a deformed tensor algebra of some G i : C i Ñ C i suc h that G i r 2 i p n ´ 1 q ` 1 s is left t -exact. Applying p S 0 q to the functor F m ` 1 : C m ` 1 Ñ D , we see that τ ě c Cone p U cont k p F m ` 1 qq “ 0 . Hence, we ha ve an equiv alence τ ě c Cone p U cont k p F 0 ,m ` 1 qq „ Ý Ñ τ ě c Cone p U cont k p F qq . THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 75 Th us, it suﬃces to pro ve that the map (5.3) τ ě c Cone p U cont k p F 0 ,m ` 1 qq Ñ τ ě c Cone p U cont l p F 0 ,m ` 1 qq is null-homotopic. W e will apply Prop osition 5.5 to the follo wing functor: Φ : r m ` 1 s ˆ r m ` 1 s Ñ Sp , Φ p i, j q “ Cone p U cont k ` 2 j ´ 1 p F 0 ,i qq . W e need to c heck that the assumptions of lo c. cit. hold. Clearly , w e hav e Φ p 0 , i q “ 0 for 0 ď i ď m ` 1 . It remains to see that for 0 ď i ď m the map τ ě c Cone p U cont k ` 2 i ´ 1 p F m ´ i,m ´ i ` 1 qq Ñ τ ě c Cone p U cont k ` 2 i ` 1 ´ 1 p F m ´ i,m ´ i ` 1 qq is null-homotopic. This follo ws from p T i q . Indeed, the functor G m ´ i r 2 m ´ i p n ´ 1 q ` 1 s : C m ´ i Ñ C m ´ i is left t -exact, the monad F R m ´ i,m ´ i ` 1 ˝ F m ´ i,m ´ i ` 1 is the deformed tensor algebra of G m ´ i and we hav e ´ 2 i ` 1 p 2 m ´ i p n ´ 1 q ` 1 q ` 2 p k ` 2 i ´ 1 q “ ´ 2 m ` 1 p n ´ 1 q ` 2 k ´ 2 ă c. Applying Proposition 5.5, we obtain that the map (5.3) is n ull-homotopic, as required. This pro v es the proposition. □ Pr o of of The or em 5.1. The realization functor F : ˇ D p C ♡ q Ñ C is p´ 2 q -co connectiv e. It follo ws from Prop osition 5.6 that for any k ě 0 and for an y c P Z there exists l ě k suc h that A p 2 , c, k , l q holds. Namely , it suﬃces to take the smallest m ě 0 such that c ě ´ 2 m ` 2 k ´ 1 , and put l “ k ` 2 m ´ 1 . Therefore, for an y c P Z we ha ve τ ě c Cone p K H cont p F qq – lim Ý Ñ k τ ě c Cone p U cont k p F qq “ 0 . In other w ords, the map K H cont p F q is an equiv alence of sp ectra. This prov es the theorem. □ 6. D ´ evissage theorems for K -theor y and homotopy K -theor y In this section w e apply the results from Sections 4 and 5 to deduce d´ evissage theorems for coheren tly assembled abelian categories. One wa y to approac h this is b y generalizing the construction from [E25b, Theorem 0.3]. Ho wev er, we choose a slightly diﬀerent but similar argument. Theorem 6.1. L et A b e a c oher ently assemble d ab elian c ate gory. L et B Ă A b e a c oher ently assemble d ab elian ful l sub c ate gory, such that the inclusion functor is str ongly c ontinuous, and B gener ates A via extensions and ﬁlter e d c olimits. (i) The map K cont j p B q Ñ K cont j p A q is an isomorphism for j ě ´ 1 , and a monomor- phism for j “ ´ 2 . (ii) The map K H cont p A q Ñ K H cont p B q is an e quivalenc e of sp e ctr a. 76 ALEXANDER I. EFIMOV Pr o of. (i) The assumption implies that the induced functor ˇ D p B q Ñ ˇ D p A q is p´ 1 q - co connectiv e. Hence, the statemen t of (i) is a special case of Theorem 4.1. (ii) Consider more generally a p´ 1 q -co connectiv e functor F : C Ñ D b etw een dualizable t -categories. W e need to prov e that the map K H cont p F q is an equiv alence. Arguing as in the pro of of Theorem 4.1 we reduce to the sp ecial case when the monad F R ˝ F is isomorphic to a deformed tensor algebra of some G : C Ñ C such that G r 1 s is left t -exact. Arguing as in loc. cit. we ﬁnd a dualizable t -category E and a p´ 2 q -co connective functor Θ : C Ñ E such that Cone p K H cont p F qq – Cone p K H cont p Θ qqr 1 s . The latter sp ectrum is zero by Theorem 5.1 since we hav e C ♡ » E ♡ b y Prop osition 2.22. □ W e will see in the next section that in the situation of Theorem 6.1 the map K cont ´ 2 p B q Ñ K cont ´ 2 p A q do es not hav e to b e surjective, ev en when A and B are locally coheren t and linear ov er a ﬁeld. Remark 6.2. One c an obtain a mor e pr e cise version of p art (ii) of The or em 6 .1 when every obje ct of A is an extension of two obje cts of B . Under this assumption a gener alization of [E25b, Theorem 0.3] shows that the functor F : ˇ D p B q Ñ ˇ D p A q satisﬁes the fol lowing pr op erty: the monad F R ˝ F is isomorphic to a deforme d tensor algebr a of some c ontinuous exact functor G : ˇ D p B q Ñ ˇ D p B q such that G r 1 s is left t -exact. By Pr op osition 5.6 for any m ě 0 , k ě 0 and c “ ´ 2 m ` 1 ` 2 k the map τ ě c Cone p U cont k p F qq Ñ τ ě c Cone p U cont k ` 2 m p F qq is nul l-homotopic. Her e U k “ Fil k K H ( k ě 0 ) is the lo c alizing invariant deﬁne d in (5.1) . 7. Sharpness of the estima tes 7.1. Estimates for theorems of the heart and d´ evissage. In this section w e pro ve the sharp- ness of our estimates from Corollary 3.2 and Theorem 6.1 (i), ev en when w e are dealing with small dg resp. ab elian categories ov er a ﬁeld. Theorem 7.1. L et k b e a ﬁeld. (i) Ther e exist smal l k -line ar ab elian c ate gories B Ă A such that the (ful ly faithful) inclusion functor B Ñ A is exact, e ach obje ct of A is an extension of two obje cts of B , and the map K ´ 2 p B q Ñ K ´ 2 p A q is not surje ctive. (ii) F or any n ě 1 ther e exists a k -line ar smal l t -c ate gory C n such that the maps Ext i C ♡ n p x, y q Ñ Ext i C n p x, y q ar e isomorphisms for i ď n and x, y P C ♡ n , and the map K ´ n ´ 2 p C ♡ n q Ñ K ´ n ´ 2 p C n q is not surje ctive. W e will need the follo wing lemma, whic h is certainly known to experts, but we do not kno w a reference. THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 77 Lemma 7.2. L et E b e an additive 8 -c ate gory, such that for any obje ct x P E the ring π 0 End E p x q is left or right artinian. Then we have N s K j p E q “ 0 for s ě 1 , j ď 0 . Pr o of. Since the functors N s K j comm ute with ﬁltered colimits, it suﬃces to consider the case when E is equiv alent to Pro j f .g . - A, the category of ﬁnite pro jective righ t modules o v er a connectiv e E 1 -ring A, suc h that π 0 p A q is left or right artinian. Let s ě 1 and j ď 0 . Then by [BGT13, Theorem 9.53] and by the nilinv ariance of K ď 0 w e hav e N s K j p A q – N s K j p π 0 p A q{ J q where J Ă π 0 p A q is the Jacobson radical. The ring π 0 p A q{ J is a ﬁnite pro duct of matrix algebras o v er sk ew-ﬁelds, hence w e ma y assume that A “ D is a sk ew- ﬁeld. In this case all the sp ectra N s K p D q v anish for s ě 1 . □ In the follo wing pro of w e minimize the computations, focusing only on pro ving the sharp- ness results. Pr o of of The or em 7.1. (i) Le t X “ t x 3 1 “ x 0 x 2 2 u Ă P 2 k b e the cuspidal cubic curve ov er k . Denote b y E “ V ect X the category of v ector bundles of ﬁnite rank on X . Denote b y Ac b p E q the dg category of acyclic b ounded complexes of ob jects of E . By [Nee21, Lemma 1.2] this category has a b ounded t -structure whose heart B 0 “ Ac b p E q ♡ is equiv alent to the category eﬀ p E q of eﬀaceable functors E op Ñ Ab (this is a sp ecial case of Prop osition 1.3). Denote by Nil p B 0 q the abelian category of pairs p x, f q , where x P B 0 and f : x Ñ x is a nilp otent endomorphism. W e ha ve a fully faithful inclusion i : B 0 Ñ Nil p B 0 q , i p x q “ p x, 0 q , and w e iden tify B 0 with its essen tial image. More generally , for k ě 0 denote b y B k Ă Nil p B 0 q the (ab elian) full sub category of pairs p x, f q such that f k “ 0 . Then the inclusion functors are exact and each ob ject of B k ` 1 is an extension of t w o ob jects of B k . W e claim that for some k ě 0 the map K ´ 2 p B k q Ñ K ´ 2 p B k ` 1 q is not surjectiv e, whic h w ould prov e (i). It suﬃces to pro ve that the map K ´ 2 p B 0 q Ñ K ´ 2 p Nil p B 0 qq is not surjectiv e. By Prop osition 2.25 (ii) we ha ve coker p K ´ 2 p B 0 q Ñ K ´ 2 p Nil p B 0 qqq – N K ´ 1 p B 0 q , so w e need to show that this nil group is non-zero. By [Nee21, Prop osition 2.4] the realization functor D b p B 0 q Ñ Ac b p E q is an equiv alence. Denoting b y E add the category E with a split exact structure, w e obtain a short exact sequence 0 Ñ D b p B 0 q Ñ St p E add q Ñ Perf p X q Ñ 0 . By Lemma 7.2 w e ha ve N K ď 0 p E add q “ 0 . Hence, N K ´ 1 p B 0 q – N K 0 p X q . The latter group is non-zero, since we ha v e a surjection N K 0 p X q ↠ coker p Pic p X q Ñ Pic p X ˆ A 1 qq – x k r x s . This prov es (i). 78 ALEXANDER I. EFIMOV (ii) T ake any pair B Ă A as in (i) such that the map K ´ 2 p B q Ñ K ´ 2 p A q is not surjective. W e will construct inductively the small k -linear t -categories C n with required prop erties, so that C ♡ n » B for all n ě 1 . First w e construct C 1 . W e use the construction from [E25b, Theorem 0.3] and recall the assertions of lo c. cit. W e denote by E A , B the quasi-ab elian category of pairs p x, y q , where x P A and y Ă x is a sub ob ject such that b oth y and x { y are in B . The forgetful functor E A , B Ñ A , p x, y q ÞÑ x, induces a quotien t functor on bounded deriv ed categories. W e deﬁne C 1 to b e its kernel, so we hav e a short exact sequence (7.1) 0 Ñ C 1 Φ Ý Ñ D b p E A , B q q Ý Ñ D b p A q Ñ 0 . W e ha v e a semi-orthogonal decomp osition D b p E A , B q “ x i 1 p D b p B qq , i 2 p D b p B qqy . Here the fully faithful functors i 1 , i 2 : D b p B q Ñ D b p E A , B q are induced by the exact functors B Ñ E A , B , which are giv en resp ectiv ely b y x ÞÑ p x, 0 q , x ÞÑ p x, x q . The category C 1 has a b ounded t -structure, and we hav e an equiv alence B „ Ý Ñ C ♡ 1 , given by x ÞÑ Cone pp x, 0 q Ñ p x, x qq . No w, the coﬁber sequence of K -theory spectra for (7.1) is of the form K p C 1 q Ñ K p B q ‘ K p B q Ñ K p A q . Both functors q ˝ i 1 and q ˝ i 2 are isomorphic to the deriv ed functor of the inclusion B Ñ A . Hence, b y assumption the map K ´ 2 p B q ‘ K ´ 2 p B q Ñ K ´ 2 p A q is not surjective. It follo ws that the map K ´ 3 p i R 2 ˝ Φ q : K ´ 3 p C 1 q Ñ K ´ 3 p B q is not a monomorphism, where i R 2 is the righ t adjoint to i 2 . But the functor i R 2 ˝ Φ : C 1 Ñ D b p B q is left in verse to the realization functor, hence the map K ´ 3 p C ♡ 1 q Ñ K ´ 3 p C 1 q is not surjectiv e, as required. No w suppose that for some n ě 1 we hav e constructed a k -linear small t -category C n with required prop erties such that the map K ´ n ´ 2 p C ♡ n q Ñ K ´ n ´ 2 p C n q is not surjective. W e will construct the t -category C n ` 1 as follo ws. Put D “ C n , and consider the exact category D r 0 ,n s Ă D with the induced exact structure. Consider another exact category r D r 0 ,n s from Prop osition 3.7 with the same underlying category as D r 0 ,n s (w e use the small category version of lo c. cit.). Recall from loc. cit. that a morphism f : x Ñ y in r D r 0 ,n s is an inclusion if and only if b oth π n ´ 1 p f q and π n p f q are monomorphisms. W e deﬁne C n ` 1 via the short exact sequence (7.2) 0 Ñ C n ` 1 Φ Ý Ñ St p r D r 0 ,n s q q Ý Ñ St p D r 0 ,n s q Ñ 0 . Our assumption on D “ C n implies that the realization functor D b p D ♡ q Ñ St p D r 0 ,n ´ 1 s q is an equiv alence: this is Prop osition 2.19 applied to Ind p D q . Hence, by Proposition 3.7 w e ha ve a semi-orthogonal decomp osition D b p r D r 0 ,n s q “ x j 1 p D b p D ♡ q , j 2 p D b p D ♡ qqqy . Here the functors j 1 and j 2 are induced by the exact functors D ♡ Ñ r D r 0 ,n s , given b y x ÞÑ x THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 79 resp. x ÞÑ x r n s . The category C n ` 1 has a b ounded t -structure and we hav e an equiv- alence D ♡ „ Ý Ñ C ♡ n ` 1 . Moreov er, b y lo c. cit. the maps Ext i C ♡ n ` 1 p x, y q Ñ Ext i C n ` 1 p x, y q are isomorphisms for i ď n ` 1 , x, y P C ♡ n ` 1 . No w the coﬁber sequence of K -theory spectra for (7.2) is of the form K p C n ` 1 q Ñ K p D ♡ q ‘ K p D ♡ q Ñ K p D r 0 ,n s q . Both q ˝ j 1 and q ˝ j 2 r´ m s are isomorphic to the realization functor D b p D ♡ q Ñ St p D r 0 ,n s q . No w, b y Theorem 3.1 we hav e an isomorphism K ´ n ´ 2 p D r 0 ,n s q „ Ý Ñ K ´ n ´ 2 p D q . Hence, our assumption on D “ C n implies that the map K ´ n ´ 2 p D ♡ q ‘ K ´ n ´ 2 p D ♡ q Ñ K ´ n ´ 2 p D r 0 ,n s q is not surjectiv e. Hence, the map K ´ n ´ 3 p j R 2 ˝ Φ q : K ´ n ´ 3 p C n ` 1 q Ñ K ´ n ´ 3 p D ♡ q is not a monomorphism, where j R 2 is righ t adjoint to j 2 . The functor j R 2 ˝ Φ is left inv erse to the realization functor, hence the map K ´ n ´ 3 p C ♡ n ` 1 q Ñ K ´ n ´ 3 p C n ` 1 q is not surjectiv e, as required. This concludes the inductiv e step of the construction and prov es the theorem. □ 7.2. Estimates for higher nil groups of ab elian categories. W e giv e a closer lo ok at the ab elian category B 0 from the pro of of Theorem 7.1 in the c haracteristic zero case. W e recall the lo calizing inv arian t D K “ Fib er p K Ñ K H q . F or a small ab elian category A we put D K p A q “ D K p D b p A qq . W e do the follo wing computation, in particular pro ving the sharpness of the estimates from Corollary 4.12. Theorem 7.3. L et k b e a ﬁeld of char acteristic zer o. L et X “ t x 3 1 “ x 0 x 2 2 u Ă P 2 k b e the cuspidal cubic curve over k , and let E “ V ect X b e the c ate gory of ve ctor bund les of ﬁnite r ank on X. L et A “ eﬀ p E q » Ac b p E q ♡ b e the (smal l) ab elian c ate gory of eﬀac e able functors E op Ñ Ab . (i) F or s ě 1 we have (7.3) N s K p A q – HC ´ , red p k r x 1 , . . . , x s s{ k qr´ 2 s . In p articular, we have N s K s ´ 2 p A q – Ω s k r x 1 ,...,x s s{ k ‰ 0 . (ii) We have (7.4) D K n p A q “ $ & % k for n ě ´ 1 o dd; 0 else. W e elaborate on the meaning of the right hand side of (7.3). First, b y HC ´ p´{ k q we mean the negativ e cyclic homology o v er k , considered as a spectrum [Tsy83, FT85, Con94]. 80 ALEXANDER I. EFIMOV The superscript “red” means the reduced part in the simplicial sense, where w e consider the assignmen t r s s ÞÑ k r x 1 , . . . , x s s as a functor ∆ op Ñ CAlg p V ect k q , see for example [E25c, Section 8]. F or the pro of of Theorem 7.3 we will need a certain short exact sequence, whic h might b e kno wn to some exp erts. W e recall the notion of a lax pullbac k: if F : A Ñ C and G : B Ñ C are functors b etw een small 8 -categories, then A ˆ Ñ B C is the category of triples p x, y , φ q , where x P A , y P B and φ : F p x q Ñ G p y q . If A , B , C are stable and F , G are exact, then the category A ˆ Ñ C B is also stable, and it has a semi-orthogonal decomp osition A ˆ Ñ C B “ x A , B y , where the inclusions are given by x ÞÑ p x, 0 , 0 q resp. y ÞÑ p 0 , y , 0 q . The same applies to k -linear stable categories and k -linear exact functors. Prop osition 7.4. L et X b e as in The or em 7.3. We identify the normalization r X with P 1 k with c o or dinate x “ x 2 x 1 . We put k r ε s “ k r x s{ x 2 , and c onsider the pul lb ack functors P erf p P 1 k q Ñ Perf p k r ε sq , Perf p k q Ñ P erf p k r ε sq . Consider the semi-fr e e (asso ciative) dg alge- br a B “ k x x, y y , wher e deg p x q “ 0 , deg p y q “ 1 , dx “ 0 , dy “ x 2 . Then we have a short exact se quenc e in Cat perf k : (7.5) 0 Ñ P erf p X q F Ý Ñ Perf p k q ˆ Ñ Perf p k r ε sq P erf p P 1 k q Ñ Perf p B q Ñ 0 . Pr o of. The fully faithful functor F in (7.5) is the familiar categorical resolution of sin- gularities from [KL15]. It is induced b y the pullback functors: to the singular p oin t P erf p X q Ñ Perf p k q and to the normalization P erf p X q Ñ Perf p P 1 k q . The essential image of F is in fact the strict pullback. Put C “ P erf p k q ˆ Ñ Perf p k r ε sq P erf p P 1 k q . W e need to compute the quotient C { F p P erf p X qq . First take the aﬃne c hart U “ X ztp 0 : 0 : 1 qu . Then Perf p U q is a quotient of Perf p X q , and F induces a fully faithful functor F : Perf p U q ã Ñ P erf p k q ˆ Ñ Perf p k r ε sq P erf p k r x sq “ : C . W e hav e an equiv alence C { F p P erf p X qq „ Ý Ñ C { F p P erf p U qq . W e consider the ob jects P 1 “ p k , k r x s , id q P C , P 2 “ p 0 , k r x s , 0 q P C . Clearly , P 1 – F p O U q , and together P 1 and P 2 generate C (as an idemp otent-complete stable sub category). Hence, it suﬃces to compute the endomorphism dg algebra of the image of P 2 in the quotien t of C by (the stable subcategory generated by) P 1 . THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 81 First, the dg algebra End C p P 1 ‘ P 2 q is discrete, i.e. its homology v anishes in non-zero degrees. Hence, w e ma y iden tify it with its H 0 . Putting R “ k r x 2 , x 3 s Ă k r x s , we ha v e Hom C p P 1 , P 1 q “ R , Hom C p P 1 , P 2 q “ x 2 k r x s , Hom C p P 2 , P 1 q “ k r x s , Hom C p P 2 , P 2 q “ k r x s . Denote by P 2 the image of P 2 in C 1 “ C {x P 1 y . By [Dr04] the dg algebra End C 1 p P 2 q is quasi-isomorphic to B 1 “ k r x s ‘ à n ě 0 p x 2 k r x s h q b p Rh q b n b k r x s , where h has degree 1 , the m ultiplication is straightforw ard and the diﬀerential is given by d p x k h q “ x k . Note that the cokernel of k r x s Ñ B is iden tiﬁed with the shifted non-reduced bar complex computing the deriv ed tensor pro duct x 2 k r x s b L R k r x s . Since the non-reduced bar complex is quasi-isomorphic to the reduced bar complex, w e ha ve a quasi-isomorphism B „ Ý Ñ B 1 , x ÞÑ x, y ÞÑ x 2 h. This prov es the prop osition. □ W e recall from [Ke64] that for an ordinary small additiv e category B its Jacobson radical is a sub-bifunctor rad B Ă B p´ , ´q : B op ˆ B Ñ Ab , rad B p x, y q “ t f P B p x, y q | @ g P B p y , x q the morphism 1 x ` g f is in vertible u . W e denote by B { rad the category with the same ob jects as in B , where the morphisms are given by p B { rad qp x, y q “ B p x, y q{ rad B p x, y q . W e will need a simple observ ation. Prop osition 7.5. L et k b e a ﬁeld and supp ose that B is an or dinary idemp otent-c omplete k - line ar smal l c ate gory with ﬁnite-dimensional morphisms. Then we have N s K p B { rad q “ 0 for s ě 1 . Pr o of. Indeed, the category B { red is equiv alen t to a direct sum of (a p ossibly inﬁnite collection of ) categories of ﬁnite-dimensional vector spaces o ver skew-ﬁelds. F or any skew- ﬁeld D we ha ve N s K p D q “ 0 for s ě 1 . □ No w Theorem 7.3 is easily deduced from the following lemma. W e ﬁrst clarify the no- tation. F or a Q -linear stable category (or for a Q -algebra, or for a scheme ov er Q ) we denote b y HH the Ho c hsc hild homology ov er Q , and similarly for HC ´ , HC and HP . 82 ALEXANDER I. EFIMOV W e will also use the notation N s HC ´ for the higher nil versions, so for C P Cat perf Q w e ha v e N s HC ´ p C q “ HC ´ , red p C b P erf p Q r x 1 , . . . , x s sqq , s ě 0 . W e also note that for s ě 1 the Q S 1 -mo dule H H red p Q r x 1 , . . . , x s sq is isomorphic to a direct sum of shifts of Q S 1 , whic h implies an isomorphism N s HC ´ p C q – H H p C q b HC ´ , red p Q r x 1 , . . . , x s sq , s ě 1 , C P Cat perf Q . Lemma 7.6. L et X, E “ V ect X and A “ eﬀ p E q b e as in The or em 7.3. As in Pr op osition 7.4 we identify the normalization r X with P 1 k , and denote by r E “ V ect P 1 k the c ate gory of ve ctor bund les on P 1 k of ﬁnite r ank. (i) Consider the universal ﬁnitary lo c alizing invariant of Q -line ar idemp otent-c omplete stable c ate gories U loc : Cat perf Q Ñ Mot loc Q . Then we have an isomorphism U loc p r E add q „ Ý Ñ U loc p r E add { rad q . This gives a natur al c ommutative squar e in the c ate- gory D p Q q B S 1 » D p Q S 1 q : (7.6) HH p E add q HH p X q HH p E add { rad q HH p P 1 k q . (ii) Denote by V the total c oﬁb er of (7.6) , c onsider e d as a c omplex of Q -ve ctor sp ac es (for getting the S 1 -action). Then for s ě 1 we have a natur al isomorphism (7.7) N s K p A q – V r´ 2 s b HC ´ , red p Q r x 1 , . . . , x s sq . Pr o of. (i) By Grothendiec k’s theorem ev ery v ector bundle on P 1 k is a ﬁnite direct sum of line bundles O p n q . Hence, as a k -linear dg category St p r E add q has a Z -indexed full exceptional collection t O p n qu n P Z . No w, the Jacobson radical of r E add is generated b y the morphisms O p n q Ñ O p n 1 q for n ă n 1 . Hence, the category St p r E add { rad q has a (completely orthogonal) Z -indexed full exceptional collection formed b y the images of O p n q . It follows that the functor St p r E q Ñ St p r E add { rad q induced an isomorphism on U loc . In particular, the map HH p r E add q Ñ HH p r E add { rad q is an isomorphism. The square (7.6) is obtained from the diagram HH p E add { rad q HH p E add q HH p X q HH p r E add { rad q HH p r E add q HH p P 1 k q . „ THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 83 (ii) Let s ě 1 b e an in teger. Recall that b y [Nee21, Prop osition 2.4] w e ha ve an equiv a- lence D b p A q » Ac b p E q . Hence, we ha v e a coﬁb er sequence of sp ectra (7.8) N s K p A q Ñ N s K p E add q Ñ N s K p X q . By Prop osition 7.5 and by the Dundas-Go o dwillie-McCarthy theorem [DGM13, Theorem 7.0.0.2] we hav e the isomorphisms (7.9) N s K p E add q – Fib er p N s K p E add q Ñ N s K p E add { rad qq – Fib er p N s HC ´ p E add q Ñ N s HC ´ p E add { rad qq – Fib er p HH p E add q Ñ HH p E add { rad qq b HC ´ , red p Q r x 1 , . . . , x s sq . Next, recall the semi-free dg k -algebra B from Prop osition 7.4: it is giv en b y B “ k x x, y y , where deg p x q “ 0 , deg p y q “ 1 , and dx “ 0 , dy “ x 2 . Applying lo c. cit. and [DGM13, Theorem 7.0.0.2] again we obtain (7.10) N s K p X q – N s K p B qr´ 1 s – Fib er p N s K p k q Ñ N s K p B qq – Fib er p N s HC ´ p k q Ñ N s HC ´ p B qq – Fib er p N s HC ´ p X q Ñ N s HC ´ p P 1 k qq – Fib er p HH p X q Ñ HH p P 1 k qq b HC ´ , red p Q r x 1 , . . . , x s sq . Com bining (7.9) and (7.10) together with the coﬁb er sequence (7.8) w e obtain the isomor- phism (7.7). □ Pr o of of The or em 7.1. (i) Consider the complex of Q -vector spaces V from Lemma 7.6. By lo c. cit. it suﬃces to pro v e that V – k . Clearly , w e ha v e Cone p HH p E add q Ñ HH p E add { rad qq P D p Q q ě 1 . Next, by Prop osition 7.4 for the dg k - algebra B from lo c. cit. we ha v e Cone p HH p X q Ñ HH p P 1 k qq – Cone p HH p k q Ñ HH p B qq P D p Q q ě 0 . Moreo v er, H 0 of the latter complex is isomorphic to k . Hence, we ha v e V P D p Q q ě 0 and H 0 p V q – k . Applying Lemma 7.6 for s “ 1 w e obtain N K p A q – V r´ 2 s b HC ´ , red p Q r x 1 sq – V r´ 1 s b Ω 1 Q r x 1 s{ Q . By Corollary 4.12 we hav e N K p A q P Sp ď´ 1 . W e conclude that V P D p Q q ď 0 , and b y the ab o ve we obtain V – k . This prov es (i). (ii) now follows almost directly from (i). Namely , w e ﬁrst obtain an isomorphism D K p A q – Fib er p HC ´ p k { k qr´ 2 s Ñ colim r n sP ∆ op HC ´ p k r x 1 , . . . , x n s{ k qr´ 2 sq . 84 ALEXANDER I. EFIMOV Since HP p k r x 1 , . . . , x n s{ k q – H P p k { k q for n ě 0 , w e obtain Fib er p HC ´ p k { k qr´ 2 s Ñ colim r n sP ∆ op HC ´ p k r x 1 , . . . , x n s{ k qr´ 2 sq – Fib er p HC p k { k qr´ 1 s Ñ colim r n sP ∆ op HC p k r x 1 , . . . , x n s{ k qr´ 1 sqq – HC p k { k qr´ 1 s . The latter isomorphism follows from the w ell-known v anishing of the colimit colim r n sP ∆ op HH p k r x 1 , . . . , x n s{ k q : this is an E 8 -algebra in whic h the unit elemen t is homotopic to zero. Therefore we obtain D K p A q – HC p k { k qr´ 1 s , whic h gives (7.4). □ Remark 7.7. It fol lows dir e ctly fr om the [E25c, Theorem 9.3] that for any C P Cat perf Q the sp e ctrum D K p C q is natur al ly a Q rr u ss -mo dule, wher e u has c ohomolo gic al de gr e e 2 (r e c al l that T C p Q q – Q rr u ss ). Mor e over, it fol lows fr om [E25c, Theorem 9.1] that u is acting lo c al ly nilp otently on D K p C q , sinc e by lo c. cit. we have π ´ 2 End Mot loc Q p r U loc p Q r x sqq “ 0 . The pr o of of The or em 7.3 shows that for the ab elian c ate gory A “ eﬀ p V ect X q we have an isomorphism D K p A q – p k pp u qq{ u k rr u ssqr´ 1 s of Q rr u ss -mo dules. 8. Examples of comp a ctl y assembled t -structures and coherentl y assembled abelian ca tegories 8.1. Sheav es on lo cally compact Hausdorﬀ spaces. F or simplicity we consider only the con- tin uously bounded t -structures, and w e only consider the categories of sheav es with v alues in a ﬁxed category (not in a presheaf of categories like in [E24, Sections 6.2-6.4]). Prop osition 8.1. L et X b e a c omp act Hausdorﬀ sp ac e and let C b e a dualizable t -c ate gory. (i) Ther e is a c omp actly assemble d c ontinuously b ounde d t -structur e on the dualizable c ate gory of pr eshe aves PSh p X ; C q , such that PSh p X ; C q ě 0 “ PSh p X ; C ě 0 q . (ii) Ther e is a unique c omp actly assemble d c ontinuously b ounde d t -structur e on the c ate gory Shv p X ; C q such that the she aﬁﬁc ation functor PSh p X ; C q Ñ Sh v p X ; C q is t -exact. Mor e over, we have Shv p X ; C q ♡ » Sh v p X ; C ♡ q . (iii) If A is a c oher ently assemble d ab elian c ate gory, then the c ate gory Shv p X ; A q is also c oher ently assemble d. Pr o of. (i) follows directly from Prop osition 2.29. (ii) It is well-kno wn that there is a unique t -structure on Sh v p X ; C q suc h that the sheaﬁﬁcation functor F ÞÑ F 7 is t -exact. Moreo ver, this t -structure is compatible with ﬁltered colimits. T o ﬁnish the pro of, by Proposition 2.12 it suﬃces to construct a functor THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 85 Φ : Sh v p X ; C q Ñ PSh p X ; C q , righ t in verse to the sheaﬁﬁcation functor, such that Φ is con tin uous, exact and left t -exact. F or a closed subset Z Ă X denote by i Z : Z Ñ X the inclusion. W e put Φ p F qp U q “ Γ c p X , i ¯ U , ˚ p i ˚ ¯ U p F qqq , F P Shv p X ; C q , U Ă X op en . Then Φ comm utes with ﬁltered colimits since so do i ˚ ¯ U , i ¯ U , ˚ and Γ c p X , ´q . F or the same reason Φ is left t -exact. It remains to show that Φ is righ t in verse to the sheaﬁﬁcation. T o see this, consider another (accessible exact) functor Ψ : Sh v p X ; C q Ñ PSh p X ; C q , giv en by Ψ p F qp U q “ Γ p ¯ U , i ˚ ¯ U p F qq , F P Shv p X ; C q U Ă X op en . W e hav e natural transformations Φ Ñ Ψ and Ψ Ñ incl , where incl is the inclusion functor. By construction, for F P Sh v p X ; C q and for an y compact Z Ă X we hav e Γ p Z, Cone p Φ p F q 7 Ñ Ψ p F q 7 q | Z q “ 0 , Γ p Z, Cone p Ψ p F q 7 Ñ F q | Z q “ 0 . Hence, the maps Φ p F q 7 Ñ Ψ p F q 7 and Ψ p F q 7 Ñ F are isomorphisms of sheav es, as re- quired. Finally , (iii) follo ws from (ii) applied to C “ ˇ D p A q . □ Corollary 8.2. Ther e exist c oher ently assemble d ab elian c ate gories for which the gr oup K cont ´ 1 is non-zer o. F or example, let k b e a ﬁeld, and take A “ Sh v p R ; V ect k q . Then A is c oher ently assemble d and K cont ´ 1 p A q “ Z . Pr o of. By Prop osition 8.1 the category A is coherently assem bled. The computation of K cont ´ 1 p A q is a special case of [E24, Prop osition 4.18]: w e ha ve K cont ´ 1 p A q – K cont ´ 1 p Sh v p R ; D p k qqq – K 0 p k q – Z . □ 8.2. Nuclear modules o v er Z p . Consider the ab elian category Solid Z p of ligh t solid mo d- ules ov er Z p from [CS20, CS24]. It is generated by a single compact pro jective ob ject P “ ś N Z p . W e denote by Nuc C S p Z p q Ă D p Solid Z p q the original version of nuclear solid mo dules ov er Z p , whose description will b e recalled in the pro of of the next prop osition. This category is known to be dualizable (but not compactly generated). Recall from [E25a] that w e ha ve a bigger dualizable category Nuc p Z p q Ą Nuc C S p Z p q , such that the inclusion functor is strongly contin uous and w e hav e Nuc p Z p q » dual lim Ð Ý n D p Z { p n q . Here the in verse limit is tak en in Cat dual st . Recall that w e ha ve Nuc p Z p q ω » Nuc C S p Z p q ω » P erf p Z p q . 86 ALEXANDER I. EFIMOV The latter category has a standard t -structure since Z p is regular. Our goal in the next prop osition is to sho w that this t -structure extends to a compactly assem bled con tinuously b ounded t -structure on Nuc C S p Z p q . One can show b y a more diﬃcult argumen t that it further extends to a compactly assembled t -structure on Nuc p Z p q with the same heart, but the latter t -structure is not contin uously bounded. The ﬁrst part of the following prop osition is kno wn but we include the proof for com- pleteness. Prop osition 8.3. (i) The c ate gory Solid Z p is lo c al ly c oher ent. The ab elian c ate gory Solid ω Z p has homolo gic al dimension 1 , and we have ˇ D p Solid Z p q » D p Solid Z p q . (ii) The standar d t -structur e on D p S ol id Z p q induc es a c omp actly assemble d c on- tinuously b ounde d t -structur e on Nuc C S p Z p q . Mor e over, the r e alization functor ˇ D p Nuc C S p Z p q ♡ q Ñ Nuc C S p Z p q is an e quivalenc e. Pr o of. (i) Consider the compact pro jectiv e generator P P Solid Z p as abov e. Putting A “ End p P q , we ha ve an equiv alence D p Solid Z p q – D p A q . Recall that the morphisms P Ñ P are simply the con tinuous maps of top ological ab elian groups ś N Z p Ñ ś N Z p . It follows that w e hav e an isomorphism A op – End Z pp À N Z q ^ p q . Recall that w e hav e an equiv alence D p - compl p Z q „ Ý Ñ D p - tors p Z q , sending p À N Z q ^ p to À N Q p { Z p r´ 1 s . Hence, we ha v e an isomorphism A op – End p À N Q p { Z p q . Now, the ob ject I “ À N Q p { Z p is injec- tiv e in the small ab elian category A “ p Mo d p - tors - Z q ω 1 . Moreo ver, any ob ject of A can b e em b edded into I . It follows that the ring A is right coherent and we hav e equiv alences Solid ω Z p » Coh - A – A op . In particular, these categories ha ve homologi- cal dimension 1 . Since Solid Z p also has a compact pro jectiv e generator, the equiv alence ˇ D p Solid Z p q » D p Solid Z p q follo ws for example from [Lur18, Prop osition C.5.8.12]. (ii) Within the abov e notation, denote b y J Ă A the ideal of trace-class maps in the sense of [CS20, Deﬁnition 13.11]. Under the ab ov e identiﬁcation A op – End Z pp À N Z q ^ p q , the ideal J op Ă A op consists of compact operators, i.e. maps X : p À N Z q ^ p Ñ p À N Z q ^ p suc h that for each n ą 0 the map X b Z { p n : À N Z { p n Ñ À N Z { p n has ﬁnite rank. W e ha ve a short exact sequence 0 Ñ Nuc C S p Z p q i Ý Ñ D p A q Ñ D p A { J q Ñ 0 . By (i), the standard t -structure on D p A q is compactly assem bled and contin uously b ounded. By Corollary 2.13 it suﬃces to pro ve that the functor i ˝ i R : D p A q Ñ D p A q is t -exact, where i R is right adjoint to i. THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 87 W e ha ve i ˝ i R – ´ b A J. Hence, w e need to pro ve that J is ﬂat as a left A -mo dule. This is kno wn, but we giv e a pro of for completeness. It is conv enient to deal with the opp osite algebra, i.e. with endomorphisms of p À N Z q ^ p . W e consider the p oset S “ t f : N Ñ N | lim n Ñ8 f p n q “ 8u with reverse order: f ď g if f p n q ě g p n q for n ě 0 . Then S is directed. F or eac h such f P S consider the op erator X f : p à N Z q ^ p Ñ p à N Z q ^ p , X f p e n q “ p f p n q e n for n ě 0 . Here t e n u n ě 0 is the top ological basis of p À N Z q ^ p . It is easy to see that J op as a right ideal in A op is given by the directed union: J op “ ď f : N Ñ N X f ¨ A op , where f runs through functions as ab o ve. No w, each X f (as an op erator) has zero kernel, and for f ď g we ha v e X f P X g ¨ A op . It follows that J op is ﬂat as a righ t A op -mo dule, i.e. J is ﬂat as a left A -mo dule. This pro ves the prop osition. □ 8.3. Chase criterion. In this subsec tion w e consider ab elian categories in the setting of the follo wing proposition. In fact they are exactly the dualizable Ab -mo dules in Pr L , as sho wn in [LLS]. They are also exactly the categories describ ed b y Ro os’ theorem [Roo65]. Namely , these are categories of the form Mo d - R { Mo d - p R { I q , where R is an associative unital ring and I Ă R is an ideal such that I 2 “ I . Prop osition 8.4. L et C b e a c omp actly assemble d additive or dinary c ate gory. Consider the c ate gory A “ F un cont , add p C , Ab q of c ontinuous additive functors. Then A is a Gr othendie ck ab elian c ate gory, satisfying (AB6) and (AB4*). Mor e over, we have an e quivalenc e (8.1) C „ Ý Ñ F un cont , ex p A , Ab q , x ÞÑ p F ÞÑ F p x qq . Her e the tar get is the c ate gory of c ontinuous exact functors. Pr o of. If C is compactly generated, then A » F un add p C ω , Ab q , and the stated axioms hold. The equiv alence (8.1) is essen tially a reform ulation of Lazard’s theorem. Indeed, w e ha v e an equiv alence F un L p A , Ab q » F un add pp C ω q op , Ab q , and the full subcategory of ﬂat functors p C ω q op Ñ Ab is iden tiﬁed with Ind p C ω q » C . No w consider the general case. Restriction to C ω 1 giv es a fully faithful functor Φ : A ã Ñ F un add p C ω 1 , Ab q . Its righ t adjoin t Φ R is induced b y the functor ˆ Y : C Ñ Ind p C ω 1 q . 88 ALEXANDER I. EFIMOV Explicitly , for x P C with ˆ Y p x q “ “lim Ý Ñ i ” x i P Ind p C ω 1 q , w e ha ve Φ R p F qp x q “ lim Ý Ñ i F p x i q , F P F un add p C ω 1 , Ab q . In particular, Φ R is exact and commutes with ﬁltered colimits. Since (AB5), (AB6) and (AB4*) hold in F un add p C ω 1 , Ab q , they also hold in A . It is also clear that A has a gener- ator, namely Φ R p G q , where G is a generator in F un add p C ω 1 , Ab q . Finally , note that by the ab ov e observ ations the functor (8.1) is a retract of the functor Ind p C ω 1 q Ñ F un cont , ex p F un p C ω 1 , Ab q , Ab q , whic h is an equiv alence b y Lazard’s theorem. □ The follo wing result is a generalization of Chase criterion of left coherence of an asso ciativ e unital ring R. F or completeness we give essentially a self-contained pro of, whic h is purely categorical and cov ers the original theorem [Ch60, Theorem 2.1]. More precisely , the result in loc. cit. corresponds to the case when C “ Flat - R is the (compactly generated) category of ﬂat righ t R -mo dules and A “ R - Mo d is the abelian category of left R -mo dules. Theorem 8.5. L et C b e a c omp actly assemble d additive or dinary c ate gory, and let A “ F un cont , add p C , Ab q . The fol lowing ar e e quivalent. (i) A is c oher ently assemble d. (ii) C has inﬁnite pr o ducts. Pr o of. (i) ù ñ (ii) . By Proposition 8.4 w e need to show that the category F un cont , ex p A , Ab q has inﬁnite pro ducts, provided that A is c oheren tly assembled. Re- striction to A ω 1 deﬁnes a fully faithful functor Ψ : F un cont , ex p A , Ab q ã Ñ F un ex p A ω 1 , Ab q . No w, exactness of ˆ Y A : A Ñ Ind p A ω 1 q implies that Ψ has a right adjoint, giv en b y Ψ R p F qp x q “ lim Ý Ñ i F p x i q , F P F un ex p A ω 1 , Ab q , x P A , ˆ Y A p x q “ “lim Ý Ñ i ” x i . Since the category of ab elian groups satisﬁes (AB4*) (exactness of inﬁnite pro ducts), the cat- egory F un ex p A ω 1 , Ab q has inﬁnite pro ducts. Hence, so do es the category F un cont , ex p A , Ab q , as required. (ii) ù ñ (i) W e ﬁrst consider the case when C is compactly generated, and we iden tify A » F un add p C ω , Ab q . Cho ose a suﬃciently large regular cardinal κ with the following prop erties: the set of isomorphism classes of ob jects in C ω is κ -small, and for an y x, y P C ω the group Hom C ω p x, y q is κ -small. Then the category A κ is ab elian. Moreo ver, this small category has a collection of pro jectiv e generators. More precisely , let B Ă A κ THEOREM OF THE HEAR T F OR WEIBEL’S HOMOTOPY K -THEOR Y 89 b e the full (additive) sub category of ob jects of the form À j P J h _ x j , where J is a κ -small set, x j P C ω and h _ x j “ Hom p x j , ´q P A ω for j P J. Then we hav e an equiv alence Ind p A κ q » F un add p B op , Ab q . W e iden tify C ω with a full sub category of B op via x ÞÑ h _ x . No w, the colimit functor colim : Ind p A κ q Ñ A is identiﬁed with the precomp osition functor Θ : F un add p B op , Ab q Ñ F un add p C ω , Ab q , which is exact and comm utes with all colimits. Thus, it suﬃces to pro ve that the left adjoin t functor Θ L : F un add p C ω , Ab q Ñ F un add p B op , Ab q is exact. Indeed, this would imply that A is coherently assem bled since so is Ind p A κ q . No w, exactness of Θ L means that for any P P B the functor F P : p C ω q op Ñ Ab , F P p x q “ Hom B p P , h _ x q , is ﬂat. But this holds b y assumption: if P “ À j P J h _ x j , then the pro duct ś j P J x j exists in C , which exactly means that the functor F P – ś j Hom p´ , x j q is ﬂat. This pro ves that A is coheren tly assem bled, or equiv alen tly lo cally coheren t. No w consider the general case. W e ﬁrst observe that the category Ind p C ω 1 q has inﬁnite pro ducts. Indeed, it suﬃces to show that for a collection of ob jects t x j P C ω 1 u j P J their pro duct exists in Ind p C ω 1 q . This pro duct is computed as follows: w e ﬁrst take the pro duct of x j in C (whic h exists by assumption), and then tak e its image under the functor C » Ind ω 1 p C ω 1 q Ă Ind p C ω 1 q . By the ab ov e sp ecial case we kno w that the category F un add p C ω 1 , Ab q is coherently assem bled. By the pro of of Prop osition 8.4 w e ha ve a retraction A Ñ F un add p C ω 1 , Ab q Ñ A , where b oth functors are exact and comm ute with all colimits. Therefore, by Proposition 1.31 the category A is coheren tly assem bled, as required. □ 8.4. Concluding remarks. W e co v ered only some v ery basic non-trivial examples of com- pactly assembled t -structures and coheren tly assembled ab elian categories. W e brieﬂy men tion the follo wing directions. ‚ One can consider ab elian categories of almost mo dules A “ Mod - R { Mo d - p R { I q , where R is a discrete asso ciative unital ring, and I Ă R is a t wo-sided ideal such that I 2 “ I . It is in teresting to ﬁgure out some basic suﬃcient conditions (for example, in the comm utative case) which imply that A is coheren tly assem bled. If this is the case, the functor ˇ D p A q Ñ D p A q do es not ha ve to b e an equiv alence, and one can consider the contin uous K -theory K cont p A q as G -theory of the pair p R, I q . When I “ R, this means that R is righ t coherent and w e obtain the usual G -theory of R. ‚ One can consider the categories of n uclear mo dules for more general adic spaces, in particular for formal sc hemes. F or example, if R is a regular no etherian comm u- tativ e ring with an ideal I Ă R, we expect that the category Nuc C S p R ^ I q has a 90 ALEXANDER I. EFIMOV compactly assembled contin uously b ounded t -structure. If R is not regular, there should still b e a contin uously b ounded t -structure with a coherently assembled heart, so one can consider G -theory in this framew ork. W e also exp ect nice t - structures in the arc himedean setting, say for nuclear gaseous mo dules o ver v arious gaseous rings. ‚ One can construct a dualizable t -category starting from a coherently assembled exact category E . Namely , (slightly abusing the notation) denote b y E add the same category with an exact structure from Prop osition 1.32. Then we ha ve a short exact sequence in Cat dual st : 0 Ñ ˇ Ac p E q Ñ ˇ St p E add q Ñ ˇ St p E q Ñ 0 . One can sho w that the category ˇ Ac p E q has a natural compactly assem bled con- tin uously b ounded t -structure, induced b y the natural t -structure on ˇ St p E add q . If E is lo cally coheren t, then ˇ Ac p E q » Ind p Ac p E ω qq , and the induced t -structure on Ac p E ω q coincides with the (b ounded) t -structure from Proposition 1.3. In general the heart ˇ Ac p E q ♡ is equiv alent to the category of sequential limit-preserving addi- tiv e functors F : p E ω 1 q op Ñ Ab such that for an y x P E ω 1 , for any α P F p x q and for an y compact morphism f : y Ñ x with y P E ω 1 , there exists an exact pro jection g : z Ñ y in E ω 1 suc h that F p f g qp α q “ 0 . ‚ Let R b e an asso ciative unital ring, and consider the ab elian category A “ Mo d - R of righ t R -mo dules. By Prop osition 1.43, A is coherently assembled if and only if R is right coheren t. In general, one can consider the left derived functors L k ˆ Y : A Ñ Ind p A q , and deﬁne the non-coherence rank of R to b e n “ sup t k ě 0 | L k ˆ Y ‰ 0 u P N Y t8u . Supp ose that 0 ă n ă 8 . One can show that the category ˇ D p A q is dualizable and w e are in the situation of Question 0.7. 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Graduate Studies in Mathematics, v ol. 145, American Mathematical So ciet y , Providence, RI, 2013. Steklov Ma thema tical Institute of RAS, Gubkin St. 8, GSP-1, Moscow 119991, Russia Email address : efimov@mccme.ru

Theorem of the heart for Weibel's homotopy $K$-theory

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