Formal completions and idempotent completions of triangulated categories of singularities

F ORMAL COMPLETIO NS AND IDEMPOTENT COMPLETIONS OF TRIANGULA TED CA T E GORIES OF SINGULARITIES DMITRI ORLOV De dic ate d to the blesse d memory of my adviser V asily Alexe evich Iskovskikh Abstra ct. The main goal of this pap er is t o prove t hat t he idemp otent completions of the tri- angulated categories of singularities of t w o sc hemes are equiv alen t if the formal completions of these s chemes along singularities are isomorphic. W e also discuss T homason theorem on dense sub categories and a relation to the negative K-theory . 1. Introduction Let X b e a noetherian sc heme o ver a field k . Denote b y D b (coh X ) the b ounded deriv ed catego ries of coheren t sheav es on X. Sin ce X i s no etherian the natural functor from D b (coh X ) to the unb ounded deriv ed category of q u asi-coheren t shea v es D (Qcoh X ) is fu lly faithful and realizes an equ iv alence of D b (coh X ) with the full sub category D ∅ , b coh (Qcoh X ) consisting o f all cohomologi cally b ounded complexes with coherent cohomologi es ([5], Ex.I I, 2.2.2) . Denote b y Perf ( X ) ⊆ D (Qcoh X ) the full triangulated sub category of p erfect complexes. Recall that a complex on a scheme is said to b e p erf ect if it is lo cally qu asi-isomorphic to a b ounded complex of lo cally f ree shea v es of finite rank. The derived cat egory D (Qcoh X ) adm its all copro d ucts and it is w ell-kno wn that sub category of p erfect complexes P erf ( X ) coincides with su b category of compact ob jects in D (Qcoh X ) , i.e. all ob jects C ∈ D (Qcoh X ) for wh ic h the fun ctor Hom( C, − ) comm utes with arbitrary copro ducts. The category Perf ( X ) can b e considered as a f ull triangulated sub category of D b (coh X ) . Definition 1.1. We define a triangulate d c ate gory of singularities of X , denote d by D Sg ( X ) , as the quotient of the triangulate d c ate gory D b (coh X ) by the ful l triangulate d sub c ate gory Perf ( X ) . W e sa y that X satisfies a c ondition (ELF) if X is separated no etherian of finite Kru ll dimension and has enough lo cally free shea v es, i.e. for any co herent sheaf F there is an epimorphism E ։ F with a lo cally fr ee sheaf E . The last condition implies that an y p erfect complex is also globally (not only lo cally) qu asi- isomorphic to a b ounded complex of lo cally free shea v es o f finite r ank. F or example, an y quasi- pro jectiv e sc heme satisfies these cond itions. Note that an y closed and an y op en su bsc heme of X is also noetherian, finite dimensional and has enough lo cally free shea ves. It is clea r for a clo sed This wo rk is done under partial fi nancial supp ort by RFFI gr ant 08-01-00297, INT AS grant 05-10000 08-8118, and NSh grant 9969.2006.1. 1 2 subscheme while for an op en sub s c h eme U it follo w s f rom the fact that an y coheren t sheaf on U can b e obtained as the restriction of a coheren t sheaf on X ([7], ex.5.15). F urther in the p ap er we usually assu me that a sc heme X satisfies condition (ELF). It is kno w n th at if a sc heme X is regular then the category Perf ( X ) coincides w ith D b (coh X ) . In this case, the tr iangulated category of s in gularities is trivial. Let f : X → Y b e a mo rph ism of finite T or-dim en sion (for example, a flat mo rph ism or a r egular closed em b edding). In this case we ha v e an in v erse image fu nctor L f ∗ : D b (coh Y ) → D b (coh X ) . It is clear that the functor L f ∗ sends p erfect complexes on Y to p erfect complexes on X. Therefore, the functor L f ∗ induces an exact fun ctor L ¯ f ∗ : D Sg ( Y ) → D Sg ( X ) . A fundamen tal prop ert y of tr iangulated categories of singularities is a prop ert y of lo calit y in Zarisky top ology . It sa ys that for an y op en em b eddin g j : U ֒ → X , for w hic h Sin g ( X ) ⊂ U, the functor ¯ j ∗ : D Sg ( X ) → D Sg ( U ) is an equiv alence of triangulated categorie s [12]. On the other hand , t w o analytically isomorph ic singularities can ha v e non-equiv alen t triangulated catego ries of singularities. Ev en tw o differen t double p oints giv en b y equat ions f = y 2 − x 2 and g = y 2 − x 2 − x 3 ha v e non-equiv alen t categ ories of singularities. T he main reason h ere is that a triangulated category of singularities is not necessary id emp oten t c omplete. This means that not for eac h pro jector p : C → C, p 2 = p there is a decomp osition of the form C = K er p ⊕ Im p. F or an y triangulated category T w e can consider its so cal led idemp oten t completion (or Karoubian e n v elop e) T . This is a cat egory that consists o f all ke rnels of all pro j ectors. It has a natural structure of a triangulated category and the ca nonical functor T → T is an exact fu ll em b eddin g [4]. Moreo ver, the category T is idemp oten t complete no w, i.e. eac h id emp oten t p : C → C in T arises from a splitting Ker p ⊕ Im p. W e denote b y D Sg ( X ) the idemp oten t completion of the triangulated catego ries of singularities. F or any closed subscheme Z ⊂ X we can define the formal completion of X along Z as a ringed space ( Z , lim ← − O X / J n ) , w h ere J is the ideal sheaf corresp onding to Z. T he formal completion actually dep ends only on the closed sub set Supp Z and do es not dep end on a s cheme structure on Z . W e denote b y X the formal completion of X along its singularities Sin g( X ) . The main goa l of this p ap er is to pro v e that for an y t wo sc hemes X an d X ′ satisfying (ELF), if the formal completions X and X ′ along sin gu larities are isomorphic, then the idemp oten t completions of the t riangulated catego ries of sin gularities D Sg ( X ) and D Sg ( X ′ ) a re equiv alen t (Theorem 2.10). Actually , we s h o w a little bit more. W e pro v e that an y ob j ect of D Sg ( X ) is a direct summand of an ob ject from its full sub catego ry D b Sing( X ) (coh X ) / Pe rf Sing ( X ) ( X ) , w here D b Sing( X ) (coh X ) and Perf Sing( X ) ( X ) are sub categories of D b (coh X ) and Perf ( X ) resp ectiv ely , consisting of complexes with cohomologi es su pp orted on Sin g X (Prop osition 2.7). Th us, to any scheme X w e can attac h the category D Sg ( X ) and tw o subgroups in K 0 ( D Sg ( X ) ) that b y Thomason theorem [1 6] (see Theorem 4.1) one-to -one corr esp onds to the dense su b cate- gories D Sg ( X ) and D b Sing( X ) (coh X ) / Perf Sing( X ) ( X ) r esp ectiv ely . W e discuss t his c orresp ondence and a relation to th e negativ e K-theory of the category of p er f ect complexes in the last section. 3 2. Completions Let X b e a n o etherian sc heme and le t i : Z ֒ → X b e a c losed su b space. Let coh Z X ⊂ coh X b e the ab elian su b category of coheren t sh ea ves on X with su pp ort on Z . Consider the n atural fun ctor from D b (coh Z X ) to D b (coh X ) . It can b e easily sho wn that this fun ctor is fu lly faithful and giv es an equiv alence with the full sub category D b Z (coh X ) ⊂ D b (coh X ) consisting of all complexes cohomologies of whic h are supp orted on Z. (In ot her w ords, the sub category D b Z (coh X ) consists of all complexes restriction of wh ic h on the op en subset U = X \ Z is acyclic.) A t first, let u s consider ab elian category of q u asi-coheren t shea v es Qcoh X a nd its ab elian sub catego ry Qcoh Z X of quasi-coheren t s hea v es w ith s u pp ort on Z (or Z -torsion shea v es), i.e . all quasi-coheren t sheav es F suc h that j ∗ F = 0 , where j : U → X is t he op en em b edding of the co mplement U = X \ Z. The inclusion functor i : Qcoh Z X → Qcoh X has a right adjoin t Γ Z whic h asso ciates to eac h quasi-co herent sheaf F its subs heaf of sections with su pp ort in Z . It can b e shown that for a quasi-coheren t s h eaf F we ha v e an isomorp hism Γ Z ( F ) ∼ = lim − → n H om O X ( O X / J n , F ) , where J is a some ideal sh eaf suc h th at Z = Supp( O X / J ) . T he fu nctor Γ Z has a right -deriv ed functor R Γ Z : D (Qcoh X ) → D (Qco h Z X ) via h-injectiv e resolutions [15]. It is kno wn that the canonical functor i : D (Qcoh Z X ) → D (Qcoh X ) is fully faithfu l and realizes equiv alences of D (Qcoh Z X ) with the fu ll s ub category D Z (Qcoh X ) consisting of all complexes cohomologies of whic h are su pp orted on Z . It is prov ed f or no etherian s chemes for example in [3 ] (Prop. 5.2.1 and Prop. 5.3.1). (It is also true for quasi-c ompact and separated X and proregular e m b edd ed Z ⊂ X as sho wn in [2].) T o pro v e this fact it is sufficien t to sho w that for an y C · ∈ D Z (Qcoh X ) the natural map i R Γ Z ( C · ) → C · is an isomorphism (see, for example, [3] Lemm a 5.2.2). Since the functor R Γ Z is boun ded for no etherian sc hemes b y u sual ”w a y out” argum en t ([8], § 7) it is sufficien t to c hec k the isomorphism i R Γ Z ( F ) → F only for shea v es F ∈ Qcoh Z X. That is evident, b ecause R Γ Z F ∼ = Γ Z F , when F is Z -torsion. Th us, for any ob ject C · ∈ D (Qcoh X ) there is a distinguished triangle of the form i R Γ C · → C · → R j ∗ j ∗ C · , whic h sh o w s that the categories D (Qcoh U ) and D (Qcoh Z X ) are equiv alent to the quotien t catego ries D (Qcoh X ) / D Z (Qcoh X ) an d D (Qcoh X ) / D (Qcoh U ) resp ectiv ely . Lemma 2.1. L et X b e a no etherian scheme and let Z b e a close d su b sp ac e. Then the natur al functor D b (coh Z X ) → D b (coh X ) is ful ly faithful and gives an e quivalenc e with the ful l sub c ate gory D b Z (coh X ) ⊂ D b (coh X ) c onsisting of al l c om plexes c oh omolo gies of which ar e supp orte d on Z . Pro of. W e know that the n atur al functors D b (coh X ) ֒ → D (Qcoh X ) and D (Qcoh Z X ) ֒ → D (Qcoh X ) a re fu lly faithful. This implies that the functor D b (coh Z X ) → D b (coh X ) i s fu lly 4 faithful iff the functor D b (coh Z X ) → D (Qcoh Z X ) is fully faithfu l. D enote b y φ the n atural em b eddin g of coh Z X to Qcoh Z X. S ince coherent shea v es generate the ca tegory D b (coh Z X ) it is enough to sho w that for an y t wo coherent s h ea ves F , G ∈ coh Z X the natural maps Ext n ( F , G ) → Ext n ( φ ( F ) , φ ( G )) are isomorph isms. W e kno w that it i s evidently true for n = 0 . No w to ap p ly induction, it is sufficien t to chec k that for any e ∈ Ex t n ( φ ( F ) , φ ( G )) , n ≥ 1 , there is an epimorphism F ′ → F which erases e ([5], Ex. I I, Lemma 2.1.3). Any suc h elemen t e can b e r epresen ted by an exact sequence in Qcoh Z X 0 → G → E n − 1 → · · · → E 0 → F → 0 , where E i are quasi-coheren t s hea v es with supp ort on Z. The epimorphism E 0 → F erases e. Since an y qu asi-coherent sheaf on a n o etherian sc heme is a d irect colimit of its coherent subsh ea ves there is a coheren t subsh eaf F ′ ⊂ E 0 that co ve rs F . As a subsheaf of E 0 it is erase e and b elongs to coh Z X. Thus, th e natural functor D b (coh Z X ) → D b (coh X ) is fully faithfu l and giv es an equiv alence with the fu ll sub category D b Z (coh X ) ⊂ D b (coh X ) consisting of all complexes cohomologi es of whic h are supp orted on Z. ✷ The r estriction fu nctor j ∗ sends c oherent shea v es to co herent and w e get a functor from the quo- tien t catego ry D b (coh X ) / D b Z (coh X ) to the d eriv ed category D b (coh U ) . This fu nctor establishes an equiv alence b etw een these categories. Lemma 2.2. L et X b e a no etherian scheme. Then the natur al functor D b (coh X ) / D b Z (coh X ) → D b (coh U ) is an e quivalenc e. Pro of. T his fact is kno wn and w e omit a pro of. There are few different w a y s to get it. First, w e know this fact for quasi-coherent shea ves and by Lemma 2.5 it is enough to sho w that an y m ap from a b ounded complex of coheren t sh ea ves to an ob ject of D Z (Qcoh X ) admits a factorizat ion thr ough an ob ject from D b Z (coh X ) . Second, since coh U is the quotien t of the ab elian category coh X by the Serre sub category coh Z X it is p ossible to show that the fun ctor D b (coh X ) / D b Z (coh X ) → D b (coh U ) is surjectiv e on ob jects and on morphism s. This implies an equiv alence as we ll. ✷ Remark 2.3. Note that the second w a y allo ws us to pro v e a more general result. F or any Serre sub catego ry B of an ab elian category A the fu nctor F : D b ( A ) / D b B ( A ) → D b ( A / B ) is an equiv alence of triangulated categories, w here D b B ( A ) is the full sub catego ry in D b ( A ) consisting of all complexes with cohomolog ies in B (it is known but unp ublished [6]). Denote by Perf Z ( X ) the int ersection Perf ( X ) ∩ D b Z (coh X ) . Lemma 2.4 . L et X satisfies (ELF). Then an obje ct A ∈ D b Z (coh X ) b elongs to Perf Z ( X ) iff for any o bje ct B ∈ D b Z (coh X ) al l Hom( A, B [ i ]) ar e trivial exc ept for finite numb e r of i ∈ Z . Pro of. Denote b y D hf ⊂ D b Z (coh X ) the full su b category consisting of all ob jects A suc h that for an y ob ject B ∈ D b Z (coh X ) the spaces H om( A, B [ i ]) a re trivial except for finite n u m b er of 5 i ∈ Z . If an ob ject A ∈ Perf Z ( X ) th en it is qu asi-isomorphic to a b ound ed complex of v ector bund les. Since the cohomologies of any coherent sh eaf is b ou n ded b y the Kru ll dimension of the sc heme w e ha ve that for an y v ector bundle P and an y coheren t sheaf F there is an equalit y Ext i ( P , F ) = 0 when i is greater than Kr ull dimension of X . Therefore, A b elongs to the sub catego ry D hf . Supp ose now that A ∈ D hf . Th e ob ject A is a b ounded complex of coherent sh ea ves. Let us tak e lo cally free b ound ed ab o v e resolution P · ∼ → A and consid er a goo d tru ncation τ ≥− k P · for sufficien t large k ≫ 0 whic h is clearly isomorp h ic to A in D . Since A ∈ D hf , f or an y closed p oin t t : x ֒ → X th e groups Hom( A, t ∗ O x [ i ]) are zero for | i | ≫ 0 . This m eans th at for sufficien tly large k ≫ 0 the truncation τ ≥− k P · is a complex o f lo cally free shea ves at the p oin t x, and, hence, in some neigh b orho o d of x. The scheme X is quasi-compact. This imp lies that there is a common su ffi cien tly large k su c h that the trun cation τ ≥− k P · is a complex of lo cally free shea ves ev erywhere on X, i.e. A is p erfect. ✷ The natural embedd ing D b Z (coh X ) ֒ → D b (coh X ) induces a fu nctor b etw een qu otien t cat egories D b Z (coh X ) / Pe rf Z ( X ) − → D Sg ( X ) := D b (coh X ) / Pe rf ( X ) . It can b e pro v ed that this fun ctor b et we en quotien t catego ries is fully faithful to o. T o pro v e it we need the follo win g w ell-kno wn lemma. Lemma 2.5. ([18, 9]) L et D b e a triangulate d c ate gory and D ′ , N b e f ul l triangulate d sub c ate- gories. L et N ′ = D ′ ∩ N . Assume that any morphism N → X ′ (r esp. any morphism X ′ → N ) with N ∈ N and X ′ ∈ D ′ admits a factorization N → N ′ → X ′ (r esp. X ′ → N ′ → N ) with N ′ ∈ N ′ . Then th e natur al functor D ′ / N ′ − → D / N is ful ly faithful. Lemma 2.6. The fu nc tor D b Z (coh X ) / Pe rf Z ( X ) → D Sg ( X ) is ful ly faithful. Pro of. By Lemma 2.5 we should show that an y morphism P · → C · , where P · is a p erf ect complex and C · is an ob ject of D b Z (coh X ) , can b e factorized through an ob ject P · ′ ∈ Perf Z ( X ) . Since there is an equiv alence D b (coh Z X ) ∼ = D b Z (coh X ) w e can assu me that t he ob ject C · is a b ound ed complex of coherent shea ves with sup p ort on Z. This implies th at there is a su bsc heme structure i S : S ֒ → X with su pp ort on Z suc h that C · ∼ = R i S ∗ C · ′ . Consider a v ector bundle E on X that co vers the ideal sheaf J S . It exists by (ELF) condition. Denote b y K · the Koszul complex 0 → det( E ) → · · · → E → O X → 0 . W e ha v e canonical maps P · → P · ⊗ K · → P · ⊗ O S . No w an y map from P · to C · ∼ = R i S ∗ C · ′ is fact orized through P · ⊗ O S and, hence, through P · ⊗ K · . But the ob ject P · ⊗ K · is p erfect as a tensor pr o duct of tw o p erfect complexes and has cohomologies with sup p orts on Z. Th us, it b elongs to Perf Z ( X ) . ✷ No w consider the case when Z is exactly the subset of singularities of X , i.e. Z = Sing( X ) . Prop osition 2.7. Any obje ct of D Sg ( X ) i s a dir e ct summand of a n o bje ct fr om its f u l l sub c at- e gory D b Sing( X ) (coh X ) / Perf Sing( X ) ( X ) . In p articular, idemp otent c omp letions of these c ate gories ar e e quivalent. 6 Pro of. Since an y ob ject of t he categ ory D Sg ( X ) is represen ted by a coherent sheaf up to shift ([12], Lemma 1.11) it is sufficien t to consider the coheren t sheaf F . Let us tak e the lo cally free resolution P · → F and consider the br utal tru ncation σ ≥− n P · for s ufficien tly large n > dim X. Denote b y G the (-n)-th cohomology of σ ≥− n P · . Let α : F → G [ n + 1] b e th e corresp onding m ap in D b (coh X ) . Its image in the category D Sg ( X ) is an isomorphism. On the other hand , consider the f unctor j ∗ : D b (coh X ) → D b (coh U ) , where U = X \ Sing( X ) . S ince U is sm o oth and n > dim U the image j ∗ ( α ) is zero. But the category D b (coh U ) is the quotien t of the category of D b (coh X ) by the sub category D b Sing( X ) (coh X ) . Hence the morphism α is factorized through an ob j ect A of D b Sing( X ) (coh X ) . Therefore, in the quot ien t c ategory D Sg ( X ) t he ob ject F is a dir ect summand of the image of the ob ject A in D Sg ( X ) . ✷ F or an y sc heme X w e denote b y X Z the formal completion of X along a closed subspace Z and denote by κ : X Z → X the canonical morphism. Let J b e an id eal sheaf suc h that Supp( O X / J ) = Z and let J b e a corresp onding ideal of definition of the form al n o etherian sc heme X Z . W e set Γ X ( F ) := lim − → n H om O X Z ( O X Z / J n , F ) , for an y quasi-coheren t sheaf F on X Z . Th is functor dep ends only on O X Z and do es not dep end on the ideal J . W e sa y that F ∈ Qcoh X Z is a torsion sheaf if Γ X ( F ) = F . W e d en ote by coh t X Z (resp. Qcoh t X Z ) the full s u b category of Q coh X Z whose ob jects are th e (quasi)-c oherent torsion shea ves. It is easy to see that under t he in v erse image functor κ ∗ a Z - torsion (quasi)- coheren t sheaf on X go es to a torsion (quasi)-coheren t sheaf o n X Z . Indeed, applying lim − → to the isomorph isms κ ∗ H om X ( O X / J n , F ) ∼ − → H om X Z ( O X / J n , κ ∗ F ) w e get a natural isomo rphism κ ∗ Γ Z ∼ = Γ X κ ∗ . Hence, w e obtain that κ ∗ (Qcoh Z X ) ⊂ Qco h t X Z and κ ∗ (coh Z X ) ⊂ co h t X Z . No w, if F is a Z -torsion coheren t s h eaf on X then there is an in teger n suc h that F comes from Z n = Sp ec O X / J n = ( X Z , O X Z / J n ) under the closed inclusion i n : Z n → X, i.e. F = i n ∗ F ′ for some coheren t s h eaf F ′ ∈ coh Z n . Consider the cartesian diagram Z n i n − − − − → X Z      y κ Z n i n − − − − → X W e hav e a sequence of isomorphisms κ ∗ κ ∗ F ∼ = κ ∗ κ ∗ i n ∗ F ′ ∼ = κ ∗ i n ∗ F ′ ∼ = i n ∗ F ′ ∼ = F . If no w F is a torsion sh eaf on X Z then again there is an intege r n suc h that F ∼ = i n ∗ F ′ and κ ∗ κ ∗ F ∼ = κ ∗ κ ∗ i n ∗ F ′ ∼ = κ ∗ i n ∗ F ′ ∼ = i n ∗ F ′ ∼ = F . T h us, we obtain that th e f u nctors κ ∗ and κ ∗ induce inv erse equiv alences b etw een the ab elian categ ories coh Z X and coh t X Z . It i s also can b e sho w n that the functor κ ∗ sends quasi-coheren t torsion sheaf to qu asi-coheren t Z -torsion shea v es, b ecause κ ∗ comm u tes with colimits (see [3], P r op. 5.1.1, 5.1.2). Thus, w e get the follo wing p rop osition. 7 Prop osition 2.8. ([3]) L et X b e a no etherian scheme and κ : X Z → X b e a formal c ompletion of X along a close d subsp ac e Z . Then the fu nc tors κ ∗ and κ ∗ induc e inverse e quivalenc es b etwe en the c ate gories coh Z X and coh t X Z , and b etwe en the c ate gories Qcoh Z X and Qcoh t X Z . Corollary 2.9. L et X and X ′ b e two schemes satisfying (ELF). Assume that the formal schemes X Z and X ′ Z ′ ar e isomorphic. Then the derive d c ate gories D b Z (coh X ) and D b Z ′ (coh X ′ ) ( r esp. D b Z (Qcoh X ) and D b Z ′ (Qcoh X ′ ) ) ar e e qui v alent. Pro of. By Lemma 2.1 there is an equiv alence D b Z (coh X ) ∼ = D b (coh Z X ) (resp. D b Z (Qcoh X ) ∼ = D b (Qcoh Z X ) b y and by Prop osition 2.8 w e ha ve coh Z X ∼ = coh t X Z (resp. Qcoh Z X ∼ = Qcoh t X Z ). Since X Z ∼ = X ′ Z ′ w e obtain that coh Z X ∼ = coh Z ′ X ′ (resp. Qcoh Z X ∼ = Qcoh Z ′ X ′ ). Therefore, the deriv ed categ ories are equiv alen t as w ell. ✷ Theorem 2.10. L et X and X ′ b e two schemes satisfying (ELF). Assume that the formal c om- pletions X and X ′ along si ng u larities ar e isomorphic. Then th e idemp otent c omp letions of the triangulate d c ate gories of singularities D Sg ( X ) and D Sg ( X ′ ) ar e e quivalent. Pro of. By C orollary 2.9 there is an equiv alence b et w een D b Sing( X ) (coh X ) and D b Sing( X ′ ) (coh X ′ ) . By L emm a 2.4 th e sub catego ries Perf ( X ) and Perf ( X ′ ) are also equiv alen t, b ecause they can b e defined in th e internal terms of the boun ded deriv ed cate gories of coheren t shea v es with supp ort on Z and Z ′ . Hence, there is an equ iv alence b et w een qu otien t categories D b Sing( X ) (coh X ) / Pe rf Sing ( X ) ( X ) ∼ − → D b Sing( X ′ ) (coh X ′ ) / Perf Sing ( X ′ ) ( X ′ ) , It indu ces an equiv alence b et w een their idemp oten t completions, whic h by Proposition 2.7 coincide with the idemp otent completions of the triangulated catego ries of singularities. ✷ 3. Local iza tion in Nisnevich topology and isomorphisms i nfinitel y near singularities Let X b e a no etherin sc heme and i : Z ֒ → X b e a closed su bsc heme. Consider the pair ( Z, X ) . Let f : X ′ → X b e a map of schemes. Definition 3.1. We say that f is an isomorph ism infinitely ne ar Z if it is flat over Z and the fib er pr o duct Z ′ = Z × X X ′ is isomorphic to Z . It ca n b e p ro v ed that this condition do es n ot dep end on a c hoice of a closed subscheme with the underline subs pace Supp Z ([17], L emma 2.6.2.2 ). In p articular, if it h olds for Z = Sp ec O X / J it also holds for infinitesimal thic k enings Z n := Sp ec O X / J n . Th us, we ma y sa y that f is an isomorphism in finitely near th e closed s u bspace Supp Z. This imp lies that an y morphism f : X ′ → X, which is an isomorphism infinitely near Z , induces an isomorph ism b et w een the formal completions f : X ′ Z ′ ∼ → X Z . Hence, by Corollary 2.9 w e obtain that the deriv ed categories D b Z (coh X ) and D b Z ′ (coh X ′ ) are equiv alen t for an y morphism f : X ′ → X that is an isomorphism infi nitely near Z (see [17], Th. 2.6.3). Imp ortant examples of such morphisms are Nisnevic h n eigh b orho o ds of Z in X. 8 Definition 3.2. A n X -scheme π : Y → X is c al le d a Nisnevich neighb orho o d of Z in X if the morphism π i s eta le and th e fib er pr o duct Z × X Y is isomorphic to Z. Prop osition 3.3. L et a scheme X satisfy (ELF) and let Z ⊂ X b e a c lose d subscheme. Then for any mor phism f : X ′ → X , that is an isomorphism infinitely ne ar of Z, the f u nctors f ∗ : D Z (Qcoh X ) → D Z (Qcoh X ′ ) and f ∗ : D b Z (coh X ) → D b Z (coh X ′ ) ar e e quivalenc es. Pro of. The morphism f : X ′ → X induces an isomorphism b etw een the formal completions f : X ′ Z ′ ∼ → X Z ([17], Lemma 2.6.2.2 ). Hence, b y Corollary 2.9 we obtain that the deriv ed categories D b Z (coh X ) and D b Z (coh X ′ ) (resp. D Z (Qcoh X ) an d D Z (Qcoh X ′ ) ) are equiv alen t. ✷ Prop osition 3.4. L et schem es X and X ′ satisfy (ELF) an d let f : X ′ → X b e a morph ism that is an isomorphism infinitely ne ar Z . Supp ose the c omplements X \ Z and X ′ \ Z ar e smo oth. Then the func tor ¯ f ∗ : D Sg ( X ) → D Sg ( X ′ ) is ful ly faithful and, mor e over, any obje ct B ∈ D Sg ( X ′ ) is a dir e ct summand of a som e obje ct of the for m ¯ f ∗ A. Pro of. By assu m ption Sing( X ) ∼ = Sing( X ′ ) ⊆ Z , hence, f is an isomorphism infinitely near Sing( X ) . By Pr op osition 3.3 and Lemma 2.4 we obtain an equiv alence D b Sing( X ) (coh X ) / Pe rf Sing ( X ) ( X ) ∼ − → D b Sing( X ′ ) (coh X ′ ) / Perf Sing ( X ′ ) ( X ′ ) . By Prop osition 2.7 their idemp oten t completions coincide w ith idemp oten t completions of trian- gulated categories o f s ingularities. Hence, the n atural fun ctor ¯ f ∗ is fully f aithful and a ny ob ject D Sg ( X ′ ) is a direct su m mand of an ob ject from D Sg ( X ) . ✷ Corollary 3.5. L et a scheme X satisfy (ELF) and the c omplement X \ Z i s smo oth. Then for any Ni snevich neighb orho o d π : Y → X o f Z the fu nctor ¯ π ∗ : D Sg ( X ) → D Sg ( Y ) is ful ly faithful and, mor e over, any obje ct B ∈ D Sg ( Y ) is a dir e ct summand of a some o bje ct o f the for m ¯ π ∗ A. Remark 3.6. Let ( A, p ) b e a pair consisting of a comm utativ e k -alg ebra of finite type A and a prime ideal p. Consider the henselization ( A h , p h ) of this pair. By defin ition, A h = lim − → B , p h = pA h , wh ere the limit is taking by the category of all Nisnevic h neighborh o o ds Sp ec B → Sp ec A of Spec A/p in Sp ec A. In particular, w e ha v e A/p ∼ → A h /p h . Let ˆ A b e the p -adic completion of A. By one of a pplication o f Artin Approxi mation (Theorem 3.10 , [1]) for an y finitely gener- ated ˆ A -mo dule ¯ M , whic h is locally free on Spe c ˆ A outside V ( ˆ p ) , there is an A h -mo dule M suc h that ˆ M ∼ = ¯ M . Assume n o w that Sp e c A is regular outside V ( p ) . This imp lies that the natural functor from D Sg ( Sp ec A h ) to D Sg ( Sp ec ˆ A ) is an equiv alence, b ecause an y ob ject of a triangulated ca tegory of singularities can b e represen ted by a coheren t sheaf whic h is lo cally free on the complemen t to the singularities. If moreo ver, K − 1 ( A h ) = 0 then the triangulated category D Sg ( Sp ec A h ) is idemp oten t complete (see n ext section), i.e. it coincides with D Sg ( Sp ec A ) . F or example, it is tru e when A is a normal lo cal ring of d imension tw o of essen tially fin ite t yp e [19]. 9 4. Thomas on th eorem an d gr ou ps K − 1 A full triangulated sub catego ry N of a triangulated c ategory T is called dense in T if eac h ob ject of T is a direct summand of an ob ject iso morphic to an ob ject in N . There is a not so w ell-kno wn bu t amazing theorem of R. Thomason whic h allo ws us to describ e all strictly full dense sub catego ries in a triangulated category . Theorem 4.1. (R. Th omason, [16]) L et T b e a n essential ly smal l triangulate d c ate gory. Then ther e is a on e-to-one c orr esp ond enc e b etwe en the strictly ful l dense triangulate d sub c ate gories N in T and the sub gr oups H of the Gr othendie ck gr oup K 0 ( T ) . T o N c orr esp onds the sub gr oup wh ich is the i mage of K 0 ( N ) in K 0 ( T ) . T o H c orr esp onds the ful l sub c ate gory N H whose obje cts a r e those N in T such that [ N ] ∈ H ⊂ K 0 ( T ) . Remark 4.2. Recall that a full triangulated su b category N of T is called stri ctly full if it con tains every ob ject of T that is isomorphic to an ob ject of N . Th us, to an y sc heme X we can attac h the triangulat ed category D Sg ( X ) and t wo subgroups in the Grothendiec k group K 0 ( D Sg ( X ) ) whic h are r elated to the natural dense sub catego ries D Sg ( X ) and D b Sing( X ) (coh X ) / Pe rf Sing ( X ) ( X ) and w hic h by Thomason’s theorem uniqu ely determine them. The sequence of triangulated categories (1) Perf ( X ) − → D b (coh X ) − → D Sg ( X ) is exact in De finition 1 .1 of [14], i.e. the first fun ctor is a f u ll em b eddin g and the quotien t o f th is map is dense su b category in the third categ ory . F ollo win g Amn on Neeman [11] this exact sequence can b e considered as an exact sequence of tri- angulated cate gories of compact ob jects coming f r om a lo calizing sequence of compactly generated triangulated cate gories. As w e know the categ ory of p er f ect complexes Perf ( X ) is the category of compact ob jects in D (Qcoh X ) . It is pr o v ed b y H. Kr ause [10] that the category D b (coh X ) can b e considered as the category of compact ob j ect in the homotop y category of injectiv e quasi-coheren t shea v es H (Inj X ) for a no etherian sc h eme X. On the other h and, the derived category D (Qcoh X ) is equiv alen t to the fu ll sub category H inj (Qcoh X ) ⊂ H (Inj X ) of h-injectiv e complexes, i.e. such complexes I that Hom H (Qcoh X ) ( A, I ) = 0 f or all acyclic complexes A from homotop y category H (Qcoh X ) (see [15]). Since X is no etherian the category H inj (Qcoh X ) closed with resp ect to formation of copro ducts and , furthermore, the inclusion fu nctor H inj (Qcoh X ) ֒ → H (Inj X ) r e- sp ects copro d ucts. Th is means that H inj (Qcoh X ) is lo calizing sub category of H (In j X ) and we ha v e a lo calizing s equ ence H inj (Qcoh X ) − → H (Inj X ) Q − → H (Inj X ) / H inj (Qcoh X ) . Moreo ver, the quotien t fu nctor Q has a righ t adjoint , which is called Bousfield lo calizing functor. It iden tifies the quotient ca tegory H (Inj X ) / H inj (Qcoh X ) with th e triangulated category of all acyclic complexes of injectiv e ob j ects Inj X ⊂ Qco h X . The latter cate gory is called stable deriv ed 10 catego ry and will b e denoted b y S (Qcoh X ) . By Theorem 2.1 of [11] the idemp oten t comple- tion D Sg ( X ) is equiv alen t to the cat egory of all compact ob jects in the stable deriv ed category S (Qcoh X ) (for more details see [10]). By Theorem 11.10 of [14 ] the sequence (1) in duces a long exact sequ ence for K-groups K 0 ( Perf ( X )) − → K 0 ( D b (coh X )) − → K 0 ( D Sg ( X ) ) − → K − 1 ( Perf ( X )) − → 0 . Here we used Theorem 9.1 f rom [14] assertin g that K − 1 for a sm all ab elian ca tegory is trivial . Therefore, w e obtain a short exact sequence 0 − → K 0 ( D Sg ( X )) − → K 0 ( D Sg ( X ) ) − → K − 1 ( Perf ( X )) − → 0 , whic h sho w s that K − 1 ( Perf ( X )) is a measure of the difference b et w een D Sg ( X ) and its idemp o- ten t completion D Sg ( X ) . By the same r eason, w e h a ve another short exact s equence 0 − → K 0 ( D b Sing( X ) (coh X ) / Perf Sing( X ) ( X )) − → K 0 ( D Sg ( X ) ) − → K − 1 ( Perf Sing( X ) ( X )) − → 0 . No w a long exact sequence for U = X \ Sin g( X ) K 0 ( Perf ( X )) → K 0 ( Perf ( U )) → K − 1 ( Perf Sing( X ) ( X )) → K − 1 ( Perf ( X )) → 0 , whic h follo ws from the Thomason’s Lo calization Theorem 7.4 [17], sho ws a difference b et w een K − 1 ( Perf Sing( X ) ( X )) and K − 1 ( Perf ( X )) . The negativ e K-groups (which is du e to Bass) are defi ned from the follo wing exact sequences 0 → K i ( Perf ( X )) → K i ( Perf ( X [ t ])) ⊕ K i ( Perf ( X [ t − 1 ])) → K i ( Perf ( X [ t, t − 1 ])) → → K i − 1 ( Perf ( X )) → 0 . In particular, th e group K − 1 ( Perf ( X )) is isomorp hic to the cok ernel of the canonical map K 0 ( Perf ( X [ t ])) ⊕ K 0 ( Perf ( X [ t − 1 ])) → K 0 ( Perf ( X [ t, t − 1 ])) . By T h eorem 2.10 w e kno w that for an y t wo sc hemes X and X ′ , the formal completi ons of whic h along singularities are isomorphic, we ha v e D Sg ( X ) ∼ = D Sg ( X ′ ) and D b Sing( X ) (coh X ) / Pe rf Sing ( X ) ( X ) ∼ = D b Sing( X ′ ) (coh X ′ ) / Perf Sing ( X ′ ) ( X ′ ) . On the other h and, in this case the triangulated categories of singu larities D Sg ( X ) and D Sg ( X ′ ) are not necessary equiv alent as we kno w. There is also another t yp e of r elations b et ween sc hemes which giv e equiv alences for triangulated catego ries of singularitie s but under whic h the quotien t categories D b Sing( X ) (coh X ) / Perf Sing( X ) ( X ) are not necessary equiv alent . It is describ ed in [13]. Let S b e a n o etherian regular sc heme. Let E b e a ve ctor b undle on S of r an k r and let s ∈ H 0 ( S, E ) b e a section. Denote b y X ⊂ S the zero su bsc heme of s. Assume that the section s is regular, i.e. the co d im en sion of the su bsc heme X in S coincides with the rank r. Consider the pro jectiv e bu ndles S ′ = P ( E ∨ ) and T = P ( E ∨ | X ) , where E ∨ is the dual bun dle. 11 The section s indu ces a sectio n s ′ ∈ H 0 ( S ′ , O E (1)) of the Grothendiec k line bundle O E (1) on S ′ . Denote by Y the divisor on S ′ defined b y the sec tion s ′ . The natural closed em b edding of T in to S ′ go es through Y . All sc hemes defined abov e can b e included in the follo wing comm u tativ e diagram. T p   i / / Y π @ @ @ @ @ @ @ @ u / / S ′ q   X j / / S Consider the comp osition fu nctor R i ∗ p ∗ : D b (coh X ) → D b (coh Y ) and d enote it b y Φ T . Theorem 4.3. ([13]) L et schemes X, Y , and T b e as ab ove. Then the functor Φ T : D b (coh( X )) − → D b (coh( Y )) define d by the form ula Φ T ( · ) = R i ∗ p ∗ ( · ) induc es a functo r Φ T : D Sg ( X ) − → D Sg ( Y ) , which is an e quivalenc e of triangulate d c ate gories. The functor Φ T = R i ∗ p ∗ has a right adjoint fun ctor wh ic h we denote by Φ T ∗ . It can b e represent ed as a comp osition R p ∗ i ♭ , where i ♭ is r ight adjoint to R i ∗ . F un ctor i ♭ has th e form L i ∗ ( · ) ⊗ ω T / Y [ − r + 1] , where ω T / Y ∼ = Λ r − 1 N T / Y is the relativ e d ualizing sh eaf. It is easy to see that a ll singularities o f Y are concen trated o v er the sin gu larities o f X, hence the functor Φ T ∗ = R p ∗ i ♭ sends the sub category D b Sing( Y ) (coh Y ) to the sub category D b Sing( X ) (coh X ) . Therefore, w e obtain the follo w ing corollary . Corollary 4.4. Th e functor Φ T ∗ , wh ich r e alizes an e quivalenc e b etwe en the triangulate d c ate gories of singularities of Y and X , gives also a f u nctor D b Sing( Y ) (coh Y ) / Perf Sing( Y ) ( Y ) − → D b Sing( X ) (coh X ) / Perf Sing ( X ) ( X ) , and this fu nctor is fu l ly faithful. Note that the functor Φ T ∗ : D b Sing( Y ) (coh Y ) / Perf Sing ( Y ) ( Y ) → D b Sing( X ) (coh X ) / Perf Sing( X ) ( X ) is n ot an equiv alence in general. A cknowledgments I am grateful to Denis Aur oux, Ludmil K atzark ov, Ant on Kapustin, J´ anos Koll´ ar, Amnon Nee- man, T on y P an tev, L eo Alonso T arr ´ ıo, and An a Jerem ´ ıas L´ op ez for v ery u seful discussions. 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Algebra Section, S teklo v Ma thema tical Institute R AN, Gubkin str. 8, Mosco w 119991, R USSIA E-mail addr ess : orlov@mi.ra s.ru