Maximal exact structures on additive categories revisited

MAXIMAL EXA CT STR UCTURES ON ADDITIVE C A TEGORIES REVISITED SEPTIMIU CRIVEI Abstra ct. Sieg and W egner sho w ed that the stable e xact sequences define a maximal exact structure (in the sense of Quillen) in any pre-abelian category [14]. W e generalize this result for w eakly idemp otent complete additive categories. 1. Introduction Sev eral notions of exact categories ha v e b een defined in the literature, see Barr [2], Heller [4], Quillen [9] or Y oneda [16]. They p ro vide a suitable setting for dev elopping a relat iv e homologic al algebra, and ha v e imp ortan t applications in differen t fields suc h as algebraic geometry , algebraic and functional analysis, algebraic K -theory etc. (e.g., see [3] f or fu rther details). W e s h all consider here the concept of exact add itiv e category defined by Quillen [9] and refined b y Keller [6]. In an y additiv e category , the class of all split exact sequ ences defines an exact structure, and this is the smallest exact str u cture. On the other hand, the other extreme, namely the class of all kernel-co k ernel pairs, defin es an exact stru cture p ro vided the category is quasi-ab elian [11], b ut f ails to d efine an exact str ucture in arbitrary additive categories (see the example in [12]). Recen tly , S ieg a nd W egner [14] sho w ed t hat the stable exact sequences in t he sense of [10] define a maxima l exact structure in an y pre-ab elian category , i.e. an additive catego ry with k ernels and cok ernels. W e shall generalize this result to weakl y idemp oten t complete additiv e catego ries, i.e. additiv e catego ries in whic h ev ery section has a cok ernel, or equiv alentl y , every retraction has a k ernel (e.g., see [3]). Clearly , ev ery pr e-ab elian category is w eakly idemp oten t complete additiv e. But there are significant examples of wea kly id emp oten t complete catego ries whic h are not pre-ab elian. F or instance, usin g the terminology from [8], any fin itely accessible additiv e cate gory wh ic h is n ot locally finitely presen ted is w eakly id emp otent complete, but not p re-ab elian (see Example 2.1 b elo w). Let u s also p oin t ou t that th e assu mption on the additiv e catego ry to b e w eakly idemp oten t complete is rather mild . This is b ecause ev ery additiv e category has an idemp oten t-splitting completion, also called Karoubian completion (see [5, p. 75]), which in turn is w eakly idemp oten t complete [3]. 2. Preliminaries Throughout the pap er we shall use the s etting of an add itiv e categ ory C . In this section we giv e examples of w eakly idemp oten t complete additiv e categories whic h are not pre-ab elian, and w e in tro duce the needed terminology . 2.1. Examples. F ollo wing th e terminology fr om [8], an additiv e category C is called finitely ac c e ssible if it has direct limits, the class of fi nitely pr esen ted ob jects is sk eletally small, and ev ery ob ject is a direct limit of finitely presen ted ob jects. Also, C is calle d lo c al ly finitely pr esente d if it is fin itely accessible and co complete (i.e., it has all colimits), or equ iv alen tly , it is finitely accessible and complete (i.e., it h as all limits). Date : December 30, 201 0. 2000 Mathematics Subje ct Classific ation. 18E10, 18G50. Key wor ds and phr ases . Additive categ ory , exact category , w eakly idemp otent complete category , p re-ab elian category , stable exact sequen ce. This work was sup p orted by the Romanian grant PN-I I-ID -PCE-2008-2 pro ject ID 2271. I would like to thank Silva na Bazzoni for inspiring d iscussions on exact categories and kind hospitality during my visit at t h e Department of Mathematics of Univers it´ a di Pado v a in F ebruary-March 2010. Also, I w ould lik e to thank Dennis Sieg and Sv en-Ake W egner for pro viding their paper [14]. 1 2 SEPTIMIU CRIVEI Example 2.1. (1) Let C b e a finitely accessible add itive categ ory whic h is not lo cally finitely present ed. F or instance, tak e the category of flat righ t m o dules o v er a ring whic h is not left coheren t (see [8]). Then C is weakly idemp oten t complete, bu t not pre-ab elian. In deed, s ince C is finitely accessible, it h as split idemp otents [1, 2.4], and so it is weakly idemp oten t complete [3]. On the other hand , a fi nitely accessible category is lo cally finitely p resen ted if and only if it has coke rnels [8, Corollary 3.7]. Hence C is not pr e-ab elian. (2) An y triangulated category is w eakly idemp otent complete additiv e, and its maximal exact structure is the trivial one. Hence it is pre-ab elian if and only if it is semi-simple (in the sense that every morp hism factors into a retraction follo w ed by a section). (3) An y non-ab elian category of fi nitely present ed mo d ules ov er a ring is w eakly id emp otent complete additiv e, b ut not pre-ab elian. It has cok ernels, h ence split idemp oten ts, bu t not enough k ernels. 2.2. Pullbac ks. W e sh all need the follo wing t w o results on pullbac ks, whose duals for pushouts hold as well. Lemma 2.2. [7, Lemma 5.1] Consider the fol lowing diagr am in which the squar es ar e c ommu- tative and the right squar e is a pul lb ack: A ′ f   i ′ / / B ′ g   d ′ / / C ′ h   A i / / B d / / C Then the left squar e is a pul lb ack if and only if so is the r e ctangle. Lemma 2.3. [10, Theorem 5] L et d : B → C and h : C ′ → C b e morphisms such that d has a kernel i : A → B , and the pul lb ack of d and h exists. Then ther e is a c ommutat ive diagr am A i ′ / / B ′ g   d ′ / / C ′ h   A i / / B d / / C in which the right squar e is a pul lb ack and i ′ : A → B ′ is the kernel of d ′ . 2.3. Stable exact sequences. T he follo wing sp ecial kernels and coke rnels w ill b e of fund a- men tal imp ortance for our topic. W e extend their definition from the setting of pr e-ab elian catego ries, as given in [10], to arbitrary additiv e categories. Definition 2.4. A cok ernel d : B → C is called a semi-stable c okernel if the pu llbac k of d along an arbitrary morp hism h : C ′ → C exists and is again a cok ernel, i.e. there is a pu llbac k square B ′ g   d ′ / / C ′ h   B d / / C with th e morphism d ′ : B ′ → C ′ a cokernel. The n otion of semi-stable kernel is defin ed dually . A short exact sequence, i.e. a k ernel-cok ernel pair, A i → B d → C is called stable if i is a semi-stable k ernel and d is a semi-stable cok ernel. Let u s note some useful remarks, whose dual v ersions hold as well. Remark 2.5. (i) Ev ery semi-stable cok ernel d : B → C has a k ernel (namely , its pullbac k along the morphism 0 → C ). Hence ev ery semi-stable co k ernel is the coke rnel of its kernel (e.g., by the dual of [15, Chapter IV, Prop osition 2.4], whose pro of wo rks in arbitrary additiv e categories). (ii) The p u llbac k of a semi-stable coke rnel along an arbitrary morphism exists and is again a semi-stable cok ernel by L emma 2.2. (iii) Every isomorph ism is a semi-stable cok ernel. MAXIMAL EXACT STRUCTURES ON ADDITIVE CA TEGORIES REVISITED 3 2.4. Exact categories. W e sh all consider the f ollo w ing concept of exact category giv en b y Quillen [9 ] and refi n ed by Keller [6]. Definition 2.6. By an e xact c ate g ory w e mean an add itive catego ry C en d o w ed with a d is- tinguished class E of short exact sequences satisfying the axioms [ E 0], [ E 1], [ E 2] and [ E 2 op ] b elo w. The s hort exact s equ ences in E are called c onflatio ns , wh ereas the kernels and cok ernels app earing in suc h exact sequences are called i nflations and deflations r esp ectiv ely . [ E 0] T he id en tit y morphism 1 0 : 0 → 0 is a defl ation. [ E 1] T he comp osition of t w o d efl ations is again a defl ation. [ E 2] T he pullb ac k of a deflation along an arbitrary m orphism exists and is again a d eflation. [ E 2 op ] The pushout of an inflation along an arbitrary morphism exists and is again an inflation. Note th at the duals of th e axioms [ E 0] and [ E 1] hold as well (see [6 ]). Some examples of exact catego ries are the follo wing. Example 2.7. (1) It is w ell-kno wn that in an y additiv e categ ory th e split sh ort exact sequences define an exact str ucture, and th is is the minimal one. (2) Recall that an additive catego ry is called quasi-ab elian if it is pr e- ab elian (i.e. it has k ernels and cok ernels), the pu s hout of any kernel along an arbitrary m orphism is a k ernel, and the pullbac k of an y cok ernel along an arbitrary morphism is a cok ernel. In an y quasi-ab elian catego ry the sh ort exact sequen ces d efine an exact structure, and this is the maximal one [11 ]. (3) In any pre-ab elian category the stable exact s equences define an exact stru cture, and th is is the maximal one [14]. 3. The maximal e x act structure In this s ection we s hall extend the main result of [14] from pre-ab elian categories to w eakly idemp otent complete additiv e categories. W e shall state and p ro v e some essen tial results on semi-stable cok ernels. Note that their d ual v ersions for semi-stable k ernels hold as well. The setting will b e th at of an add itive category C , if not sp ecified otherwise. The follo wing resu lt is mo d elled after [10, Th eorem 2]. W e include a p ro of for completeness. Prop osition 3.1. The c omp osition of two semi- stable c okernels is a semi-stable c okernel. Pr o of. Let d : B → C and p : C → D b e semi-stable cok ernels. Then d = Cok er( i ) and p = C ok er( h ), where i = Ker( d ) : A → B and h = Ker( p ) : C ′ → C . F orming the pullbac k of d and h , by Lemma 2.3 we ha v e the f ollo wing d iagram with commutativ e squares: A i ′ / / B ′ g   d ′ / / C ′ h   A i / / B pd   d / / C p   D D in w hic h d ′ is a cok ernel. W e claim that pd = Cok er ( g ). Let u : B → E b e a morp hism su c h that ug = 0. Since ui = 0 and d = Coke r( i ), there is a uniqu e morphism v : C → E suc h that v d = u . Since v hd ′ = 0 and d ′ is an epimorp hism, w e h av e v h = 0. But p = Coker( h ), and so ther e is a uniqu e morphism w : D → E suc h that w p = v . Now w e ha v e wp d = u . The fact th at p is an epimorphism ensur es the uniqueness of su c h a morp hism w with w p = v . Therefore, pd = Cok er ( g ). In order to get the pullback of an arbitrary morphism k : F → D and pd : B → D , construct the pu llbac k of k and p : C → D , and th en the p ullbac k of the resulting m orphism and d : B → C . Both of them yield semi-stable cok ernels by Remark 2.5. No w the p ullbac k of pd along an arbitrary morph ism exists by L emm a 2.2, and it is the resulting rectangle. Moreo ver, by the first part of the pr o of, it is a cok ernel as the comp osition of t w o semi-stable cok ernels.  Lemma 3.2. The dir e ct sum of two semi-stable c okernels is a semi-stable c okernel. 4 SEPTIMIU CRIVEI Pr o of. Let d : B → C and d ′ : B ′ → C ′ b e semi-stable coke rnels. Consider the pu llbac k square B ⊕ B ′ [ 1 0 ]   h d 0 0 1 i / / C ⊕ B ′ [ 1 0 ]   B d / / C Then  d 0 0 1  : B ⊕ B ′ → C ⊕ B ′ is a semi-stable cok ern el by Remark 2.5. Similarly ,  1 0 0 d ′  : C ⊕ B ′ → C ⊕ C ′ is a semi-stable cok ernel. Therefore, their comp osition  d 0 0 d ′  : B ⊕ B ′ → C ⊕ C ′ , that is d ⊕ d ′ , is a semi-stable cok ernel by Pr op osition 3.1.  Corollary 3.3. Every pr oje ction onto a dir e ct su mmand is a semi-stable c okernel. Pr o of. Consider a p ro jection [ 0 1 ] : B ⊕ D − → D . By a d iagram as in the pr o of of Lemm a 3.2 with C = 0, it f ollo ws that there exists the pullbac k of the cok ern el B → 0 and any morp hism B ′ → 0. Moreo v er, the resulting morphism [ 0 1 ] : B ⊕ B ′ → B ′ is a cok ernel, as the d irect s u m of the cok ernels B → 0 and 1 B ′ . Hence B → 0 is a semi-stable coke rnel. T o conclude, note that the morphism B ⊕ D [ 0 1 ] − → D is the direct su m of the semi-stable coke rnels B → 0 and 1 D , and use Lemm a 3.2.  Recall that an additiv e cate gory is called we akly idemp otent c ompl ete if ev ery retraction has a k ernel (equiv alen tly , ev ery section has a cok ern el) (e.g., see [3]). Th e next result will b e a key step in the proof of our main theorem. It generalizes [7, Prop osition 5.12] and [13, Prop osition 1.1.8 ]. Prop osition 3.4. L et C b e we akly idemp otent c omplete. L et d : B → C and p : C → D b e morphism s such that pd : B → D is a semi- stable c okernel. Then p is a semi-stable c okernel. Pr o of. W e fir s t show that p h as a k ernel. S ince pd is a s emi-stable cok ernel, there is a pu llbac k square L t / / q   C p   B pd / / D The pullback prop ert y implies the existence of a m orphism r : B → L suc h th at tr = d and q r = 1 B . Hence q is a retraction, and th us, b y assumption, it h as a kernel l : C ′ → L . Using again the p ullbac k prop er ty , it follo ws easily that h = t l : C ′ → C is the kernel of p . No w let g : B ′ → B b e the k ernel of pd . Since pd is a semi-stable cok ernel, we ha v e pd = Cok er( g ). W e obtain the follo wing comm utativ e left diagram: B ′ d ′ / / g   C ′ h   B d / / pd   C p   D D B ⊕ C ′ [ d h ] / / [ 1 0 ]   C p   B pd / / D W e claim that the right diagram is a pullbac k. T o this end, let α : E → C and β : E → B b e morphisms such that pα = pdβ . Since p ( dβ − α ) = 0 and h = K er( p ), th er e is a un ique morphism δ : E → C ′ suc h th at dβ − α = hδ . Then it is easy to c hec k th at h β − δ i is the uniqu e morphism [ u v ] : E → B ⊕ C ′ suc h that [ d h ] [ u v ] = α and [ 1 0 ] [ u v ] = β , and so the square is a pullbac k. No w [ d h ] is a coke rnel, b ecause pd is a semi-stable cok ernel. W e claim that p = Cok er( h ). Let w : C → F b e a morphism suc h that w h = 0. S in ce wdg = 0 an d pd = C ok er( g ), there is a morphism t : D → F s u c h th at tpd = w d . It follo ws that ( tp − w ) [ d h ] = 0, whence we ha v e tp = w , b ecause [ d h ] is an epimorp h ism. S ince p is an epimorp hism, we ha v e th e un iqu eness of th e morph ism t : D → F suc h that tp = w . Hence p = Coker( h ). MAXIMAL EXACT STRUCTURES ON ADDITIVE CA TEGORIES REVISITED 5 No w let c : G → D b e a morph ism. W e shall show that there exists the pu llb ac k of p and c . W e may write [ p 0 ] as the comp osition of the follo wing morph ism s: C ⊕ B h 1 − d 0 1 i / / C ⊕ B h 1 0 0 pd i / / C ⊕ D h 1 0 p 1 i / / C ⊕ D [ 0 1 ] / / D The first and the third m orphisms are isomorphisms, and so they are semi-stable co k ernels. The second morphism is a semi-stable cok ern el by Lemma 3.2. The last morph ism is a semi-stable cok ernel b y Corollary 3.3. Therefore, their comp osition [ p 0 ] is also a semi-stable cok ernel by Prop osition 3.1. Hence [ p 0 ] and c hav e a pullb ac k squ are as follo ws: Y γ / / h α ′ β ′ i   G c   C ⊕ B [ p 0 ] / / D Consider the m orp hism [ 0 1 ] : B → C ⊕ B . S ince [ p 0 ] [ 0 1 ] = 0 = c 0, by the pu llbac k prop ert y there is a un ique morphism δ : B → Y suc h that h α ′ β ′ i δ = [ 0 1 ] and γ δ = 0. In particular, β ′ δ = 1 B , and so β ′ is a retraction. Since C is we akly idemp otent complete, β ′ has a k ernel, sa y i : K → Y . Let u s show now that th e follo wing squ are K γ i / / α ′ i   G c   C p / / D is a pullbac k of p and c . T o this end, let a : E ′ → C and b : E ′ → G b e morphisms suc h that pa = cb . T hen [ p 0 ] [ a 0 ] = cb , hence the p ullbac k square of [ p 0 ] and c imp lies the existence of a uniqu e morp hism v ′ : E ′ → Y such that h α ′ β ′ i v ′ = [ a 0 ] and γ v ′ = b . Since β ′ v ′ = 0 and i = Ker( β ′ ), there is a u nique m orp hism w ′ : E ′ → K such that v ′ = iw ′ . Then we ha v e α ′ iw ′ = α ′ v ′ = a and γ iw ′ = γ v ′ = b . Let us sho w th at the morp hism w ′ : E → K is unique with these prop erties. Su p p ose that there is another m orphism w ′′ : E ′ → K suc h that α ′ iw ′′ = a and γ iw ′′ = b . It follo ws that h α ′ β ′ i ( iw ′ − iw ′′ ) = [ 0 0 ] and γ ( iw ′ − iw ′′ ) = 0. Then w e h a v e iw ′ − iw ′′ = 0 b y the pullbac k p rop erty of [ p 0 ] and c , and so w ′ = w ′′ , b ecause i is a monomorphism. No w consid er the p ullbac k of pd and c , sa y K ′ γ ′ / / α ′′   G c   B pd / / D The pullbac k prop erty of p and c implies the factorization of γ ′ through γ i . Since pd is a semi-stable cok ernel, so is γ ′ . Moreo v er, b y Lemma 2.3 γ i h as a k ernel, b ecause p has a kernel. Then γ i must b e a cok ern el b y an argument similar to the fi rst part of the pro of. Hence p is a semi-stable cok ernel.  No w we are in a p osition to pro v e our m ain resu lt, whic h generalizes [14, Theorem 3.3]. Ha ving prepared the setting, w e shall follo w a sim ilar path as in the cited result, slightly simplifying the pro of of axiom [ E 1]. Theorem 3.5. L e t C b e a we akly idemp otent c omp lete additive c ate gory. Then the stable exact se quenc es define an exact structur e on C . Mor e over, this is the maximal exact structur e on C . Pr o of. [ E 0] This is clear. [ E 2] Let A i → B d → C b e a stable exact sequence, and let h : C ′ → C b e a morph ism. Sin ce d is a semi-stable cok ernel, we ma y consider the pu llbac k of d and h , and by Lemma 2.3 w e ha v e 6 SEPTIMIU CRIVEI a commutativ e diagram A i ′ / / B ′ g   d ′ / / C ′ h   A i / / B d / / C in wh ic h i ′ = Ker( d ′ ) and d ′ is a s emi-stable cok ernel, and so the u pp er row is a sh ort exact sequence. S ince i = g i ′ is a semi-stable kernel, i ′ is a semi-stable k ernel b y the d ual of Pr op osition 3.4. [ E 2 op ] Dual to [ E 2]. [ E 1] Let A i → B d → C and A ′ i ′ → C d ′ → D b e stable exact sequ ences. By P rop osition 3.1, d ′ d : B → D is a s emi-stable cok ernel. W e shall show that its k ernel, say j : K → B , is semi- stable. Since d ′ d j = 0, there is a un ique morph ism p : K → A ′ suc h th at d j = i ′ p . W e also ha v e d ′ d = Coke r( j ). Note that we hav e the follo wing equalit y:  d 1  j =  i ′ 0 0 1   p j  The m orphism  i ′ 0 0 1  : A ′ ⊕ B → C ⊕ B is a semi-stable kernel by Lemma 3.2. Consequ ently , if w e pr ov e that  p j  : K → A ′ ⊕ B is also a semi-stable kernel, then j will b e a semi-stable ke rnel b y the d uals of Prop ositions 3.1 and 3.4, and we are done. W e claim fir s t that there is a commutativ e diagram: A k / / K j   p / / A ′ i ′   A i / / B d / / C in wh ich the fi rst row is a stable exact sequence. W e s h all sho w that the righ t s q u are is a pullbac k. T o this end, let α : E → A ′ and β : E → B b e morphisms su ch that i ′ α = dβ . Since d ′ dβ = 0 and j = Ker( d ′ d ), there is a uniqu e morph ism γ : E → K such that j γ = β . W e hav e i ′ pγ = d j γ = dβ = i ′ α , whence pγ = α , b ecause i ′ is a monomorph ism. Moreo v er, it is easy to see th at γ is the u nique m orp hism with th e r equired prop erties of the pullbac k. No w the existence of the r equired comm utativ e diagram follo ws by Lemma 2.3 and [ E 2]. Next let us sh ow that the follo wing commutativ e square is a push out: A k   i / / B [ 0 1 ]   K h p j i / / A ′ ⊕ B T o this end, let α ′ : K → F and β ′ : B → F b e suc h that α ′ k = β ′ i . Since ( α ′ − β ′ j ) k = 0 and p = Cok er( k ), there is a un ique morp hism δ : A ′ → F su c h that δ p = α ′ − β ′ j . Then it follo w s that [ δ β ′ ] is th e u nique morph ism [ u v ] : A ′ ⊕ B → F suc h that [ u v ]  p j  = α ′ and [ u v ] [ 0 1 ] = β ′ . Hence the sq u are is a push out. No w  p j  is a semi-stable kernel by Remark 2.5. 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Sc hneiders, Quasi-ab elian c ate gories and she aves , M ´ em. So c. Math. F r. (N.S.) 76 (1999), pp. 1–14 0. [14] D. Sieg and S.-A. W egner, Maxim al exact structur es on additive c ate gories , Math. Nac hr., 2010, to appear. [15] B. Stenstr¨ om, R ings of quotients , Springer, Berlin, Heidelberg, New Y ork, 1975 . [16] N. Y oneda, On Ext and exact se quenc es , J. F ac. Sci. U niv. T okyo Sect. I 8 (1960), 507–57 6. F ac ul ty of Ma them a tics and Computer Science, “Babes ¸ -Bol y ai ” University, Str. Mihail Ko g ˘ alniceanu 1, 400084 Cluj-Napoca, Romania E-mail addr ess : crivei@ma th.ubbcluj. ro