Exact Categories

Exact Categ ories Theo B ¨ uhler FIM, HG G39.5, R¨ amistr asse 101, 8092 ETH Z¨ urich, Switzerland Abstract W e surv ey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas ar e pr ov ed dir ectly fro m the axioms, notably the ﬁve lemma, the 3 × 3- lemma a nd the snake lemma. W e brieﬂy discuss exact functors, idemp otent completion and weak idemp otent co mpleteness. W e then show that it is possible to construct the derived category of an exact categor y without any embedding into abelian categories and we sketc h Deligne’s approach to derived functors. The construction o f classical derived functors with v alues in an ab elian c ategory painlessly translates to e xact catego ries, i.e., we give pro ofs of the comparison theo rem for pro jectiv e resolutions and the horsesho e lemma. A fter discuss ing so me exa mples we elab orate on Thomason’s pro of of the Gabriel- Quillen em b edding theorem in an app endix. Key wor ds: Exact Ca tegories, Diag ram Lemmas, Homolo gical Algebra, Derived F unctors, Derived Categories , E m bedding Theo rems 2000 MSC: Primary: 18-02, Secondary: 18E 10, 18E 30 Con tents 1 In tro duction 2 2 Deﬁnition and B asi c Prop erties 5 3 Some Diagram Lemmas 11 4 Quasi-Ab eli an Categories 17 5 Exact F unctors 18 6 Idemp otent Comple ti on 19 7 W eak Idemp otent Compl eteness 23 8 Admissibl e Morphism s and the Snak e Lemma 25 9 Chain Compl exes and Chain Homotopy 30 Pr eprint submitte d t o Elsevier Novemb er 26, 2024 10 Acyclic Complexes and Quasi-Isomorphisms 32 10.1 The Homo top y Ca tegory of Acyclic Complexes . . . . . . . . . . . . . . . 33 10.2 Boundedness Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 10.3 Quasi-Isomo rphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 10.4 The Deﬁnition of the Deriv e d Categ ory . . . . . . . . . . . . . . . . . . . 38 10.5 Derived Categories of F ully Exact Subca tegories . . . . . . . . . . . . . . 38 10.6 T otal Deriv ed F unctor s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 11 Pro jectiv e and Injective Ob jects 41 12 Reso lutions and Classical Deriv ed F unctors 43 13 Examples and Appli cations 48 13.1 Additiv e Categ ories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 13.2 Quasi-Ab elian Categ ories . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 13.3 F ully Exact Sub categor ies . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 13.4 F rob enius Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 13.5 F urther Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 13.6 Higher Algebraic K -Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 51 A The Embeddi ng Theorem 52 A.1 Separated P resheav es a nd Shea ves . . . . . . . . . . . . . . . . . . . . . . 53 A.2 Outline of the Pro of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 A.3 Sheafiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 A.4 Pro of o f the Embedding Theorem . . . . . . . . . . . . . . . . . . . . . . . 6 1 B H eller’s Axioms 63 1. In tro duction There a re several notions of exact categories. On the one ha nd, there is the notion in the context of additive categor ies commonly a ttributed to Q uillen [50 ] with which the present article is co ncerned; on the other hand, there is the non-additive notion due to Barr [3], to men tion but the t wo mo st prominent ones. While Barr’s deﬁnition is in tr insic and an a dditiv e categor y is ex act in his sens e if and only if it is ab elian, Quillen’s deﬁnition is extrinsic in that one has to sp ecify a distinguished class of s hort exact sequences (an exact structure) in or der to obtain an exa ct categ ory . F rom now on we shall only deal with additive catego ries, so functors a re ta citly as- sumed to b e additive. On every additive category A the cla ss of all split ex act sequences provides the smallest exact structure, i.e., ev er y other exact structure must contain it. In general, an exact structure co nsists of kernel-cokernel pairs sub ject to some clo sure requirements, so the class of a ll kernel-cokernel pairs is a candidate for the largest ex- act structure. It is quite often the case that the class of a ll k ernel- cokernel pairs is an exact structure, but this fails in general: Rump [52] cons tructs an example of an a ddi- tiv e ca tegory with kernels and cokernels whose kernel-cokernel pairs fail to b e an exact structure. 2 It is commonplace that basic ho mological algebra in catego ries of mo dules ov er a ring extends to ab elian categor ies. By using the F r eyd-Mitc hell full em b edding theo rem ([17] and [46]), diagra m lemmas can be tra nsferred from module categ ories to general ab elian catego ries, i.e., o ne may argue by chasing e lemen ts a round in diagra ms. There is a point in p roving the fundamental dia gram lemmas dir ectly , and b e it only to familiarize oneself with the axioms. A careful study of what is actually needed reveals that in most situations the axioms of exact categorie s are suﬃcien t. An a p osteriori reas on is pr ovided b y the Gabriel-Q uillen embedding theorem which reduces homologica l algebra in exact categories to the case of abelian categories, the slogan is “relative homolo gical algebra made absolute” (F rey d [16]). More sp eciﬁcally , the embedding theore m asserts that the Y oneda functor em b eds a small exa ct catego ry A fully faithfully into the ab elian category B of left exact functors A op → Ab in s uc h a way tha t the ess en tial image is closed under extensions and that a short sequence in A is short exact if and only if it is sho rt exact in B . Co n versely , it is no t hard to see that an e xtension-closed sub category of an ab elian ca tegory is exact—this is the basic reco gnition pr inciple for exact categories. In app endix A we present Tho mason’s pro of o f the Gabriel-Quillen embedding theorem for the sake of completeness, but we will not apply it in these notes. The author is convinced that the embedding theore m should b e us ed to transfer the intuition from ab elian categories to exact categories r ather than to prove (simple) theorems with it. A direct pro of from the axioms provides m uch more insig h t than a reduction to ab elian categories . The interest of exact categories is manifold. First of all they ar e a natural genera liza- tion of ab elian categories and there is no need to argue that ab elian categories a re b oth useful a nd importa n t. There ar e several reasons for going b eyond abelia n categories. The fact that one may cho ose an exa ct structure gives more ﬂexibility which turns o ut to b e essential in many contexts. Even if one is working with ab elian catego ries o ne so on ﬁnds the need to consider o ther exact structures than the canonical one, for instance in rela- tiv e homolog ical a lgebra [29]. Beyond this, there are quite a few “cohomolo gy theories ” which inv olve functional analytic ca tegories lik e lo cally con vex mo dules over a top olo gi- cal group [3 0, 9], lo ca lly co mpact abelia n gr oups [3 1] o r Banach mo dules ov er a Banach algebra [32, 25] where there is no obvious abelian ca tegory around to which one could re- sort. In more adv anced topics o f algebra a nd r epresentation theory , (e.g. ﬁltered ob jects, tilting theory), exact categor ies arise na turally , while the theory of ab elian categ ories simply do es not ﬁt. It is a n observ atio n due to Happ el [24] that in guise o f F r ob enius c at- e gories, exact categorie s give rise to triangulated categories by passing to the asso c iated stable categories , see section 13.4. F urther ﬁelds of application are algebraic geometry (certain catego ries of vector bundles), alg ebraic analysis ( D -mo dules) and, of course, alge- braic K - theory (Quillen’s Q -construction [50], Balmer’s Witt gro ups [2] and Sc hlich ting’s Grothendieck-Witt groups [53]). The r eader will ﬁnd a slight ly more e xtensive di scussion of some o f the topics ment ioned a bove in sec tion 13. The author hope s to convince the reader that the axioms of exact categories are quite conv enient for giving relatively painless pro ofs of the basic results in homological algebr a and that the g ain in generality co mes with a lmost no eﬀor t. Indeed, it even seems that the axio ms of exact categories a re more adequa te for proving the fundamental diagra m lemmas than Grothendieck’s axioms for ab elian categories. F or instance, it is q uite a challenge to ﬁnd a complete pro o f (directly fr om the axio ms) of the snake lemma for ab elian categories in the literature. 3 That b eing said, we turn to a shor t description of the conten ts of this pa per . In sec tion 2 we sta te and discuss the axioms and draw the basic consequences, in particular we give the characterization of pull-back squares and Keller’s pro of of the obscure axiom. In section 3 we prov e the (shor t) ﬁve lemma, the No ether isomorphism theorem and the 3 × 3 -lemma. Section 4 brieﬂy discusses qua si-ab elian categor ies, a so urce of many examples of exact categories . Cont rary to the notio n of an exact categ ory , the prop er t y of b eing quasi-ab elian is intrinsic. Exact functors a re brieﬂy touched upo n in section 5 and after that we treat the idempo ten t completion and the prop erty o f weak idempo ten t completeness in se ctions 6 and 7. W e co me clos er to the hea rt of homological algebra when disc ussing a dmissible mor- phisms, long exact sequences, the ﬁve lemma and the sna ke lemma in sec tion 8 . In order for the snake lemma to hold, the assumption of weak idempo ten t completeness is neces- sary . After that we brieﬂy remind the reader o f the notions of chain complexes and chain homotopy in section 9, before w e turn to acyclic complexes a nd q uasi-isomor phisms in section 10. Notably , we give an elementary pr o of of Neeman’s crucia l re sult that the category o f acyclic complexes is triangulated. W e do not indulge in the details of the construction of the derived categor y of an exact category b ecause this is w ell trea ted in the literature. W e give a brief summary of the derived ca tegory of fully exac t sub categ ories and then sketch the main p oints of Deligne’s approach to total der ived functors on the level of the der ived ca tegory as exp ounded b y K eller [3 8]. On a more leisurely level, pr o jective and injectiv e o b jects ar e discussed in section 11 preparing the g rounds for a treatmen t of classica l der ived functors (with v alues in an ab elian categ ory) in section 12, where w e state and prove the resolution lemma, the comparison theorem and the horse sho e lemma, i.e., the three basic ingredients for the classical construction. W e end with a short list o f examples and applications in section 13. In app endix A we give Thomason’s proo f of the Gabriel-Quillen embedding theorem of an exact catego ry into an ab elian one. Fina lly , in a ppendix B we give a pro o f of the folklore fact that under the assumption of w eak idempotent completeness Heller’s axioms for an “ ab elian” ca tegory a re equiv alent to Quillen’s axioms for an exact ca tegory . Historical Note. Quillen’s no tion of an exact category ha s its predecesso rs e.g. in Heller [26], B uc hsbaum [10], Y o neda [60], B utler-Horro cks [13] and Mac Lane [4 3, XII.4 ]. It sho uld be noted that Buchsbaum, Butler-Horro cks and Mac Lane assume the ex is- tence of an amb ient ab elian category and miss the crucial push-out and pull-back ax- ioms, while Heller and Y oneda anticipate Quillen’s deﬁnition. According to Quillen [50, p. “92/ 16/10 0”], assuming idempo ten t completeness, Heller’s notion of a n “ab elian ca t- egory” [26, § 3], i.e., an additiv e category equipped with an “a belia n class o f shor t exa ct sequences”coincides with the present deﬁnition of an exa ct ca tegory . W e give a pro of of this a ssertion in a ppendix B. Y oneda’s quasi-a belia n S -ca tegories are nothing but Quillen’s exact ca tegories and it is a rema rk able fact that Y oneda pr ov es that Quillen’s “obscure axiom” follows fro m his deﬁnit ion, see [60, p. 525, Corollary], a fact r ediscov er ed thirt y y ears later by Keller in [3 6, A.1 ]. 4 Prerequisites. The prer equisites ar e kept a t a minim um. The r eader should k now what a n a dditiv e catego ry is and b e familiar with fundamen tal ca tegorical concepts such as k ernels, pull-backs, pro ducts and duality . Acquaintance with bas ic categor y theory as pr esented in Hilton-Stamm bach [28, Chapter II] or W eib el [59, App endix A] should amply suﬃce for a complete understanding of the text, up to section 10 where w e assume some familiarit y with the theo ry of tria ngulated catego ries. Disclaimer. This article is written for the reader who wants to learn about exact categories and knows why . V ery few motiv ating exa mples a re g iven in this text. The author makes no claim to origina lit y . All the results ar e well-kno wn in s ome form and they are scattered aro und in the literature. The r aison d’ˆ etr e of this article is the la ck of a systematic elementary exposition of the theory . The works o f Heller [26], Keller [36, 38] and Thomason [57] heavily inﬂuenced the present paper a nd many pro ofs given here can b e found in their pap er s. 2. Deﬁnition and Basic Prop erties In this section w e introduce the notion of an exact category and dra w the basic conse- quences of the axioms. W e do not use the minimal axiomatics as provided by Keller [3 6, Appendix A] but pre fer to use a co n venien t self-dual pr esent a tion of the axioms due to Y oneda [60, § 2 ] (modulo some of Y oneda ’s n umerous 3 × 2-lemmas and our Prop osi- tion 2.12) . The author hopes that the Bourbakists a mong the readers will pardon this faux p as . W e will discuss that the pr esent axioms are equiv alent to Quillen’s [50, § 2] in the course of even ts. The main po in ts o f this section are a characterization o f push-out squares (Prop osition 2.12) and the obscure axio m (Pr op osition 2.16). 2.1 Definition. Let A be an additive catego ry . A kernel-c okernel p air ( i, p ) in A is a pair of c ompo sable morphisms A ′ i − → A p − → A ′′ such that i is a kernel of p and p is a co kernel of i . If a class E of kernel-cokernel pa irs on A is ﬁxed, an admissible monic is a morphism i for which there ex ists a morphism p such that ( i, p ) ∈ E . Ad missible epics a re deﬁned dually . W e depict admissible monics b y ֌ and a dmissible epics by ։ in diagrams. An ex act s t ructur e on A is a cla ss E of kernel-cok ernel pairs which is closed under isomorphisms a nd satisﬁes the following axioms: [E0] F or all ob jects A ∈ A , the identit y morphism 1 A is an a dmissible monic. [E0 op ] F or all ob jects A ∈ A , the identit y morphism 1 A is an a dmissible epic. [E1] The class of admissible monics is closed under comp osition. [E1 op ] The class of admissible epics is closed under comp osition. [E2] The push- out of an admissible monic along an arbitrary m orphism exists and yields an a dmissible monic. [E2 op ] The pull-back o f an admissible epic along an arbitrar y mo rphism exists and yields an a dmissible epic. 5 Axioms [E2] and [E2 op ] are subsumed in the diagrams A   / / / / PO B   A ′ / / / / B ′ and A ′   / / / / PB B ′   A / / / / B resp ectively . An exact c ate gory is a pair ( A , E ) cons isting of an a dditiv e catego ry A and an exa ct structure E on A . Elements of E are called short exact se qu enc es . 2.2 Remark. Note that E is a n exact structure on A if and only if E op is an exact structure on A op . This allo ws for reasoning b y dualization. 2.3 Remark. Isomorphisms are admissible monics and admissible epics. Indeed, this follows from the co mm utative diag ram A ∼ = 1 A   f ∼ = / / B / / ∼ = f − 1   0 ∼ =   A / / 1 A / / A / / / / 0 , the fact that ex act structures a re assumed to b e close d under iso morphisms and that the axioms are self-dual. 2.4 Remark (K eller [36, App. A]) . The ax ioms are somewhat redundan t and can be weak ened. F or instance, let us assume instead of [E0] and [E0 op ] that 1 0 , the identit y of the zero o b ject, is an admissible epic. F o r a n y o b ject A there is the pull-back diag ram A   1 A / / PB A   0 1 0 / / 0 so [E2 op ] together with our assumption on 1 0 shows that [E0 op ] holds. Since 1 0 is a k ernel of itself, it is also an admissible monic, s o we conclude by [E2] that [E0 ] holds as well. More imp ortantly , Keller pr ov es in lo c. cit. (A.1, pro o f of the prop osition, step 3), that one can also dispo se of one of [E1] or [E1 op ]. Mo reov er , he mentions (A.2, Remark), that one may also weak en one of [E2] or [E2 op ]—this is a straightforw ard consequence of (the pro of of ) Pro po sition 3.1. 2.5 Remark. K eller [36, 3 8] uses c onﬂation , inﬂation and deﬂation for what we call short exact sequence, admissible monic and admissible epic. This terminology s tems from Gabriel-Ro ˘ ıter [21, Ch. 9] who give a list of axioms for ex act categories whose underlying additive categor y is weakly idemp otent complete in the sense o f sec tion 7, see K eller’s app endix to [15] for a thoroug h co mparison of the a xioms. A v ariant of the Gabr iel-Ro ˘ ıter- axioms a ppea r in F reyd’s b o ok o n ab elian categories [17, Ch. 7, Exercise G, p. 15 3] (the Gabriel-Ro ˘ ıter-axioms are obta ined fr om F r eyd’s axioms by adding the dual of F rey d’s condition (2)). 6 2.6 Exercise. An admissible epic which is additionally monic is an isomorphism. 2.7 Lemma. The se quenc e A / / [ 1 0 ] / / A ⊕ B [ 0 1 ] / / / / B is short exact. Proof. The following diag ram is a push-out s quare 0 / / / /     PO B   [ 0 1 ]   A / / [ 1 0 ] / / A ⊕ B . The top a rrow a nd the left ha nd arrow are admissible monics by [E0 op ] while the bo ttom arrow and the rig h t hand arr ow are a dmissible monics by [E 2]. The lemma now follows from the facts tha t the sequence in question is a kernel-cokernel pair and that E is closed under isomorphisms. 2.8 Remark. Lemma 2.7 shows that Q uillen’s axiom a ) [50, § 2] stating that s plit exact sequences b elong to E follows from our axioms. Conv er sely , Quillen’s axiom a) obviously implies [E0 ] a nd [E0 op ]. Quillen’s axiom b) co incides with our axioms [E1], [E1 op ], [E2 ] and [E2 op ]. W e will prov e that Quillen’s axiom c) follo ws from our axioms in Prop osition 2.1 6. 2.9 Proposition. The dir e ct su m of two short exact se quenc es is short exact. Proof. Let A ′ ֌ A ։ A ′′ and B ′ ֌ B ։ B ′′ be tw o short exact sequences . First observe that for every ob ject C the seq uence A ′ ⊕ C ֌ A ⊕ C ։ A ′′ is exact—the second morphism is an admissible epic b ecause it is the comp osition of the admissible epics [ 1 0 ] : A ⊕ C ։ A a nd A ։ A ′′ ; the ﬁrst morphism in the s equence is a kernel of the second one, hence it is an admissible monic. Now it follows from [E1] that A ′ ⊕ B ′ ֌ A ⊕ B is an admiss ible monic beca use it is the co mpo sition of the tw o admiss ible monics A ′ ⊕ B ′ ֌ A ⊕ B ′ and A ⊕ B ′ ֌ A ⊕ B . It is ob vio us that A ′ ⊕ B ′ ֌ A ⊕ B ։ A ′′ ⊕ B ′′ is a kernel-cokernel pa ir, hence the prop osition is proved. 2.10 Corollar y. The exact st ructur e E is an additive sub c ate gory of the additive c at- e gory A →→ of c omp osable morphisms of A . 2.11 Remark. In Exercis e 3.9 the reader is asked to sho w that E is exact with resp ect to a na tural ex act str ucture. 7 2.12 Proposition. Consider a c ommutative squar e A / / i / / f   B f ′   A ′ / / i ′ / / B ′ in which t he horizontal arr ows ar e admissible monics. The fol lowing assertions ar e e quiv- alent: (i) T he s qu ar e is a push-out. (ii) The se quenc e A / / h i − f i / / B ⊕ A ′ [ f ′ i ′ ] / / / / B ′ is short exact. (iii) T he squar e is bic artesian, i.e., b oth a push-out and a pul l-b ack. (iv) The squar e is p art of a c ommutative diagr am A / / i / / f   B f ′   p / / / / C A ′ / / i ′ / / B ′ p ′ / / / / C with exact r ows. Proof. (i) ⇒ (ii): The push-out prop erty is equiv alent to the ass ertion that [ f ′ i ′ ] is a cokernel o f  i − f  , so it suﬃces to prove that the latter is an admissible monic. But this follows from [E1] since  i − f  is equa l to the comp osition of the morphisms A / / [ 1 0 ] / / A ⊕ A ′ h 1 0 − f 1 i ∼ = / / A ⊕ A ′ / / [ i 0 0 1 ] / / B ⊕ A ′ which a re all admissible mo nics by Lemma 2.7, Remar k 2.3 and Prop osition 2.9, r esp ec- tiv ely . (ii) ⇒ (iii) and (iii) ⇒ (i): obvious. (i) ⇒ (iv): Let p : B ։ C be a cok ernel of i . The push-out pro per t y of the square yields that there is a unique morphism p ′ : B ′ → C such that p ′ f ′ = p and p ′ i ′ = 0. Observe that p ′ f ′ = p implies that p ′ is epic. In order to see that p ′ is a cok ernel of i ′ , let g : B ′ → X be such that g i ′ = 0. Then g f ′ i = g i ′ f = 0, so g f ′ factors uniquely ov er a morphism h : C → X suc h that g f ′ = hp . W e claim that hp ′ = g : this follows fro m the push-out prop erty of the s quare b ecause hp ′ f ′ = hp = g f ′ and hp ′ i ′ = 0 = g i ′ . Since p ′ is epic, the factorization h o f g is unique, so p ′ is a co kernel of i ′ . (iv) ⇒ (ii): F orm the pull-back ov er p and p ′ in or der to obtain the commutativ e 8 diagram A   j   A   i   A ′ / / j ′ / / D PB q ′ / / / / q     B p     A ′ / / i ′ / / B ′ p ′ / / / / C with exact rows a nd columns using the dual of the implication (i) ⇒ (iv). Since the square B f ′   B p     B ′ p ′ / / / / C is co mm utativ e, there is a unique morphism k : B → D such that q ′ k = 1 B and q k = f ′ . Since q ′ (1 D − k q ′ ) = 0, ther e is a unique mor phism l : D → A ′ such tha t j ′ l = 1 D − k q ′ . Note that l k = 0 b ecause j ′ l k = (1 D − k q ′ ) k = 0 and j ′ is monic, while the calculation j ′ l j ′ = (1 D − k q ′ ) j ′ = j ′ implies l j ′ = 1 A ′ , again b ecause j ′ is monic. F urther more i ′ l j = ( q j ′ ) lj = q (1 D − k q ′ ) j = − ( q k )( q ′ j ) = − f ′ i = − i ′ f implies lj = − f since i ′ is monic. The mor phisms [ k j ′ ] : B ⊕ A ′ → D and h q ′ l i : D → B ⊕ A ′ are m utually inv erse since [ k j ′ ] h q ′ l i = k q ′ + j ′ l = 1 D and h q ′ l i [ k j ′ ] = h q ′ k q ′ j ′ lk lj ′ i = h 1 B 0 0 1 A ′ i . Now [ f ′ i ′ ] = q [ k j ′ ] a nd  i − f  = h q ′ l i j show that A h i − f i − − − − → B ⊕ A ′ [ f ′ i ′ ] − − − − → B ′ is isomorphic to A j − → D q − → B ′ . 2.13 Remark. Consider the push-out dia gram A ′ / / i ′ / / a   PO B ′ b   A / / i / / B . If j ′ : B ′ ։ C ′ is a co kernel of i ′ then the unique mor phism j : B → C ′ such tha t j i = 0 and j b = j ′ is a cokernel o f i . If j : B ։ C is a co kernel of i then j ′ = j b is a cok er nel of i ′ . The ﬁrst statement was es tablished in the pro of o f the implication (i) ⇒ (iv) of Prop osition 2.12 and we leave the easy v eriﬁcation of the second statemen t as an exer cise for the rea der. 9 The following s imple observ a tion will only b e used in the pr oo f of Lemma 10.3. W e state it her e for ease of r eference. 2.14 Corollar y. The surr ounding r e ctangle in a diagr am of t he form A a   PB f / / / / B b   PO / / g / / C c   A ′ f ′ / / / / B ′ / / g ′ / / C ′ is bic artesian and A / / h − a gf i / / A ′ ⊕ C [ g ′ f ′ c ] / / / / C ′ is short exact. Proof. It follows from Prop osition 2.12 and its dual that b oth squares are bicartesian. Gluing tw o bicartesian s quares alo ng a common ar row yields another bicartesian square, which entails the ﬁrst part and the fact that the sequence of the seco nd part is a kernel- cokernel pair. The equation [ g ′ f ′ c ] = [ g ′ c ] h f ′ 0 0 1 C i exhibits [ g ′ f ′ c ] as a comp osition of admissible e pics b y Prop os ition 2.9 and P rop osition 2.1 2. 2.15 Proposition. Th e pul l-b ack of an admissible monic along an admissible epic yields an admissible monic. Proof. Consider the diagram A ′ e ′     PB i ′ / / B ′ e     pe / / / / C A / / i / / B p / / / / C. The p ull-back square exists by axio m [E2 op ]. Let p b e a cokernel of i , so it is an a dmissible epic and pe is an admissible epic b y axiom [E1 op ]. In a n y category , the pull-bac k of a monic is a monic (if it exists). In order to see that i ′ is an admissible monic, it suﬃces to prov e that i ′ is a kernel of pe . Supp ose that g ′ : X → B ′ is such that peg ′ = 0. Since i is a kernel of p , there exists a unique f : X → A such that eg ′ = if . Applying the univ ersal prop erty of the pull-back square, we ﬁnd a unique f ′ : X → A ′ such that e ′ f ′ = f a nd i ′ f ′ = g ′ . Since i ′ is monic, f ′ is the unique morphism such that i ′ f ′ = g ′ and w e are done. 2.16 Pr oposition (Obscure Axiom) . Supp ose that i : A → B is a morphism in A admitting a c okernel. If ther e exists a morphism j : B → C in A such t hat the c omp osite j i : A ֌ C is an admissible monic then i is an admissible monic. 2.17 Remark. The statemen t of the previo us prop osition is given as axiom c) in Quillen’s deﬁnition of an exact catego ry [5 0, § 2]. At that time, it was alr eady proved to be a c on- sequence of the other axioms by Y oneda [60, Corollar y , p. 52 5]. The r edundancy of the obscure a xiom was rediscov ered b y Keller [36, A.1]. Thoma son baptized a xiom c) the “obscure axiom” in [57, A.1.1]. A conv enient and quite pow e rful stre ngthening o f the obs cure a xiom holds under the rather mild additional hypo thesis of weak idemp otent completeness, see Prop osition 7 .6. 10 Proof of Pr oposition 2.1 6 (Keller). Let k : B → D be a cokernel o f i . F r om the push-out diagram A / / j i / / i   PO C   B / / / / E and Pr op osition 2.12 w e conclude that  i j i  : A ֌ A ⊕ B is an admissible monic. Because  1 B 0 − j 1 C  : B ⊕ C → B ⊕ C is an isomor phism it is in particular an admissible monic, hence [ i 0 ] =  1 B 0 − j 1 C   i j i  is an a dmissible monic a s w ell. Beca use  k 0 0 1 C  is a cokernel of [ i 0 ], it is an admissible epic. Consider the follo wing diagr am A i / / B [ 1 0 ]   k / / PB D [ 1 0 ]   A / / [ i 0 ] / / B ⊕ C h k 0 0 1 C i / / / / D ⊕ C. Since the rig h t hand square is a pull-back, it follows tha t k is an admissible epic a nd that i is a kernel of k , so i is an a dmissible monic. 2.18 Corollar y. L et ( i, p ) and ( i ′ , p ′ ) b e two p airs of c omp osable morphisms. If the dir e ct su m ( i ⊕ i ′ , p ⊕ p ′ ) is exact t hen ( i, p ) and ( i ′ , p ′ ) ar e b oth exact. Proof. It is clea r that ( i, p ) and ( i ′ , p ′ ) are kernel-cokernel pairs. Since i has p as a cokernel and since [ 1 0 ] i =  i 0 0 i ′  [ 1 0 ] is an a dmissible monic, the obscure a xiom implies that i is an admissible monic. 2.19 Exercise. Suppo se that the commutativ e square A ′ / / f ′ / / a   PO B ′   b   A / / f / / B is a push-o ut. Pr ov e that a is an admissible monic. Hint: Let b ′ : B ։ B ′′ be a cokernel o f b : B ′ ֌ B . Prov e that a ′ = b ′ f : A → B ′′ is a cokernel of a , then apply the obscure axiom. 3. Some Di agram Lemmas In this section we will prove v ariants of diagra m lemmas which are w ell-known in the context of ab elian catego ries, in particular we will prove the ﬁve lemma and the 3 × 3- lemma. F urther familiar diagr am lemmas will b e prov ed in section 8. The pro ofs will b e based on the following simple observ ation: 11 3.1 Pr oposition. L et ( A , E ) b e an exact c ate gory. A morphism fr om a short exact se quenc e A ′ ֌ B ′ ։ C ′ to another short exact se quenc e A ֌ B ։ C factors over a short exact se qu enc e A ֌ D ։ C ′ A ′ / / f ′ / / a   BC B ′ g ′ / / / / b ′   C ′ A / / m / / D e / / / / b ′′   BC C ′ c   A / / f / / B g / / / / C in such a way that the two squ ar es marke d BC ar e bic artesian. In p articular ther e is a c anonic al isomorphism of the push-out A ∪ A ′ B ′ with t he pul l-b ack B × C C ′ . Proof. F orm the push-out under f ′ and a in order to obtain the o b ject D a nd the morphisms m and b ′ . Let e : D → C ′ be the unique morphism such that eb ′ = g ′ and em = 0 and let b ′′ : D → B b e the unique morphism D → B suc h that b ′′ b ′ = b : B ′ → B and b ′′ m = f . It is easy to see that e is a cokernel of m (Remark 2.13) and hence the result follows fro m Prop osition 2.12 since the squa re D C ′ B C is co mm utative [this is beca use a a nd b ′′ b ′ determine c uniquely]. 3.2 Corollar y (Five Lemma, I) . Co nsider a morphism of short exact se qu enc es A ′ / / / / a   B ′ / / / / b   C ′ c   A / / / / B / / / / C. If a and c ar e isomorphisms (or admissible monics, or admissible epics) then so is b . Proof. Assume ﬁrst that a a nd c a re isomorphisms. Because iso morphisms are preser v ed b y push-outs and pull-backs, it follows from the diagram of Pro po sition 3.1 that b is the comp osition of tw o isomo rphisms B ′ → D → B . If a and c a re both admissible monics, it follows from the diagra m of Pro po sition 3.1 together with [E2] and Pr op osition 2.15 that b is the comp osition of t wo admissible monics. The case of admissible epics is dual. 3.3 Exercise. If in a morphism A ′ / / / / a   B ′ / / / / b   C ′ c   A / / / / B / / / / C. of s hort exa ct sequences as in the ﬁve lemma 3.2 tw o out of a, b, c are isomo rphisms then so is the third. Hint: Use e.g. that c is uniquely determined by a and b . 3.4 Remark. The reader insisting that Coro llary 3.2 should be called “three lemma” rather than “ﬁv e lemma” is cordially in vited to give the details of the proof of Lemm a 8.9 and to so lve E xercise 8 .10. W e will how ever use the mor e customar y name ﬁve lemma. 12 3.5 Lemma (“No ether Isomorphism C /B ∼ = ( C / A ) / ( B / A )”) . Consider the diagr am A / / / / B     / / / / X     A / / / / C / / / /     Y     Z Z in which the ﬁrst two hori zontal ro ws and the middle c olumn ar e short exact. The n t he thir d c olumn exists, is short exact, and is uniquely determine d by the r e quir ement that it makes t he diagr am c ommutative. Mor e over, the u pp er right hand squar e is bic artesian. Proof. The morphism X → Y exists since the ﬁr st row is exact a nd the comp osition A → C → Y is zero while the morphism Y → Z exists since the se cond ro w is exa ct and the co mpos ition B → C → Z v anishes. By Pro po sition 2.1 2 the s quare co n taining X → Y is bicar tesian. It follows that X → Y is an admissible monic and that Y → Z is its cokernel. The uniqueness assertion is o b vious. 3.6 Corollar y (3 × 3-Lemma) . Conside r a c ommutative diagr am A ′ f ′ / /   a   B ′ g ′ / /   b   C ′   c   A f / / a ′     B g / / b ′     C c ′     A ′′ f ′′ / / B ′′ g ′′ / / C ′′ in which t he c olumns ar e exact and assume in addition that one of the fol lowing c onditions holds: (i) the midd le ro w and either one of the outer ro ws is short exact; (ii) the two outer r ows ar e short exact and g f = 0 . Then t he r emaining r ow is short exact as wel l. Proof. Let us prov e (i). The tw o po ssibilities are dua l to ea ch other, so we need only consider the case that the ﬁrst tw o r ows are exact. Apply Prop osition 3 .1 to the ﬁrst tw o rows so as to obtain the comm utative diag ram A ′ / / f ′ / /   a   BC B ′ g ′ / / / /   i   C ′ A / / ¯ f / / D ¯ g / / / /   j   BC C ′   c   A / / f / / B g / / / / C 13 where j i = b —notice that i and j are admissible monics by axiom [E2 ] and Pro po si- tion 2 .15, resp ectively . By Rema rk 2.13 the morphism i ′ : D → A ′′ determined by i ′ i = 0 and i ′ ¯ f = a ′ is a co kernel of i a nd the morphism j ′ : B ։ C ′′ given by j ′ = c ′ g = g ′′ b ′ is a c okernel of j . If w e knew that the diagra m B ′ / / i / / D i ′ / / / /   j   A ′′ f ′′   B ′ / / b / / B b ′ / / / / j ′     B ′′ g ′′   C ′′ C ′′ is commutativ e then we would conclude from the No ether is omorphism 3.5 that ( f ′′ , g ′′ ) is a short exact s equence. It therefor e remains to pr ov e that f ′′ i ′ = b ′ j since the other commut ativity r elations b = j i and g ′′ b ′ = j ′ hold b y co nstruction. W e are going t o apply the push-out pr op erty of the square A ′ B ′ AD . W e hav e ( f ′′ i ′ ) i = 0 = b ′ b = ( b ′ j ) i and ( b ′ j ) ¯ f = b ′ f = f ′′ a ′ = ( f ′′ i ′ ) ¯ f which together with ( f ′′ i ′ ¯ f ) a = ( f ′′ i ′ i ) f ′ = 0 and ( b ′ j ¯ f ) a = f ′′ a ′ a = 0 = b ′ bf ′ = ( b ′ j i ) f ′ show that bo th f ′′ i ′ and b ′ j are solutions to the same push-out problem, hence they are equal. This settles cas e (i). In order to prove (ii) w e start b y forming the push-out under g ′ and b so that we hav e the follo wing co mm utative dia gram with exact rows and co lumns A ′ / / f ′ / / B ′ g ′ / / / /   b   PO C ′   k   A / / i / / B j / / / / b ′     D k ′     B ′′ B ′′ in whic h the cokernel k ′ of k is deter mined by k ′ j = b ′ and k ′ k = 0, while i = bf ′ is a kernel o f the a dmissible epic j , see Remar k 2.13 and Prop osition 2.15. The push- out prop erty of the square B ′ C ′ B D applied to the square B ′ C ′ B C yields a unique morphism d ′ : D → C such that d ′ k = c and d ′ j = g . The diagram C ′ / / k / / D d ′   k ′ / / / / B ′′ g ′′     C ′ / / c / / C c ′ / / / / C ′′ 14 has exact rows and it is commu tative: Indeed, c = d ′ k holds b y construction o f d ′ , while c ′ d ′ = g ′′ k ′ follows from c ′ d ′ j = c ′ g = g ′′ b ′ = g ′′ k ′ j and the fact that j is epic. W e conclude fro m P rop osition 2.12 that D C B ′′ C ′′ is a pull-back, so d ′ is a n admissible epic and so is g = d ′ j . The unique mor phism d : A ′′ → D suc h that k ′ d = f ′′ and d ′ d = 0 is a kernel of d ′ . By the pull-back prop erty of DC B ′′ C ′′ the diagram A ′   a   A ′   i   A f / / a ′     B g / / / / j     C A ′′ / / d / / D d ′ / / / / C is commutativ e as k ′ ( da ′ ) = f ′′ a ′ = b ′ f = k ′ ( j f ) and d ′ ( da ′ ) = 0 = g f = d ′ ( j f ). Notice that the hypothesis that g f = 0 enters at this p oint of the ar gument . It follows from the dual o f Pro po sition 2.12 that AB A ′′ D is bicartesian, so f is a k er nel of g by Prop osition 2.15. 3.7 Exercise. Consider the s olid a rrow diagram A ′ / / / /     B ′     / / / / C ′     A / / / /     B / / / /     C     A ′′ / / / / B ′′ / / / / C ′′ with exact rows and columns. Strengthen the No ether isomorphism 3 .5 to the statement that there exist unique maps C ′ → C a nd C → C ′′ making the diag ram commutativ e and the s equence C ′ ֌ C ։ C ′′ is sho rt ex act. 3.8 Exer cise. In the situation of the 3 × 3-lemma prov e that there are tw o exact se- quences A ′ ֌ A ⊕ B ′ → B ։ C ′′ and A ′ ֌ B → B ′′ ⊕ C ։ C ′′ in the sense that the morphisms → factor as ։ ֌ in such a way that consecutive ֌ ։ are short exact [com- pare also with Deﬁnition 8.8]. Hint: Apply Prop osition 3.1 to the ﬁrst tw o rows in order to obtain a short exact sequence A ′ ֌ A ⊕ B ′ ։ D using P rop osition 2.12. Conclude fro m the push-out prop erty of DC ′ B C tha t D ֌ B has C ′′ as cokernel. 15 3.9 Exercise (Heller [26 , 6.2]) . Let ( A , E ) b e an ex act categor y a nd c onsider E as a full sub categ ory of A →→ . W e ha ve shown that E is additiv e in Corollary 2.10. Let F be the class of s hort s equences ( A ′ / / / /     B ′ / / / /     C ′ )     ( A / / / /     B / / / /     C )     ( A ′′ / / / / B ′′ / / / / C ′′ ) ov er E with short ex act columns [w e write ( A ֌ B ։ C ) to indicate that we think o f the sequence as an ob ject of E ]. P rov e that ( E , F ) is an exa ct categ ory . 3.10 Remark. The catego ry of sho rt exa ct sequences in a nonzero ab elian ca tegory is not abelian, see [43, XI I.6, p. 375]. 3.11 Exercise (K¨ unzer’s Axiom, c f. e.g . [41]) . (i) If f : A → B is a morphism and g : B ։ C is an a dmissible epic such tha t h = g f : A ֌ C is an admissible monic then f is an admissible monic and the morphism Ker g → Co ker f is an admissible monic as well. Hint: F orm the pull-bac k P ov er h and g , use Prop osition 2.1 5 a nd factor f over P to see the ﬁr st part (see also Remark 7.4). F or the second part use the No ether isomorphism 3.5. (ii) Let E b e a class of kernel-cokernel pa irs in the a dditiv e category A . Ass ume that E is closed under isomor phisms a nd contains the split exact sequences. If E enjo ys the proper t y o f p oint (i) and its dual then it is an exact structure. Hint: Let f : A ֌ B be an admissible monic and let a : A → A ′ be arbitrar y . The push-out axiom follows fro m the commutativ e diagram A / / f / / # # [ a f ] # # G G G G G G G G G B A ′ ⊕ B [ 0 1 ] ; ; ; ; w w w w w w w w w [ f ′ − b ] # # # # G G G G G G G G G A ′ / / f ′ / / ; ; [ 1 0 ] ; ; w w w w w w w w B ′ in whic h [ a f ] and f ′ are admissible monics by (i) and [ f ′ − b ] is a cokernel of [ a f ]. Next observe that the dual o f (i) implies that if in addition a is an a dmissible epic then so is b . In order to pr ov e the comp osition axiom, let f and g b e admissible monics and ch o ose a cok ernel f ′ of f . F or m the push-out under g and f ′ and verify that g f is a kernel of the push-out of f ′ . 16 4. Quasi-Ab elian Categories 4.1 Definition. An additive catego ry A is called quasi-ab elian if (i) E very mo rphism has a kernel and a co kernel. (ii) The cla ss of k er nels is stable under push-out alo ng arbitrary morphisms a nd the class of cok ernels is stable under pull-ba ck along arbitra ry morphisms. 4.2 Remark. The concept of a quasi-abe lian categor y is self-dual, that is to sa y A is quasi-ab elian if a nd only if A op is qua si-ab elian. 4.3 Exercise. Let A b e an additive catego ry with kernels. Pr ov e that every pull-back of a kernel is a k ernel. 4.4 Pr oposition (Schneiders [54, 1.1.7]) . The class E max of al l kernel-c okernel p airs in a quasi-ab elian c ate gory is an exact st ructur e. Proof. It is clear that E max is closed under isomorphisms and t hat the classes of kernels and cok ernels contain the iden tity morphisms. The pull-back and push-out ax ioms are part of the deﬁnition of quasi- ab elian categories . By dualit y it only remains to show that the c lass o f cokernels is closed under compo sition. So let f : A ։ B a nd g : B ։ C b e cokernels and put h = g f . In the diagr am Ker f u / / Ker h v / /   ker h   Ker g   ker g   Ker f / / ker f / / A f / / / / h   B g     C C there exist unique morphisms u and v making it comm utative. The upp er r ight hand square is a pull-back, so v is a co kernel and u is its kernel. But then it follows by dualit y that the upp er right hand square is also a push-out and this together with the fact that h is epic implies that h is a cokernel of ker h . 4.5 Remark. Note that we have just r e-prov ed the No ether isomo rphism 3.5 in the sp ecial case of quasi-a be lian categor ies. 4.6 Definition. The c oimage of a mor phism f in a catego ry with kernels and cokernels is Coker (k e r f ), while the image is deﬁned to b e Ker (cok er f ). The analysis (cf. [43, IX.2]) of f is the comm utative diag ram A f / / coim f ) ) ) ) R R R R R R B coker f ) ) ) ) R R R R R R Ker f 6 6 ker f 6 6 m m m m m Coim f ˆ f / / Im f 6 6 im f 6 6 n n n n n Coker f in whic h ˆ f is uniquely determined b y requiring the diagram to b e comm utativ e. 17 4.7 Remark. The diﬀerence b etw een quasi-a belia n categor ies and ab elian categor ies is that in the quasi-ab elian case the canonical morphism ˆ f in the analysis f is not in general an isomorphism. Indeed, it is easy to see that a quasi-ab elian category is a belia n pr ovide d that ˆ f is alwa ys a n isomor phism. Equiv alentl y , not every monic is a kernel and not every epic is a cokernel. 4.8 Proposition ([54 , 1.1.5]) . L et f b e a m orphism in the quasi-ab elian c ate gory A . The c anonic al morphism ˆ f : Coim f → Im f is m onic and epic. Proof. By dualit y it suﬃces to chec k that the mo rphism ¯ f in the diagra m A f / / j ) ) ) ) R R R R R R B Coim f ¯ f 5 5 l l l l l l is monic. Let x : X → Coim f b e a morphism such that ¯ f x = 0. The pull-back y : Y → A of x along j satisﬁes f y = 0, so y factors ov er Ker f and hence j y = 0. But then the map Y ։ X → Coim f is zero as w ell, so x = 0. 4.9 Remark. Ev ery morphism f in a quas i-ab elian categ ory A has tw o epic-monic factorizations, one o ver Co im f and o ne ov er Im f . The quas i-ab elian category A is ab elian if a nd only if the tw o factoriza tions coincide for a ll morphisms f . 4.10 Re mark. An additive catego ry with kernels a nd cokernels is ca lled semi-ab elian if the cano nical morphism Coim f → Im f is always monic and epic. W e hav e just prov ed that quasi-ab elian catego ries ar e semi-ab elian. It may seem ob vio us that the concept of semi-ab elian c ategories is strictly weaker than the concept of a quasi-ab elian catego ry . How ever, it is surprisingly delicate to come up w ith an explicit exa mple. This led Ra ˘ ıko v to conjecture that ev e ry semi-abelian categor y is qua si-ab elian. A coun terexa mple to this conjecture was recent ly found by Rump [52]. 4.11 Remark. W e do not develop the theor y of quasi-ab elian categories any further. The interested reader may consult Schneiders [54], Rump [51] and the references therein. 5. Exact F unctors 5.1 Definition. Let ( A , E ) and ( A ′ , E ′ ) b e exact ca tegories. An (additive) functor F : A → A ′ is called exact if F ( E ) ⊂ E ′ . The functor F r eﬂe cts exactness if F ( σ ) ∈ E ′ implies σ ∈ E for all σ ∈ A →→ . 5.2 Proposition. An exact functor pr eserves push-outs along admissible monics and pul l-b acks along admissible epics. Proof. An exact functor preserves admissible monics and admissible epics, in particular it preserves dia grams o f type   / / / /   / / / / and   / / / /   / / / / so the re sult follo ws immediately from Pr op osition 2.12 and its dua l. 18 The following exercises show how one can induce new exact str uctures using functors satisfying certain exa ctness pr op erties. 5.3 Exercise (Heller [26, 7 .3]) . Let F : ( A , E ) → ( A ′ , E ′ ) be an exact functor and let F ′ be another exact structure on A ′ . Then F = { σ ∈ E : F ( σ ) ∈ F ′ } is an exa ct structure on A . 5.4 Remark (Heller) . The prototypical application o f the pr evious exercise is the fol- lowing: A (unital) r ing homo morphism ϕ : R ′ → R yields an exact functor ϕ ∗ : R Mo d → R ′ Mo d of the asso ciated mo dule ca tegories. Let F ′ be the cla ss of split exa ct sequences on R ′ Mo d . The induced structure F on R Mo d co nsisting of sequences σ such that ϕ ∗ ( σ ) is split exact is the r elative exact structu re with respect to ϕ . This structure is used in particular to deﬁne the relative derived functors such as the rela tiv e T or and Ext functors. 5.5 Exercise (K¨ unzer ) . Let F : ( A , E ) → ( A ′ , E ′ ) b e a funct or which preser ves admissible kernels, i.e., for ev ery morphism f : B → C with an admissible monic k : A ֌ B as kernel, the morphism F ( k ) is an admissible mo nic and a kernel of F ( f ). Let F = { σ ∈ E : F ( σ ) ∈ E ′ } b e the largest sub class of E on which F is exact. Pr ov e that F is an exact structure . Hint: Axioms [E 0], [E 0 op ] and [E1 op ] ar e easy . T o chec k axiom [E 1] use the obscure axiom 2.16 and the 3 × 3-lemma 3.6. Axio m [E2] follo ws from the obscure axiom 2.16 and Prop osition 2.1 2 (iv), while axio m [E2 op ] follows from the fact that F prese rves certain pull-back squares. 5.6 Ex er cise. Let P b e a set of ob jects in the exa ct ca tegory ( A , E ). Consider the class E P consisting of the sequences A ′ ֌ A ։ A ′′ in E such that Hom A ( P, A ′ ) ֌ Hom A ( P, A ) ։ Hom A ( P, A ′′ ) is an e xact sequence of abelia n groups for all P ∈ P . Prove that E P is an exact structure on A . Hint: Use Exer cise 5.5. 6. Idemp o te n t Completion In this section we dis cuss Karoubi’s construction of ‘the’ idempotent completi on of an additive ca tegory , see [33, 1.2] and sho w ho w to extend it to exact categories . Admittedly , the constructions and arguments presented here ar e quite obvious (once the deﬁnitions are given) and thu s rather bo ring, but as the author is unaware of a reasonably com- plete expos ition it seems worthwhile to outline the details. A diﬀerent ac count for small categories (not necessa rily additive) is given in Borc eux [5, Prop osition 6.5 .9, p. 274]. It app ears that the latter approach is due to M. B unge [1 2]. 6.1 Definition. An additive catego ry A is idemp otent c omplete [33, 1.2.1 , 1.2.2] if for every idemp otent p : A → A , i.e. p 2 = p , there is a decompo sition A ∼ = K ⊕ I o f A such that p ∼ = [ 0 0 0 1 ]. 6.2 Remark. The a dditiv e ca tegory A is idemp otent complete if and only if every idempo ten t has a kernel. 19 Indeed, supp ose that every idemp otent has a kernel. Let k : K → A be a kernel of the idemp otent p : A → A and let i : I → A b e a kernel of the idempo ten t 1 − p . Because p (1 − p ) = 0 , we hav e (1 − p ) = k l for a unique l : A → K and b ecaus e (1 − p ) p = 0 we hav e p = ij for a unique j : A → I . Since k is mo nic and k l i = (1 − p ) i = 0 we ha ve li = 0 and beca use k l k = (1 − p ) k = pk + (1 − p ) k = k we hav e lk = 1 K . Similarly , j k = 0 and j i = 1 I . There fore [ k i ] : K ⊕ I → A and  l j  : A → K ⊕ I are inv erse to each other and  l j  p [ k i ] =  l j  ij [ k i ] =  0 0 0 1 I  as desired. Notice that we hav e constructed an analysis of p : A p / / j & & & & N N N N N N A l ' ' ' ' O O O O O O K 7 7 k 7 7 p p p p p I 8 8 i 8 8 p p p p p K, in particular k : K ֌ A is a kernel of p and i : I ֌ A is a n ima ge of p . The converse direction is even mor e o b vious. 6.3 Remark. Every a dditiv e category A can be fully faithfully embedded into an idem- po ten t complete a dditiv e ca tegory A ∧ . The ob jects of A ∧ are the pairs ( A, p ) consis ting of a n ob ject A o f A and an idem- po ten t p : A → A while the sets of mor phisms ar e Hom A ∧ (( A, p ) , ( B , q )) = q ◦ Hom A ( A, B ) ◦ p with the co mpo sition induced by the co mpo sition in A . It is eas y to see tha t A ∧ is additive with bipro duct ( A, p ) ⊕ ( A ′ , p ′ ) = ( A ⊕ A ′ , p ⊕ p ′ ) and obviously the functor i A : A → A ∧ given by i A ( A ) = ( A, 1 A ) and i A ( f ) = f is fully faithful. In or der to see that A ∧ is idempo ten t co mplete, supp ose pf p is an idemp otent of ( A, p ) in A ∧ . A fortio ri pf p is an idemp otent of A and the ob ject ( A, p ) is iso morphic to the direct sum ( A, p − pf p ) ⊕ ( A, pf p ) via the morphisms h p − pf p pf p i and [ p − pf p pf p ]. The eq uation h p − pf p pf p i pf p [ p − pf p pf p ] =  0 0 0 pf p  prov es A ∧ to b e idemp otent co mplete. 6.4 Def inition. The pair ( A ∧ , i A ) constructed in Remark 6.3 is called the idemp oten t c ompletion of A . The next goa l is to characterize the pair ( A ∧ , i A ) b y a universal prop erty (Pr op osi- tion 6.10). W e ﬁr st work out some nice pro per ties of the explicit co nstruction. 6.5 Remark. If A is idempotent complete then i A : A → A ∧ is a n equiv alence of categories . In o rder to construct a quasi- in verse functor of i A : A → A ∧ , choose for each idempo ten t p : A → A a kernel K p , a n imag e I p and mo rphisms i p , j p , k p , l p as in Remark 6.2 and map the ob ject ( A, p ) of A ∧ to I p . A morphism ( A, p ) → ( B , q ) of A ∧ can b e written as q f p a nd map it to j q q f pi p = j q f i p . Obviously , this y ields a quasi-inv er se functor o f i A : A → A ∧ . 6.6 Remark. A functor F : A → B y ields a functor F ∧ : A ∧ → B ∧ , simply by setting F ∧ ( A, p ) = ( F ( A ) , F ( p )) and F ∧ ( q f p ) = F ( q ) F ( f ) F ( p ). Obviously , F ∧ i A = i B F and ( GF ) ∧ = G ∧ F ∧ . 6.7 Remark. A natura l tra nsformation α : F ⇒ G of functors A → B yields a unique natural transformation α ∧ : F ∧ ⇒ G ∧ . 20 Observe ﬁr st that a natura l transformatio n α ′ : F ′ ⇒ G ′ of functors A ∧ → B ∧ is completely determined b y its v a lues o n i A ( A ) b y the following ar gument . E very o b ject ( A, p ) of A ∧ is ca nonically a r etract of ( A, 1 A ) via the morphisms s : ( A, p ) → ( A, 1 A ) and r : ( A, 1 A ) → ( A, p ) given by p ∈ Ho m A ( A, A ). Therefore, by na turality , we must hav e α ′ ( A,p ) = α ′ ( A,p ) F ′ ( r ) F ′ ( s ) = G ′ ( r ) α ′ ( A, 1 A ) F ′ ( s ) , so α ′ is completely deter mined by its v alues on i A ( A ). Now given a na tural trans forma- tion α : F ⇒ G of functors A → B put α ∧ ( A,p ) = G ∧ ( r ) i B ( α A ) F ∧ ( s ) which is simply the element G ( p ) α A F ( p ) in Hom B ∧ ( F ∧ ( A, p ) , G ∧ ( A, p )) = G ( p ) ◦ Hom B ( F ( A ) , G ( A )) ◦ F ( p ) . It is easily check ed that this deﬁn ition of α ∧ indeed yields a natural tra nsformation F ∧ ⇒ G ∧ as desired. 6.8 Remark. The assignment α 7→ α ∧ is compatible with vertical and horizo n tal co m- po sition (see e.g. [44, I I.5, p. 42 f ]): F or functors F , G, H : A → B and natural trans- formations α : F ⇒ G and β : G ⇒ H we have ( β ◦ α ) ∧ = β ∧ ◦ α ∧ while for functor s F, G : A → B and H , K : B → C with natura l transfor mations α : F ⇒ G and β : H ⇒ K we hav e ( β ∗ α ) ∧ = β ∧ ∗ α ∧ . 6.9 Remark. A functor F : A ∧ → I to an idemp otent complete ca tegory is determined up to unique iso morphism by its v alues on i A ( A ). A natural tra nsformation α : F ⇒ G of functors A ∧ → I is determined by its v alues on i A ( A ). Indeed, exhibit each ( A, p ) as a retract of ( A, 1) as in Rema rk 6.7. Cons ider p as an idempo ten t of ( A, 1), s o F ( p ) is a n idemp otent o f F ( A, 1). Cho osing an imag e I F ( p ) of F ( p ) as in Remark 6.2, it is clear that the functor F m ust map ( A, p ) to I F ( p ) and is th us determined up to a unique isomo rphism. The claim ab out natural transformations is analogous to the argument in Remark 6 .7. 6.10 Prop osition. Th e functor i A : A → A ∧ is 2 -universal among funct ors fr om A to idemp otent c omplete c ate gories: (i) F or every funct or F : A → I to an idemp otent c omplete c ate gory I ther e ex ists a functor e F : A ∧ → I and a n atur al isomorphism α : F ⇒ e F i A . (ii) Giv en a functor G : A ∧ → I and a n atu r al tr ansformation γ : F ⇒ Gi A ther e is a unique n atur al tra nsformation β : e F ⇒ G su ch that γ = β ∗ α . Sketch of the Proof. Given a functor F : A → I , the cons truction outlined in Remark 6.9 y ields candidates for e F : A ∧ → I and α : F ⇒ e F i A and we leav e it to the reader to chec k that this works. O nce e F and α are ﬁxed, e β := γ ∗ α − 1 yields a natura l transformation e F i A ⇒ Gi A and the pr o cedure in Remar k 6 .9 s hows what an extension β : e F ⇒ G of e β must lo ok like and again w e lea ve it to the reader to chec k that this works. 21 6.11 Corollar y. L et A b e a smal l a dditive c ate gory. Th e functor i A : A → A ∧ induc es an e quivalenc e of functor c ate gories ( i A ) ∗ : Hom ( A ∧ , I ) ≃ − → Hom ( A , I ) for every idemp otent c omplete c ate gory I . Proof. Poin t (i) of Prop osition 6 .10 states that ( i A ) ∗ is essentially surjectiv e and it follows from po in t (ii) that it is fully faithful, hence it is an equiv alence of categor ies. 6.12 Exa mple. Let F be the categor y o f (ﬁnitely gener ated) free mo dules ov er a r ing R . Its idemp otent completion F ∧ is equiv alent to the ca tegory of (ﬁnitely generated) pro jective mo dules o ver R . Let now ( A , E ) b e an exact category . Call a sequence in A ∧ short exact if it is a direct summand in ( A ∧ ) →→ of a sequence in E and deno te the class o f short exa ct sequences in A ∧ b y E ∧ . 6.13 Proposition. The class E ∧ is an exact stru ctur e on A ∧ . The inclusion functor i A : ( A , E ) → ( A ∧ , E ∧ ) pr eserves and r eﬂe cts exactness and is 2 -universal among exact functors to idemp otent c omplete exact c ate gories: (i) L et F : A → I b e an exact fun ctor to an idemp otent c omplete exact c ate gory I . Ther e exists an exact functor e F : A ∧ → I to gether with a natur al isomorp hism α : F ⇒ e F i A . (ii) Giv en another exact functor G : A → I to gether with a natur al t r ansformation γ : F ⇒ Gi A , ther e is a u n ique natu r al tra n sformation β : e F ⇒ G such that γ = β ∗ α . Proof. T o prov e that E ∧ is an exact structure is straightforw ard but rather tedious, so we skip it. 1 Given this, it is clear that the functor A → A ∧ is exact and reﬂects exactness. If F : A → I is a n exact functor to an idempo ten t complete exact category then, as every sequence in E ∧ is a direct summand of a sequence in E , an extension e F of F as provided b y Pro po sition 6.10 must carry exa ct sequences in A ∧ to exact sequences in I . Thus statements (i) and (ii) follow from the co rresp onding statements in Prop osition 6.1 0. 6.14 Corollar y. F or a smal l exact c ate gory ( A , E ) , t he exact funct or i A induc es an e quivalenc e of the c ate gories of exact functors ( i A ) ∗ : Hom Ex (( A ∧ , E ∧ ) , I ) ≃ − → Hom Ex (( A , E ) , I ) to every idemp otent c omplete exact c ate gory I . 1 Thomason [57, A.9.1 (b)] gives a s hort ar gumen t relying on the embedding into an abeli an category , but it can be done b y completely elementary means as well. 22 7. W eak Idemp oten t Completeness Thomason in tro duced in [5 7, A.5 .1] t he notion o f an exact catego ry with “weakly split idempo ten ts” . It turns out that this is a pro per t y o f the underlying additiv e categ ory rather than the exa ct structure. Recall that in an arbitrar y category a mo rphism r : B → C is called a r etr action if there exists a se ction s : C → B o f r in the sense that rs = 1 C . Dually , a morphism c : A → B is a c or etr action if it admits a section s : B → A , i.e., sc = 1 A . Observe that retractions are epics and coretractions ar e monics. Moreover, a section of a retraction is a c oretraction and a section of a co retraction is a retraction. 7.1 Lemma. In an additive c ate gory A the fol lowing ar e e quivalent: (i) Ev ery c or etr action has a c okernel. (ii) Every r etr action has a kernel. 7.2 Definition. If the conditions of the previo us lemma hold then A is said to b e we akly idemp otent c omplete . 7.3 Remark. F reyd [19] uses the more descriptive terminology r etra ct s have c omple- ments for weakly idemp otent co mplete categories . He pr ov es in particular that a weakly idempo ten t complete catego ry with countable copro ducts is idemp otent complete. 7.4 Remark. Assume that r : B → C is a retra ction with se ction s : C → B . Then sr : B → B is an idemp otent. Le t us prov e that this idempotent giv es rise to a s plitting of B if r admits a k ernel k : A → B . Indeed, since r (1 B − sr ) = 0, there is a unique morphism t : B → A such that k t = 1 B − sr . It follows that k is a cor etraction b ecause k tk = (1 B − sr ) k = k implies that tk = 1 A . Mor eov er kts = 0, so ts = 0, hence [ k s ] : A ⊕ C → B is a n isomorphism with in verse [ t r ]. In pa rticular, the sequence s A → B → C a nd A → A ⊕ C → C a re isomorphic. Proof of Lemma 7.1. By dualit y it suﬃces to prov e that (ii) implies (i). Let c : C → B b e a cor etraction with section s . Then s is a retra ction and, assum- ing (ii), it admits a k ernel k : A → B . By the discus sion in Remar k 7.4, k is a co retraction with section t : B → A and it is obvious that t is a cokernel of c . 7.5 Corollar y. L et ( A , E ) b e an exact c ate gory. The fol lowing ar e e quivalent: (i) T he additive c ate gory A is we akly idemp otent c omplete. (ii) Every c or etr action is an admissible monic. (iii) Ev ery r etr action is an admissible epic. Proof. It follows from Remark 7.4 that every r etraction r : B → C a dmitting a kernel gives rise to a s equence A → B → C which is isomorphic to the split e xact sequence A ֌ A ⊕ C ։ C , hence r is an admissible epic by Lemma 2 .7, whence (i) implies (iii). By duality (i) implies (ii) as well. Conversely , every admissible monic has a cokernel and every admissible epic has a kernel, hence (ii) and (iii) b oth imply (i). 23 In a weakly idemp otent complete exact categor y the obscure axiom (Prop osition 2.16) has an e asier sta temen t—this is Heller’s ca ncellation a xiom [2 6, (P 2), p. 492 ]: 7.6 Proposition. L et ( A , E ) b e an exact c ate gory. The fol lowing ar e e quivalent: (i) T he additive c ate gory A is we akly idemp otent c omplete. (ii) Consi der two morphisms g : B → C and f : A → B . If g f : A ։ C is an admissible epic then g is an admissible epic. Proof. (i) ⇒ (ii): F orm the pull-back over g and g f and consider the diag ram A 1 A % % f   ∃ ! B ′ / / / / g ′   PB B g   A gf / / / / C which pro ves g ′ to be a retraction, so g ′ has a kernel K ′ → B ′ . Because the diagr am is a pull-bac k, the c ompo site K ′ → B ′ → B is a k ernel of g and now the dual of Prop osition 2.16 applies to yield that g is an admissible epic. F or the implication (ii) ⇒ (i) simply o bserve that (ii) implies that r etractions are admissible e pics. 7.7 Cor o llar y. An exact c ate gory is we akly id emp otent c omplete if and only if it has the fo l lowing pr op erty: al l morph isms g : B → C for which t her e is a c ommut ative diagr am A f / / / / gf     @ @ @ @ @ @ @ B g   C ar e admissible epics. Proof. A weakly idemp otent complete exact catego ry enjoys the stated prop erty b y Prop osition 7.6. Conv ersely , let r : B → C and s : C → B b e such that rs = 1 C . W e want to show that r is an a dmissible epic. The sequences B / / h 1 − r i / / B ⊕ C [ r 1 ] / / / / C and C / / h − s 1 i / / B ⊕ C [ 1 s ] / / / / B are split ex act, so [ r 1 ] and [ 1 s ] are admissible epics. But the diag ram B ⊕ C [ 1 s ] / / / / [ r 1 ] # # # # F F F F F F F F F B r   C is comm utative, hence r is an admissible epic. 24 7.8 Re mark ([47, 1.12]) . Ev ery small a dditiv e category A has a we ak idemp otent c om- pletion A ′ . O b jects o f A ′ are the pairs ( A, p ), where p : A → A is an idemp otent factoring as p = cr for some retraction r : A → X a nd coretraction c : X → A with rc = 1 B , while the morphisms are g iven b y Hom A ′ (( A, p ) , ( B , q )) = q ◦ Hom A ( A, B ) ◦ p. It is easy to see that the functor A → A ′ given on ob jects by A 7→ ( A, 1 A ) is 2-universal among functors from A to a weakly idempo ten t complete category . Mor eov er, if ( A , E ) is exa ct then s o is ( A ′ , E ′ ), where the se quences in E ′ are the direct summands in A ′ of s equences in E , and the functor A → A ′ preserves and reﬂects exactness and is 2-universal among exact functors to weakly idemp otent complete catego ries. 7.9 Remark. C ont rary to the co nstruction o f the idemp otent completion, there is the set-theoretic subtlet y that the w e ak idemp otent completion might not be well-deﬁned if A is not small: it is not clear a priori that the o b jects ( A, p ) form a cla ss—essentially for the sa me reason that the monics in a categ ory need not form a class, see e.g. the discussion in Bo rceux [6, p. 373f ]. 8. Admiss ible Morphisms and the Snak e Lemma 8.1 Definition. A morphism f : A → B in an exact category A f ◦ / / e ' ' ' ' N N N N N N B I 7 7 m 7 7 o o o o o is called admissible if it factor s as a co mpos ition of an admissible monic with an admiss ible epic. Admissible mor phisms will sometimes be displayed as ◦ / / in diagrams. 8.2 Remark. Let f b e an admissible morphism. If e ′ is an admissible epic and m ′ is an admissible monic then m ′ f e ′ is a dmissible if the co mpo sition is deﬁned. How ever, admissible morphisms are not closed under compositio n in gener al. Indeed, cons ider a morphism g : A → B whic h is not admissible. The mor phisms  1 g  : A → A ⊕ B and [ 0 1 ] : A ⊕ B → B are admissible since they are pa rt of split exact sequences. But g = [ 0 1 ]  1 g  is not admissible by hypothesis. 8.3 Re mark. W e choo se the terminology admissible morphism even though strict mor- phism seems to b e more sta ndard (see e.g. [51, 54]). By Exercis e 2.6 an admissible monic is the s ame thing as an admissible mor phism which happ ens to be monic. 8.4 Lemma ([26, 3.4]) . The factorization of an admissible morphism is unique u p to unique isomorphism. Mor e pr e cisely: In a c ommutative diagr am of the form A e / / / / e ′     I   m   i   I ′ / / m ′ / / i ′ ? ? B ther e exist un ique m orphisms i , i ′ making the diagr am c ommut ative. In p articular, i and i ′ ar e mut ual ly inverse isomorphisms. 25 Proof. Let k be a kernel of e . Since m ′ e ′ k = mek = 0 and m ′ is monic we hav e e ′ k = 0, hence there e xists a unique morphism i : I → I ′ such that e ′ = ie . Mo reov er , m ′ ie = m ′ e ′ = me and e epic imply m ′ i = m . Dually for i ′ . 8.5 Remark. An admissible morphism has an analysis (cf. [43, IX.2]) A f ◦ / / e ' ' ' ' N N N N N N B c ' ' ' ' O O O O O O K 7 7 k 7 7 o o o o o I 7 7 m 7 7 o o o o o C where k is a kernel, c is a co kernel, e is a coimag e a nd m is an image of f and the isomorphism classes o f K , I and C are w ell- deﬁned b y Lemma 8 .4. 8.6 Exercise . If A is an exact categ ory in which every morphism is a dmissible then A is abe lian. [A solution is given by F r eyd in [18, Prop osition 3 .1]]. 8.7 Lemma. A dmissible morphisms ar e stable under push-out along admissible monics and pul l-b ack along admissible epics. Proof. Let A ։ I ֌ B b e a n admissible epic-admissible monic facto rization of an ad- missible morphism. T o prove the claim ab out push-o uts cons truct the diagra m A / / / /     PO I / / / /     PO B     A ′ / / / / I ′ / / / / B ′ . Prop osition 2.15 yields that A ′ → I ′ is an a dmissible epic and the rest is clear. 8.8 Definition. A sequence of admissible morphisms A ′ f ◦ / / e ' ' ' ' N N N N N N A f ′ ◦ / / e ′ ' ' ' ' N N N N N N A ′′ I 8 8 m 8 8 p p p p p I ′ 7 7 m ′ 7 7 o o o o o is exact if I ֌ A ։ I ′ is short exact. Longer sequences of admissible m orphisms are exact if the se quence given by any t wo co nsecutiv e morphisms is exa ct. Since the term “exact” is hea v ily ov er loaded, we also use the synonym “acyclic” , in particular in connectio n with chain complexes. 8.9 Lemma (Fiv e Lemma, I I) . If the c ommut ative diagr am A 1 ∼ =   ◦ / / A 2 ∼ =   ◦ / / A 3 f   ◦ / / A 4 ∼ =   ◦ / / A 5 ∼ =   B 1 ◦ / / B 2 ◦ / / B 3 ◦ / / B 4 ◦ / / B 5 has exact r ows then f is an isomorphi sm. 26 Sketch of the Proof. By hypothesis w e may c ho o se factorizations A i ։ I i ֌ A i +1 of A i → A i +1 and B i ։ J i ֌ B i +1 of B i → B i +1 for i = 1 , . . . , 4. Using Lemma 8.4 a nd Exercise 3.3 there ar e iso morphisms I 1 ∼ = J 1 and I 2 ∼ = J 2 which one may insert int o the diagram without destroying its comm utativit y . Dually for I 4 ∼ = J 4 and I 3 ∼ = J 3 . The ﬁve lemma 3.2 then implies that f is an isomorphism. 8.10 Exercise. Assume that A is weakly idemp otent complete (Deﬁnition 7.2). (i) (Shar p F our Le mma) Consider a comm utative diagram A 1     ◦ / / A 2 ∼ =   ◦ / / A 3 f   ◦ / / A 4     B 1 ◦ / / B 2 ◦ / / B 3 ◦ / / B 4 with exact rows. Pr ov e that f is a n admissible monic. Dualize. (ii) (Shar p Fiv e Lemma) If the commutativ e diagram A 1     ◦ / / A 2 ∼ =   ◦ / / A 3 f   ◦ / / A 4 ∼ =   ◦ / / A 5     B 1 ◦ / / B 2 ◦ / / B 3 ◦ / / B 4 ◦ / / B 5 has exact rows then f is an is omorphism. Hint: Use Prop os ition 7.6, Exercise 2 .6, Exercise 3.3 as well a s Co rollary 3.2. 8.11 Proposition (Ker–Coker–Sequence) . Assume that A is a we akly idemp otent c om- plete exact c ate gory. Le t f : A → B and g : B → C b e admissib le morphi sms such that h = g f : A → C is admissible as wel l. Ther e is an exact se quenc e Ker f / / / / Ker h ◦ / / Ker g ◦ / / Coker f ◦ / / Coker h / / / / Coker g dep ending natura l ly on the diagr am h = g f . Proof. Observe that the morphism Ker f ֌ A f actors ov er K er h ֌ A via a unique morphism K er f → Ker h which is an admissible mo nic by Pro po sition 7.6. Let Ker h ։ X be a cokernel of Ker f ֌ Ker h . Dually , there is an admissible epic Coker h ։ Coker g with Z ֌ Cok er h a s kernel. The Noether isomorphism 3.5 implies that there are tw o commut ative diagrams with exa ct rows and columns Ker f / / / / Ker h     / / / / X     Ker f / / / / A / / / /     Im f     Im h Im h and Im h / / / / Im g / / / /     Z     Im h / / / / C / / / /     Coker h     Coker g Coker g . 27 It is easy to see that there is a n admissible monic X ֌ Ker g whose cokernel w e denote b y Ker g ։ Y . Therefor e the 3 × 3-lemma yields a commutativ e dia gram with exact rows and columns X     / / / / Ker g     / / / / Y     Im f     / / / / B     / / / / Coker f     Im h / / / / Im g / / / / Z. The desir ed K er–Coker – sequence is now obtained by splicing: Start w ith the ﬁrst r ow of the ﬁrst diagram, splice it with the ﬁrst r ow of the third dia gram, a nd contin ue with the third row of the third diag ram and the third r ow of the second diagram. The naturalit y assertion is o b vious. 8.12 Lemma. L et A b e an exact c ate gory in which e ach c ommutative triangle of admis- sible morphisms yield s an exact Ker – Coker –se qu enc e wher e exactness is understo o d in the sense of admissible morphisms. Then A is we akly idemp otent c omplete. Proof. W e chec k the criterion in Coro llary 7.7. W e need to show tha t in every commu- tative diagra m of the form A f / / / / h     @ @ @ @ @ @ @ B g   C the morphism g is an admissible epic. Given such a diagram, consider the diagr am Ker f / / / / ◦ E E E E 0 " " E E E E A h     C whose asso ciated Ker–Co ker –seq uence is 0 ◦ / / Ker f ◦ / / Ker h ◦ / / B ◦ g / / C ◦ / / 0 so that g is an admissible epic. The Ker–Coker – sequence immediately yields the following version of the snake lemma, the neat pr o of o f which was p ointed out to the author by Matthias K ¨ unzer. 8.13 Cor ollar y (Snak e Lemma, I) . L et A b e we akly idemp otent c omplete. Consider a morphism of short ex act se quenc es A ′ ֌ A ։ A ′′ and B ′ ֌ B ։ B ′′ with admissible 28 c omp onents. Ther e is a c ommutative diagr am K ′     k / / K     k ′ / / K ′′     A ′ / / / / ◦   A / / / / ◦   A ′′ ◦   B ′ / / / /     B / / / /     B ′′     C ′ c / / C c ′ / / C ′′ with exact ro ws and c olumns and ther e is a c onne cting morphism δ : K ′′ → C ′ ﬁtting into an exact se quenc e K ′ / / k / / K ◦ k ′ / / K ′′ δ ◦ / / C ′ ◦ c / / C c ′ / / / / C ′′ dep ending natura l ly on the morphism of short ex act se quenc es. 8.14 Remark. U sing the notations of the pro of o f Lemma 8.12 consider the diag ram Ker f 0 ◦   / / / / A h ◦   / / / / B ◦   C C / / 0 . The sequence Ker f / / / / Ker h ◦ / / B ◦ g / / C ◦ / / 0 / / / / 0 provided by the snake lemma shows that g is an a dmissible epic. It follows that the s nake lemma can only hold if the categ ory is weakly idemp otent complete. Proof of Cor olla r y 8.13. By P rop osition 3.1 and Lemma 8.7 we get the co mm uta- tiv e diagram A ′ ◦   / / / / BC A ◦   / / / / A ′′ B ′ / / / / D ◦   / / / / BC A ′′ ◦   B ′ / / / / B / / / / B ′′ [more explicitly , the diagram is obtained by forming the push-out A ′ AB ′ D ]. The Ker– Coker–sequence of the co mm utative tr iangle of admissible morphisms A ◦ / / ◦ @ @ @ @ @ @ @ @ D ◦   B yields the des ired r esult by Remark 2.13. 29 8.15 Exer cise (Snake Lemma, II, cf. [26, 4.3 ]) . F ormulate and prov e a sna k e lemma for a dia gram of the form A ′ ◦ / / ◦   A ◦ / / ◦   A ′′ / / ◦   0 0 / / B ′ ◦ / / B ◦ / / B ′′ in weakly idempotent complete ca tegories. Prove Ker ( A ′ → A ) = K er ( K ′ → K ) and Coker ( B → B ′′ ) = Coker ( C → C ′′ ). Hint: Reduce to Corollary 8 .13 by using Pro po sition 7.6 and the No ether isomor- phism 3.5. 8.16 Remark. Heller [26, 4 .3] gives a dir ect pro of o f the snake lemma starting from his axioms. Using the No ether isomorphism 3.5 a nd the 3 × 3-lemma 3 .6 as well as Prop osition 7.6, Heller’s pro of is easily a dapted to a proo f from Quillen’s axioms. 9. Chain Complexes and Chain Homotopy The notion of chain co mplexes ma kes sense in ev ery additive category A . A (chai n ) c omplex is a diagram ( A • , d • A ) · · · − → A n − 1 d n − 1 A − − − → A n d n A − − → A n +1 − → · · · sub ject to the condition t hat d n d n − 1 = 0 for all n and a ch ain map is a morphism of suc h diagrams. The category of complexes and c ha in maps is denoted by Ch ( A ). Obviously , the category Ch ( A ) is additive. 9.1 Lemma. If ( A , E ) is an exact c ate gory t hen Ch ( A ) is an exact c ate gory with r esp e ct to the class Ch ( E ) of short se qu enc es of chain maps which ar e exact in e ach de gr e e. If A is ab elian then so is Ch ( A ) . Proof. The p oint is that (as in every functor ca tegory) limits and colimits of diagrams in Ch ( A ) a re obtained b y taking the limits and colimits p oint wise (in each degree), in particular push-outs under admissible monics a nd pull-bac ks ov er admissible epics exist and yield admissible monics and epics. The rest is ob v ious. 9.2 Definition. The mapping c one of a chain map f : A → B is the complex cone ( f ) n = A n +1 ⊕ B n with diﬀeren tial d n f = h − d n +1 A 0 f n +1 d n B i . Notice that d n +1 f d n f = 0 precis ely b eca use f is a c hain map. It is plain that the mapping cone deﬁnes a functor from the ca tegory o f morphisms in Ch ( A ) to Ch ( A ). The tr anslation funct or on Ch ( A ) is deﬁned to be Σ A = cone ( A → 0). More explic- itly , Σ A is the co mplex with comp onents (Σ A ) n = A n +1 and diﬀerentials d n Σ A = − d n +1 A . If f is a chain ma p, its translate is given by (Σ f ) n = f n +1 . Clearly , Σ is an additiv e automorphism of Ch ( A ). 30 The st rict triangle o ver the chain map f : A → B is the 3-p erio dic (or ra ther 3- helicoidal, if you insist) sequence A f − → B i f − → cone ( f ) j f − → Σ A Σ f − − → Σ B Σ i f − − → Σ cone ( f ) Σ j f − − → · · · , where the chain map i f has components [ 0 1 ] and j f has components [ 1 0 ]. 9.3 Remark. L et f : A → B b e a chain map. Observe that the sequence of chain maps B i f − → cone ( f ) j f − → Σ A splits in eac h deg ree, how ever it need not be a split exact sequence in Ch ( A ), bec ause the degreewise splitting maps need not assemble to chain maps. In fact, it is straightforward to verify that the a bove sequence is split exa ct in Ch ( A ) if and o nly if f is chain homotopic to z ero in the sense of Deﬁnition 9.5. 9.4 Exercise. Assume that A is an abelian catego ry . Prov e that the strict tria ngle o ver the c ha in map f : A → B gives r ise to a long exact ho mology se quence · · · − → H n ( A ) H n ( f ) − − − − → H n ( B ) H n ( i f ) − − − − → H n (cone ( f )) H n ( j f ) − − − − − → H n +1 ( A ) − → · · · . Deduce that f induces an isomor phism of H ∗ ( A ) with H ∗ ( B ) if and o nly if cone ( f ) is acyclic. 9.5 Definition. A chain map f : A → B is chain homotopic t o zer o if there exist morphisms h n : A n → B n − 1 such that f n = d n − 1 B h n + h n +1 d n A . A c hain co mplex A is called nul l-homotopic if 1 A is c hain ho motopic to zero. 9.6 Remark. The maps which a re chain homotopic to zero form an idea l in Ch ( A ), that is to say if h : B → C is chain homo topic to zero then so are hf and g h for all morphisms f : A → B and g : C → D , if h 1 and h 2 are c hain ho motopic to zer o then so is h 1 ⊕ h 2 . The set N ( A, B ) of c hain maps A → B which are c ha in homoto pic to zero is a s ubgroup of the ab elian group Ho m Ch ( A ) ( A, B ). 9.7 Definition. The homotopy c ate gory K ( A ) is the category with th e chain complexes ov er A as ob jects and Hom K ( A ) ( A, B ) := Ho m Ch ( A ) ( A, B ) / N ( A, B ) as mo rphisms. 9.8 Remark. Notice t hat ev er y n ull-homotopic complex is isomorphic to the zero o b ject in K ( A ). It turns out that K ( A ) is additiv e, but it is v er y ra rely ab elian or ex act with resp ect to a non-trivial exact s tructure (see V er dier [58, Ch.I I, 1 .3.6]). How ever, K ( A ) has the structure of a triangulate d c ate gory induced by the strict triangles in Ch ( A ), see e.g. V erdier [58], Be ˘ ılinson-Bernstein-Deligne [4], Gelfand-Manin [23], Grivel [8, Chap- ter I], Ka shiwara-Sc hapira [34], Keller [38], Neeman [48] or W eibe l [59]. 9.9 Remark. F o r each ob ject A ∈ A , deﬁne cone ( A ) = cone (1 A ). Notice that co ne ( A ) is n ull- homotopic with [ 0 1 0 0 ] as co n tracting homo topy . 9.10 Remark. If f and g are c ha in homotopy equiv alent, i.e., f − g is chain homotopic to zer o, then cone ( f ) and co ne ( g ) ar e isomorphic in Ch ( A ) but the isomorphism and its homotop y class will generally depend on the choice of a chain homoto p y . In pa rticular, the mapping cone co nstruction do es not yield a functor deﬁned on mor phisms of K ( A ). 31 9.11 Remark. A chain map f : A → B is c hain homotopic to zero if and only if it factors as hi A = f o ver h : cone ( A ) → B , where i A = i 1 A : A → co ne ( A ). Moreov er , h has comp onents [ h n +1 f n ], where the family of morphisms { h n } is a chain homo topy of f to ze ro. Similarly , f is chain homoto pic to zero if and only if f factors through j Σ − 1 B = j 1 Σ − 1 B : cone (Σ − 1 B ) → B . 9.12 Remark. The mapping cone co nstruction yie lds the push-out diagra m A f / / i A   PO B i f   cone ( A ) h 1 0 0 f i / / cone ( f ) in Ch ( A ). Now supp ose that g : B → C is a chain map such that g f is chain homotopic to zero. By Remark 9 .11, g f factors ov er i A and using the pu sh-out prop erty of the above diagram it follo ws that g facto rs o ver i f . This construction will dep end on the c hoice of an explicit c hain homo topy g f ≃ 0 in genera l. In particular, cone( f ) is a we ak c okernel in K ( A ) of the ho motopy clas s of f in that it has the factor ization prop er t y of a cokernel but w ithout uniqueness. Similar ly , Σ − 1 cone ( f ) is a we ak kernel of f in K ( A ) . 10. Acyclic Complexes and Quasi-Isom orphisms The pres en t section is probably only of interest to reader s acquainted with tria ngu- lated categ ories or a t le ast with the c onstruction of the der ived categor y o f a n ab elian category . Af ter giving the fundament al deﬁnition of a cyclicit y of a complex ov er an exact category , we may formulate the in timately connected notion of quasi- isomorphisms. W e will give an elemen tar y pro of of the fact that the homotopy category Ac ( A ) of acyclic complexes over an e xact catego ry A is a tr iangulated catego ry . It turns out that Ac ( A ) is a strictly fu ll subc ategory of the homotop y category of ch ain complexes K ( A ) if and only if A is idemp otent complete, a nd in this case Ac ( A ) is even thic k in K ( A ). Since thic k sub categories are strictly full b y deﬁnition, Ac ( A ) is thick if and o nly if A is idempotent co mplete. By [48, Cha pter 2], the V e rdier quotient K / T is deﬁned for any (strictly full) tri- angulated sub category T o f a tria ngulated catego ry K and it coincides with the V erdier quotient K / ¯ T , where ¯ T is the t hick closur e of T . The ca se we ar e interested in is K = K ( A ) and T = Ac ( A ). The V erdier quotien t D ( A ) = K ( A ) / Ac ( A ) is the derive d c ate gory of A . If A is idemp otent complete then Ac ( A ) = Ac ( A ) and it is clear that quasi-is omorphisms are then precisely the chain maps with a cyclic ma pping cone. If A fails to b e idemp otent complete, it turns out that the thick closur e Ac ( A ) of Ac ( A ) is the same a s the closur e of Ac ( A ) under isomor phisms in K ( A ), so a chain map f is a quasi-isomor phism if and only if co ne ( f ) is ho motopy eq uiv a lent to an acyclic complex. Similarly , the deriv ed categories of bounded, left b ounded o r righ t bounded complexes are constructed a s in the ab elian setting. It is useful to notice that for ∗ ∈ { + , − , b } the category Ac ∗ ( A ) is thick in K ∗ ( A ) if and only if A is w eakly idempotent co mplete, which leads to an easie r description of quasi-isomor phisms. 32 If B is a fully exact sub category of A , the inclusion B → A yields a c anonical functor D ( B ) → D ( A ) and we state conditions which ensure that this functor is essentially surjective or fully faithful. W e end the section with a short discussion of Deligne’s a pproach to total derived functors. 10.1. The Homotopy Cate gory of Acy clic Complexes 10.1 Definition. A chain complex A ov er an exact c ategory is called acyclic (or exact ) if each diﬀerential factors as A n ։ Z n +1 A ֌ A n +1 in such a wa y that each sequence Z n A ֌ A n ։ Z n +1 A is e xact. 10.2 Remark. An acyclic complex is a complex with admiss ible diﬀeren tials (Deﬁni- tion 8.1) which is exac t in the sense of Deﬁnition 8.8. In particular, Z n A is a kernel of A n → A n +1 , an imag e and coimag e of A n − 1 → A n and a co kernel o f A n − 2 → A n − 1 . 10.3 Lemma (N eeman [4 7, 1.1]) . The mapping c one of a chain map f : A → B b etwe en acyclic c omplexes is acyclic. Proof. An easy dia gram chase shows that the dotted morphisms in the diagram A n − 1 d n − 1 A / / j n − 1 A ( ( ( ( Q Q Q Q Q Q f n − 1   A n d n A / / j n A ( ( ( ( Q Q Q Q Q Q f n   A n +1 f n +1   Z n A 7 7 i n A 7 7 o o o o o o ∃ ! g n   Z n +1 A 6 6 i n +1 A 6 6 l l l l l l ∃ ! g n +1   Z n B ' ' i n B ' ' O O O O O O Z n +1 B ( ( i n +1 B ( ( R R R R R R B n − 1 d n − 1 B / / j n − 1 B 6 6 6 6 n n n n n n B n d n B / / j n B 6 6 6 6 n n n n n n B n +1 exist and ar e the unique morphisms g n making the diag ram commutativ e . By P rop osition 3 .1 we ﬁnd ob jects Z n C ﬁtting into a commutativ e diagram A n − 1 d n − 1 A / / j n − 1 A ( ( ( ( Q Q Q Q Q Q Q f ′ n − 1   A n d n A / / j n A ( ( ( ( Q Q Q Q Q Q Q f ′ n   A n +1 f ′ n +1   Z n A 7 7 i n A 7 7 o o o o o o g n   BC Z n +1 A 6 6 i n +1 A 6 6 l l l l l l g n +1   BC Z n − 1 C h n − 1 6 6 6 6 m m m m m m f ′′ n − 1   BC Z n C h n 6 6 6 6 m m m m m m f ′′ n   BC Z n +1 C f ′′ n +1   Z n B 7 7 k n 7 7 o o o o o o ' ' i n B ' ' O O O O O O Z n +1 B 6 6 k n +1 6 6 l l l l l l ( ( i n +1 B ( ( R R R R R R B n − 1 d n − 1 B / / j n − 1 B 6 6 6 6 m m m m m m m B n d n B / / j n B 6 6 6 6 m m m m m m m B n +1 where f n = f ′′ n f ′ n and the quadrila terals marked BC are bica rtesian. Recall that the ob jects Z n C are obtained by forming the push-outs under i n A and g n (or the pull-backs ov er j n B and g n +1 ) and tha t Z n B ֌ Z n C ։ Z n +1 A is s hort exact. 33 It follo ws from Coro llary 2.14 that for each n the se quence Z n − 1 C / / » − i n A h n − 1 f ′′ n − 1 – / / A n ⊕ B n − 1 [ f ′ n k n j n − 1 B ] / / / / Z n C is short exact and the comm utative diag ram A n ⊕ B n − 1 » − d n A 0 f n d n − 1 B – / / [ f ′ n k n j n − 1 B ] L L L & & & & L L L A n +1 ⊕ B n " − d n +1 A 0 f n +1 d n B # / / [ f ′ n +1 k n +1 j n B ] M M M & & & & M M M A n +2 ⊕ B n +1 Z n C 9 9 » − i n +1 A h n f ′′ n – 9 9 r r r r r r r r r r Z n +1 C 7 7 » − i n +2 A h n +1 f ′′ n +1 – 7 7 o o o o o o o o o o prov es that co ne ( f ) is acyclic. 10.4 Remark. Ret aining the notations of the pro of we have a s hort ex act se quence Z n B ֌ Z n C ։ Z n +1 A. This seq uence exhibits Z n C = Ker h − d n +1 A 0 f n +1 d n B i as an extension of Z n +1 A = Ker d n +1 A b y Z n B = K er d n B . Let Ac ( A ) b e the full s ubca tegory of the homotopy category K ( A ) consisting of acyclic complexes o ver the exact category A . It follo ws fro m Propo sition 2.9 that the direct sum of tw o acyclic complexes is acyclic. Th us Ac ( A ) is a full additive sub categ ory of K ( A ) . The previous lemma implies that even more is tr ue: 10.5 Corollar y. The homotopy c ate gory of acyclic c omplexes Ac ( A ) is a triangulate d sub c ate gory of K ( A ) . 10.6 Remark. F o r reaso ns of conv enience, many authors assume that triangulated sub- categories are not only full but strictly ful l . W e do not do so b ecause Ac ( A ) is closed under isomor phisms in K ( A ) if a nd only if A is idempotent complete, see Pro po si- tion 10.9. 10.7 Lemma. Assume that ( A , E ) is idemp otent c omplete. Every r etr act in K ( A ) of an acyclic c omplex A is acyclic. Proof (c f. [36, 2.3 a)]). Let the chain map f : X → A b e a cor etraction, i.e., there is a chain map s : A → X such that s n f n − 1 X n = d n − 1 X h n + h n +1 d n X for some mor phisms h n : X n → X n − 1 . Obviously , the co mplex I X with comp onents ( I X ) n = X n ⊕ X n +1 and diﬀerential [ 0 1 0 0 ] is acyclic. T here is a c hain map i X : X → I X given by i n X = h 1 X n d n X i : X n → X n ⊕ X n +1 and the chain ma p h f i X i : X → A ⊕ I X 34 has the chain map [ s n − d n − 1 X h n − h n +1 ] : A n ⊕ X n ⊕ X n +1 → X n as a left inv erse. Hence, o n replacing the acyclic complex A by the acyclic complex A ⊕ I X , we may a ssume that f : X → A has s as a left in verse in Ch ( A ). But then e = f s : A → A is an idempotent in C h ( A ) and it induces an idemp otent o n the exa ct sequences Z n A ֌ A n ։ Z n +1 A witnessing that A is acy clic as in the ﬁrst diagra m of the pro of of Lemma 10.3. This means that Z n A ֌ A n ։ Z n +1 A decomp oses as a direct sum of tw o short exact sequences (Corollary 2.18) since A is idemp otent complete. Therefore the a cyclic complex A = X ′ ⊕ Y ′ is a direct sum of the ac yclic complexes X ′ and Y ′ , a nd f induces an isomorphism from X to X ′ in Ch ( A ). The det ails are lef t to the reader. 10.8 Exer cise. Prov e that the s equence X → cone ( X ) → Σ X from Remar k 9 .3 is isomorphic to a sequence X → I X → Σ X in Ch ( A ). 10.9 Proposition ([38, 11.2 ]) . The fol lowing ar e e quivalent: (i) Ev ery n ul l-homotopic c omplex in Ch ( A ) is acyclic. (ii) The c ate gory A is idemp otent c omplete. (iii) T he class of acyclic c omplexes is close d under isomorphisms in K ( A ) . Proof (Ke ller). Let us prov e that (i) implies (ii). Let e : A → A b e an idempo ten t of A . Consider the complex · · · 1 − e − − → A e − → A 1 − e − − → A e − → · · · which is n ull-homotopic. By (i) this complex is acyclic. This means by deﬁnition that e has a k er nel and hence A is idemp otent complete. Let us prov e that (ii) implies (iii). Assume that X is isomorphic in K ( A ) to a n acyclic complex A . Using the construction in the pro of of Lemma 10.7 o ne shows that X is a direct summand in Ch ( A ) of the acyclic complex A ⊕ I X and we conclude by Lemma 10.7. That (iii) implies (i) follows from the fact that a n ull-homoto pic complex X is iso- morphic in K ( A ) to the (acyclic) zero complex and hence X is acyclic. 10.10 Re mark. Recall that a sub categor y T o f a tria ngulated category K is called thick if it is strictly full a nd X ⊕ Y ∈ T implies X, Y ∈ T . 10.11 Corollar y. The t riangulate d su b c ate gory Ac ( A ) of K ( A ) is t hick if and only if A is idemp otent c omplete. 10.2. Bounde dness Conditions A complex A is called left b ounde d if A n = 0 for n ≪ 0 , right b ounde d if A n = 0 for n ≫ 0 and b ounde d if A n = 0 for | n | ≫ 0. 10.12 Definition. Denote by K + ( A ), K − ( A ) and K b ( A ) the full sub categ ories of K ( A ) g enerated by the left b ounded complexes, right b ounded complexes and b ounded complexes o ver A . 35 Observe tha t K b ( A ) = K + ( A ) ∩ K − ( A ). Note further that K ∗ ( A ) is not closed under isomorphisms in K ( A ) for ∗ ∈ { + , − , b } unless A = 0. 10.13 Definition. F or ∗ ∈ { + , − , b } we deﬁne Ac ∗ ( A ) = K ∗ ( A ) ∩ Ac ( A ). Plainly , K ∗ ( A ) is a full triang ulated subca tegory of K ( A ) and Ac ∗ ( A ) is a full triangulated subcateg ory o f K ∗ ( A ) b y Lemma 10 .3. 10.14 Proposition. The fol lowing assertions ar e e quivalent: (i) T he s u b c ate gories Ac + ( A ) and Ac − ( A ) of K + ( A ) and K − ( A ) ar e thick. (ii) The sub c ate gory Ac b ( A ) of K b ( A ) is thick. (iii) T he c ate gory A is we akly idemp otent c omplete. Proof. Since Ac b ( A ) = Ac + ( A ) ∩ Ac − ( A ), w e see that (i) implies (ii). Let us prove that (ii) implies (iii). Let s : B → A and t : A → B be morphisms of A such that ts = 1 B . W e need to prov e that s has a cokernel a nd t has a kernel. The complex X given by · · · − → 0 − → B s − → A 1 − st − − − → A t − → B − → 0 − → · · · is a direct summand of X ⊕ Σ X and the latter complex is acyclic since there is an isomorphism in C h ( A ) B 1   [ 1 0 ] / / B ⊕ A h 0 − t s 1 − st i   [ 0 0 0 1 ] / / A ⊕ A h − 1+ st st st 1 − st i   [ 1 0 0 0 ] / / A ⊕ B h 1 − st − s t 0 i   [ 0 1 ] / / B 1   B [ 0 s ] / / B ⊕ A h − s 0 0 1 − st i / / A ⊕ A h − 1+ st 0 0 t i / / A ⊕ B [ − t 0 ] / / B where the upp er row is obviously acyclic a nd the low er row is X ⊕ Σ X . Since Ac b ( A ) is thic k, w e conclude that X is acyclic, so that s has a cokernel and t has a k ernel. Therefore A is w ea kly idemp otent co mplete. Let us prov e tha t (iii) implies (i). Assume that X is a direct summand in K + ( A ) of a complex A ∈ Ac + ( A ). This means that we a re given a chain map f : X → A for which there exists a chain map s : A → X and mor phisms h n : X n → X n − 1 such that s n f n − 1 X n = d n − 1 X h n + h n +1 d n X . On replacing A by the a cyclic complex A ⊕ I X as in the pr o of of Lemma 10.7, w e ma y assume that s is a left in verse of f in Ch + ( A ). In particular, s ince A is a ssumed to be w eakly idempo ten t complete, P rop osition 7 .6 implies that each f n is an admissible monic and that each s n is an admissible epic. Moreov er , as b oth complexes X and A are left b ounded, w e ma y assume that A n = 0 = X n for n < 0 . It follows that d 0 A : A 0 ֌ A 1 is a n a dmissible mo nic since A is ac yclic. But then d 0 A f 0 = f 1 d 0 X is a n admissible monic, hence Prop osition 7.6 implies that d 0 X is a n admissible monic as well. Let e 1 X : X 1 ։ Z 2 X b e a cokernel of d 0 X and let e 1 A : A 1 ։ Z 2 A 36 be a cokernel of d 0 A . The do tted mo rphisms in the diagram X 0 / / d 0 X / /   f 0   X 1 e 1 X / / / /   f 1   Z 2 X   g 2   A 0 / / d 0 A / / s 0     A 1 e 1 A / / / / s 1     Z 2 A t 2     X 0 / / d 0 X / / X 1 e 1 X / / / / Z 2 X are uniquely determined b y requiring the resulting diag ram to be commut ative. Since s 0 f 0 = 1 X 0 and s 1 f 1 = 1 X 1 it follows that t 2 g 2 = 1 Z 2 X , so t 2 is an admissible epic and g 2 is a n admissible monic by Pro po sition 7 .6. Now since A and X are complexes, there are unique maps m 2 X : Z 2 X → X 2 and m 2 A : Z 2 A → A 2 such that d 1 X = m 2 X e 1 X and d 1 A = m 2 A e 1 A . Note that m 2 A is an admissible monic since A is acyclic. The upp er square in the diagr am Z 2 X m 2 X / /   g 2   X 2   f 2   Z 2 A / / m 2 A / / t 2     A 2 s 2     Z 2 X m 2 X / / X 2 is comm utative b ecause e 1 X is epic and the low er squar e is commutativ e because e 1 A is epic. F r om the commut ativity of the upp er square it follo ws in particular that m 2 X is a n admissible monic by Prop osition 7.6. An easy induction now shows that X is acyclic. The assertion a bo ut Ac − ( A ) follows by dualit y . 10.15 Remark. The isomorphism o f complexes i n the proof that (ii) implies (iii) appears in Neeman [4 7, 1 .9]. 10.3. Quasi-Isomorphi s m s In ab elian c ategories, quasi- isomorphisms are deﬁned to b e chain maps inducing an isomorphism in homo logy . T aking the observ ation in Exe rcise 9 .4 a nd Prop osition 10.9 in to account, one ar rives at the following gener alization for general exact ca tegories: 10.16 Definition. A chain ma p f : A → B is called a quasi -isomorphism if its mapping cone is ho motopy equiv alent to an acyclic co mplex. 10.17 Remark. Assume that A is idemp otent complete. By Prop osition 10.9, a chain map f is a quasi-isomor phism if and only if cone ( f ) is acyclic. In particular, for abelia n categories , a qua si-isomorphism is the s ame thing as a chain map inducing an isomor- phism on homology . 37 10.18 Remark. If p : A → A is an idemp otent in A which does not split, then the complex C g iven b y · · · 1 − p − − → A p − → A 1 − p − − → A p − → · · · is n ull- homotopic but not acyclic. Ho wev er, f : 0 → C is a chain homotopy equiv alence, hence it sho uld b e a quasi-isomorphism, but cone ( f ) = C fails to b e acyclic. 10.4. The Deﬁnition of the Derive d Cate gory The derive d c ate gory of the exa ct categ ory A is deﬁned to b e the V er dier quotient D ( A ) = K ( A ) / Ac ( A ) as described e.g. in Neeman [48, Chapter 2] o r Keller [38, §§ 10, 1 1]. F or the des cription of deriv ed functors given in section 10.6 it is useful to recall that the V erdier quotient can b e explicitly descr ibed by a calculus of fractions. A morphism A → B in D ( A ) ca n be repr esented by a fr action ( f , s ) A f − → B ′ s ← − B where f : A → B is a morphism in K ( A ) and s : B → B ′ is a qua si-isomorphism in K ( A ). Two fr actions ( f , s ) and ( g , t ) are equiv alent if there ex ists a fraction ( h, u ) and a c ommu tative diagram B ′   A f 7 7 o o o o o o h / / g ' ' O O O O O O B ′′′ B s g g O O O O O O u o o t w w o o o o o o B ′′ O O or, in words, if the fractions ( f , s ) and ( g , t ) hav e a common expansio n ( h, u ). W e re fer to Keller [38 , §§ 9, 10] for further details. When dealing with the boundedness co ndition ∗ ∈ { + , − , b } we deﬁne D ∗ ( A ) = K ∗ ( A ) / Ac ∗ ( A ) . It is not diﬃcult to prove that the cano nical functor D ∗ ( A ) → D ( A ) is an equiv alence betw een D ∗ ( A ) and the full sub catego ry of D ( A ) generated b y th e complexes satisfying the boundedness condition ∗ , se e Keller [38, 11.7]. 10.19 Remark. If A is idempotent co mplete then a chain map becomes an is omorphism in D ( A ) if and only if its cone is ac yclic by Corolla ry 10 .11. If A is weakly idemp otent complete then a chain map in Ch ∗ ( A ) beco mes an iso morphism in D ∗ ( A ) if a nd only if its co ne is acyclic by Prop osition 10.14. 10.5. Derive d Cate gories of F ul ly Exact S u b c ate gories The pr o of of the following lemma is straightforward and left to the reader as a n exercise. That a dmissible monics and epics are closed under compo sition follows from the Noether isomo rphism 3.5. 38 10.20 Lemma. L et A b e an exact c ate gory and supp ose that B is a ful l additive su b- c ate gory of A which is close d u n der ex tensions in t he sense that the existenc e of a short exact se qu enc e B ′ ֌ A ։ B ′′ with B ′ , B ′′ ∈ B implies that A is isomorphic to an obje ct of B . The se quenc es in B which ar e exact in A form an exact stru ctur e on B . 10.21 Definition. A ful ly exact su b c ate gory B of an exact category A is a ful l additive subc ategory whic h is closed under extensions and equipp ed with the exact structure from the previous lemma . 10.22 Theorem ([38, 1 2.1]) . L et B b e a ful ly exact sub c ate gory of A and c onsider t he functor D + ( B ) → D + ( A ) induc e d by the inclusion B ⊂ A . (i) Assume that for every obje ct A ∈ A ther e ex ist s an admissible monic A ֌ B with B ∈ B . F or every left b ounde d c omplex A over A ther e ex ists a left b ounde d c omplex B over B and a quasi-isomorphism A → B . In p articular D + ( B ) → D + ( A ) is essential ly surje ctive. (ii) Assume that for every short exact se quenc e B ′ ֌ A → A ′′ of A with B ′ ∈ B ther e exists a c ommutative diagr am with exact r ows B ′ / / / / A / / / /   A ′′   B ′ / / / / B / / / / B ′′ . F or every quasi-isomorphism s : B → A in K + ( A ) with B a c omplex over B ther e exists a morphism t : A → B ′ in K + ( A ) such that ts : B → B ′ is a quasi- isomorphi sm . In p articular, D + ( B ) → D + ( A ) is ful ly faithful. 10.23 Remark. The co ndition in (ii) holds if condition (i) holds a nd, moreov er, for a ll short exac t sequences B ′ ֌ B ։ A ′′ with B ′ , B ∈ B it follows that A ′′ is isomor phic to an ob ject in B . T o see this, start with a short exa ct sequence B ′ ֌ A ։ A ′′ , then choos e an admissible monic A ֌ B , for m the push- out AA ′′ B B ′′ and apply Prop osition 2.12 and Prop osition 2 .15. 10.24 Example. Let I be the full sub catego ry spanned by the inje ctive ob jects of the e xact ca tegory ( A , E ), see Deﬁnition 11.1. Clearly , I is a fully exa ct sub catego ry of A (the induced exact structure consists of the split exact sequences) and it s atisﬁes condition (ii) o f Theore m 10 .22. If I satisﬁes c ondition (i) then there are enough inje c- tives in ( A , E ), see Deﬁnition 11.9. A quasi- isomorphism o f left bounded complexes of injectiv es is a chain homotopy equiv alence, hence K + ( I ) is equiv alent to D + ( I ). B y Theorem 10.22 K + ( I ) is equiv alent to the full sub category of D ( A ) spanned by the left bo unded complexes with injectiv e co mpo nen ts. Moreover, if ( A , E ) has enough injectives, then the functor K + ( I ) → D + ( A ) is an equiv alence of tr iangulated categories. 10.6. T otal Derive d F u nctors With these constructions a t hand one can now introduce (total) derived functors in the sense of Grothendieck-V erdier and Deligne, see e.g . K eller [38, §§ 1 3-15] o r any o ne of the references giv en in Remark 9.8. W e follo w Keller’s exp osition of the D eligne approach. 39 The problem is the follo wing: An additiv e functor F : A → B from an exact categ ory to another induces functor s Ch ( A ) → Ch ( B ) and K ( A ) → K ( B ) in an obvious way . By abuse of no tation w e still denote these functors by F . The next question to ask is whether the functor descends to a functor of derived ca tegories, i.e., we lo ok for a commut ative diagram K ( A ) Q A   F / / K ( B ) Q B   D ( A ) ∃ ? / / D ( B ) . If the functor F : A → B is exa ct, this pro blem has a solution by the universal prop erty of the der ived c ategory since then F ( Ac ( A ) ) ⊂ Ac ( B ). How ever, if F fails to b e exact, it will not s end acyclic complexes to a cyclic co mplexes, or, in other w o rds, it will not send quasi-isomorphisms to quasi-isomorphisms and our na ¨ ıv e question will have a negative answ er . Deligne’s solution consists in constructing for each A ∈ D ( A ) a functor r F ( − , A ) : ( D ( B )) op → Ab . If the functor r F ( − , A ) is represe n table, a representing ob ject will be denoted by R F ( A ) and R F is said to be deﬁne d at A . T o b e a little more s peciﬁc, for B ∈ D ( B ) we deﬁne the abelian gr oup r F ( B , A ) b y the equiv alence cla sses of diagr ams B f / / F ( A ′ ) A ′ A s o o where f : B → F ( A ′ ) is a morphism of D ( B ) and s : A → A ′ is a quasi-isomorphism in K ( A ). Obs erve the a nalogy to the description of mo rphisms in D ( A ); it is useful to think of the diagr am as “ F -fractions”. Accordingly , t wo F -frac tions ( f , s ) and ( g , t ) ar e said to b e e quivalent if there exist commutativ e diagrams F ( A ′ ) F ( v )   A ′ v   B f < < y y y y y y y y y h / / g " " E E E E E E E E E F ( A ′′′ ) A ′′′ A s _ _ @ @ @ @ @ @ @ @ u o o t   ~ ~ ~ ~ ~ ~ ~ ~ F ( A ′′ ) F ( w ) O O A ′′ w O O in D ( B ) a nd K ( A ) , where ( h, u ) is a nother F -fraction. On morphisms o f D ( B ) deﬁne r F ( − , A ) by pre-co mpo sition. By deﬁning r F o n morphisms of D ( A ) o ne obtains a functor from D ( A ) to the categor y of functors ( D ( B ) ) op → Ab . Let T ⊂ D ( A ) be the full sub catego ry of ob jects at which R F is deﬁned and choose for eac h A ∈ T a representing ob ject R F ( A ) and an isomorphism Hom D ( B ) ( − , R F ( A )) ∼ − → r F ( − , A ) . These ch oices force the deﬁnition of R F on morphisms and th us R F : T → D ( B ) is a functor. Even more is true: 40 10.25 Theorem (Deligne) . L et F : K ( A ) → K ( B ) b e a functor and let T b e the ful l sub c ate gory of D ( A ) at which R F is deﬁne d. L et S b e the ful l sub c ate gory of K ( A ) sp anne d by the obje cts of T . Denote by I : S → K ( A ) the inclusion. (i) T he c ate gory T is a triangulate d sub c ate gory of D ( A ) and S is a triangulate d sub c ate gory of K ( A ) . (ii) The functor R F : T → D ( B ) is a triangle functor and ther e is a morphism of triangle functors Q B F I ⇒ R F Q A I . (iii) F or every morphi sm µ : F ⇒ F ′ of triangle functors K ( A ) → K ( B ) ther e is an induc e d morphism of triangle functors R µ : R F ⇒ R F ′ on the interse ction of the domains of R F and R F ′ . The o nly s ubtle part of the previous theorem is the fact that T is tria ngulated. The rest is a straightforw ard but ra ther tedious v er iﬁcation. The essen tial details and references are given in Keller [38, § 1 3]. The next question that aris es is whether one can get some information on T . A complex A is said to b e F - s plit if R F is deﬁned a t A and the canonical morphism F ( A ) → R F ( A ) is inv ertible. An o b ject A o f A is said to be F -acycli c if it is F -split when considered a s c omplex co ncent rated in degree zero. 10.26 Lemma ([38 , 15 .1, 15.3 ]) . L et C b e a ful ly exact sub c ate gory of A satisf ying hyp othesis (ii) of The or em 10.22. Assume t hat the r estriction of F : A → B to C is exact. Then e ach obje ct of C is F -acyclic. Conversely, let C b e the ful l sub c ate gory of A c onsisting of the F -acyclic obje ct s. Then C is a ful ly exact sub c ate gory of A , it satisﬁes c ondition (ii) of The or em 10.22 and t he r estriction of F to C is ex act. Now let C b e a fully ex act subc ategory of A co nsisting of F -acyclic ob jects a nd suppo se that C satisﬁes co nditions (i) a nd (ii) o f Theorem 1 0.22. By these assumptions, the inclusion C → A induces an equiv alence D + ( C ) → D + ( A ). As the re striction of F to C is exact, it yields a triangle functor F : D + ( C ) → D + ( B ). T o choo se a quasi-inv er se for the canonical functor D + ( C ) → D + ( A ) a mount s to cho osing for ea ch complex A ∈ K + ( A ) a quasi-iso morphism s : A → C with C ∈ K + ( C ) by [40, 1.6], a pro of of which is g iven in [37, 6 .7]. As we hav e just seen, C is F -s plit, hence s yields an isomorphism R F ( A ) → F ( C ) ∼ = R F ( C ). Such a quasi-iso morphism A → C exists b y the construction in the pro of of Theo rem 1 2.7 and our assumptions. The admittedly co ncise r´ esu m´ e given her e provides the bas ic to olkit for treating derived functors. W e refer to Keller [38, §§ 13 –15] for a muc h more thorough and genera l discussion and precise statements of the comp osition form ula R F ◦ R G ∼ = R ( F G ) and adjunction form ulæ of left and r ight der ived functor s o f a djoin t pairs of functors, 11. Pro jective and Injectiv e Ob jects 11.1 Definition. An ob ject P of a n exa ct catego ry A is ca lled pr oje ctive if the r ep- resented functor Hom A ( P, − ) : A → Ab is exa ct. An ob ject I of an exact catego ry is called inje ctive if the cor epresented functor Hom A ( − , I ) : A op → Ab is exact. 41 11.2 Remark. The co ncepts of pr o jectivity and inject ivity are dual to each other in the sense that P is pro jective in A if and only if P is injective in A op . F or our purp oses it is therefore suﬃcient to deal with pro jective ob jects. 11.3 Pr oposition. A n obje ct P of an ex act c ate gory is pr oje ctive if and only if any one of the fol lowing c onditions holds: (i) F or al l admissible epics A ։ A ′′ and al l morphisms P → A ′′ ther e exists a solution to the lifting pr oblem P ∃   A A A A A A A A A / / / / A ′′ making the diagr am ab ove c ommutative. (ii) The functor Hom A ( P, − ) : A → Ab s en ds admissible epics to surje ctions. (iii) Ev ery admissible epic A ։ P splits (has a right inverse). Proof. Since Hom A ( P, − ) transforms exact sequences to left exact seq uences in Ab for all o b jects P (see the pro of of Coro llary A.8), it is clear that conditions (i) and (ii) are equiv alent to the pro jectivity of P . If P is pro jective and A ։ P is a n admissible epic then Hom A ( P, A ) ։ Hom A ( P, P ) is surjectiv e, a nd every pre-image of 1 P is a splitting map o f A ։ P . Co n versely , let us prov e that condition (iii) implies condition (i): given a lifting problem as in (i), form the following pull-back diagram D f ′   a ′ / / / / PB P f   A a / / / / A ′′ . By h yp othesis, there ex ists a right inv erse b ′ of a ′ and f ′ b ′ solves the lifting problem beca use af ′ b ′ = f a ′ b ′ = f . 11.4 Corollar y. If P is pr oje ctive and P → A has a right inverse t hen A is pr oje ctive. Proof. This is a trivial consequence of condition (i) in Prop osition 11.3. 11.5 Remark. If A is weakly idempotent complete, the a bove corollary amoun ts to the familiar “direct summands of pro jective ob jects are pro jective” in ab elian categories. 11.6 Cor o llar y . A sum P = P ′ ⊕ P ′′ is pr oje ctive if and only if b oth P ′ and P ′′ ar e pr oje ctive. More generally: 11.7 Cor ollar y. L et { P i } i ∈ I b e a family of obje cts for which the c opr o duct P = ` i ∈ I P i exists in A . The obje ct P is pr oje ctive if and only if e ach P i is pr oje ctive. 11.8 Remark. The dual of the previous result is that a pro duct (if it exis ts) is injective if and o nly if each of its fac tors is injective. 42 11.9 Definition. An exact catego ry A is said to hav e enough pr oje ctives if for every ob ject A ∈ A there exis ts a pro jective ob ject P and an a dmissible epic P ։ A . 11.10 Exer cise (Heller [2 6, 5.6]) . Assume that A has enoug h pro jectives. Prov e that A ′ → A → A ′′ is sho rt ex act if and only if Hom A ( P, A ′ ) ֌ Hom A ( P, A ) ։ Hom A ( P, A ′′ ) is short exact for all pro jective ob jects P . Hint: F or suﬃciency prove ﬁrst that A ′ → A is a mo nomorphism, then prov e that it is a k ernel of A → A ′′ and ﬁnally apply the obscure axio m 2.16. In all three steps use that there are enough pro jectives. 11.11 Exercise (Heller [26, 5.6]) . Assume that A is weakly idemp otent complete and has enough pro jectives. Pr ov e that the sequence A n → A n − 1 → · · · → A 1 → A 0 → 0 is an exact sequence of admissible mor phisms if and only if for all pro jectives P the sequence Hom A ( P, A n ) → Hom A ( P, A n − 1 ) → · · · → Hom A ( P, A 1 ) → Ho m A ( P, A 0 ) → 0 is an exa ct sequence of ab elian g roups. 12. Res o lutions and Classical Deriv ed F unctors 12.1 Definition. A pr oje ctive r esolut ion of the ob ject A is a p ositive complex P • with pro jective comp onents together with a morphism P 0 → A s uc h that the augment e d c omplex · · · → P n +1 → P n → · · · → P 1 → P 0 → A is exact. 12.2 Prop osition (Resolution Lemma) . If A has enough pr oje ctives then every obje ct A ∈ A has a pr oje ctive r esolution. Proof. This is an ea sy induction. Beca use A has enough pr o jectives, there exists a pro jective ob ject P 0 and an admissible epic P 0 ։ A with P 0 . Cho ose an admissible monic A 0 ֌ P 0 such that A 0 ֌ P 0 ։ A is exact. Now choos e a pro jectiv e P 1 and an admissible epic P 1 ։ A 0 . Con tinue with an admissible monic A 1 ֌ P 1 such that A 1 ֌ P 1 ։ A 0 is exact, and s o o n. One th us obtains a sequence A 1 ' ' ' ' O O O O O · · · P 2 7 7 7 7 o o o o o o / / P 1 ' ' ' ' O O O O O O / / P 0 / / / / A A 0 7 7 7 7 o o o o o which is exact by co nstruction, s o P • → A is a pro jectiv e reso lution. 43 12.3 Remark. The deﬁning concept of pro jectivit y is not used in the previous pr o of. That is, we ha ve prov ed: If P is a cla ss in A suc h that for each ob ject A ∈ A there is an admissible e pic P ։ A with P ∈ P then each ob ject of A has a P -reso lution P • ։ A . Consider a m orphism f : A → B in A . Let P • be a complex of pro jectives with P n = 0 for n < 0 and let α : P 0 → A b e a mor phism such that the comp osition P 1 → P 0 → A is zer o [e.g . P • → A is a pro jective reso lution of A ]. Let Q • β − → B be a resolution (not necessarily pro jective). 12.4 Theorem (Compa rison Theorem) . Un der the ab ove hyp otheses ther e exists a chain map f • : P • → Q • such that t he fol lowing dia gr am c ommutes: · · · / / P 2 / / ∃ f 2   P 1 / / ∃ f 1   P 0 α / / ∃ f 0   A f   · · · / / Q 2 / / Q 1 / / Q 0 β / / / / B . Mor e over, the lift f • of f is un ique up to homotopy e quivalenc e. Proof. It is convenien t to put P − 1 = A , Q ′ 0 = Q − 1 = B a nd f − 1 = f . Existenc e: The question of existence of f 0 is the lifting problem given by the map f α : P 0 → B and the admissible epic β : Q 0 ։ B . This pro blem has a solution by pro jectivity of P 0 . Let n ≥ 0 and supp ose by induction that there are morphisms f n : P n → Q n and f n − 1 : P n − 1 → Q n − 1 such that d f n = f n − 1 d . Consider the following diagra m: P n +1 ∃ f n +1   ∃ ! f ′ n +1 ' ' d / / P n d / / f n   P n − 1 f n − 1   Q ′ n +1 ' ' ' ' N N N N N Q n +1 7 7 7 7 n n n n n n d / / Q n & & & & L L L L L L d / / Q n − 1 Q ′ n 7 7 7 7 p p p p p By induction the right hand squar e is commutativ e, s o the mo rphism P n +1 → Q n − 1 is zero b ecause the mor phism P n +1 → P n − 1 is zero. The mo rphism P n +1 → Q ′ n is zero as well b ecause Q ′ n ֌ Q n − 1 is monic. Since Q ′ n +1 ֌ Q n ։ Q ′ n is exact, there exists a unique morphism f ′ n +1 : P n +1 → Q ′ n +1 making the upp er right tr iangle in the left hand s quare commut e. Bec ause P n +1 is pr o jective and Q n +1 ։ Q ′ n +1 is a n admissible epic, there is a morphism f n +1 : P n +1 → Q n +1 such that the left ha nd squar e commutes. This settles the existence of f • . Uniqueness: Let g • : P • → Q • be a nother lift o f f a nd put h • = f • − g • . W e will construct by induction a chain con tra ction s n : P n − 1 → Q n for h . F or n ≤ 0 we put s n = 0. F or n ≥ 0 assume by induction that there are morphisms s n − 1 , s n such that h n − 1 = ds n + s n − 1 d . Because of this assumption and the fact that h is a chain map, w e 44 hav e d ( h n − s n d ) = h n − 1 d − ( h n − 1 − s n − 1 d ) d = 0 s o the following diag ram commutes P n h n − s n d   ∃ ! s ′ n +1 w w ∃ s n +1   0 & & M M M M M M M M M M M M M M M M Q ′ n +1 ' ' ' ' N N N N N Q n +1 7 7 7 7 n n n n n n d / / Q n d / / & & & & L L L L L L Q n − 1 Q ′ n 7 7 7 7 p p p p p and w e get a morphism s n +1 : P n → Q n +1 such that ds n +1 = h n − s n d as in th e existence pro of. 12.5 Cor ollar y. A ny two pr oje ctive r esolutions of an obje ct A ar e chain homotopy e quivalent. 12.6 Corollar y. L et P • b e a right b ounde d c omplex of pr oje ctives and let A • b e an acyclic c omplex. Then Hom K ( A ) ( P • , A • ) = 0 . In or der to dea l with der ived functors on the level o f the derived category , one needs to sharp en b oth the resolution lemma and the comparison theor em. 12.7 Theorem ([36, 4.1, Lemma, b)]) . L et A b e an exact c ate gory with en ough pr o- je ctives. F or every right b ounde d c omplex A over A exists a right b oun de d c omplex with pr oje ctive c omp onents P and a quasi-isomorphism P α − → A . Proof. Renum b ering if necessary , we may supp ose A n = 0 for n < 0. The complex P will b e constructed by induction. F or the inductive formulation it is co n venien t to deﬁne P n = B n = 0 for n < 0. P ut B 0 = A 0 , choo se an admissible epic p ′ 0 : P 0 ։ B 0 from a pro jective P 0 and deﬁne p ′′ 0 = d A 0 . Let B 1 be the pull-back ov e r p ′ 0 and p ′′ 0 . Consider the following commutativ e diagram: P 0 p ′ 0     ? ? ? ? ? ? ? ? 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? B 1 i ′′ 0     ? ? ? ? ? ? ? ? i ′ 0 ? ?         PB B 0 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?          PB 0 A 3 d A 2 / / 0 / / A 2 d A 1 / / 0 4 4 ∃ ! p ′′ 1 ? ? A 1 d A 0 / / p ′′ 0 ? ?         A 0 ? ?          The morphism p ′′ 1 exists by the univ ersal prop erty o f the pull-back and mo reov er p ′′ 1 d A 2 = 0 beca use d A 1 d A 2 = 0. Suppos e by induction that in the following diag ram everything is constr ucted except B n +1 and the mor phisms terminating or issuing fro m there. Assume further that P n is 45 pro jective and that p ′′ n d A n +1 = 0. P n p ′ n " " " " E E E E E E E E P n − 1 p ′ n − 1 # # # # G G G G G G G G B n +1 i ′′ n # # # # H H H H H H H H i ′ n ; ; v v v v v v v v PB B n i ′′ n − 1 " " " " E E E E E E E i ′ n − 1 < < y y y y y y y PB B n − 1 A n +3 d A n +2 / / A n +2 d A n +1 / / ∃ ! p ′′ n +1 ; ; A n +1 d A n / / p ′′ n < < y y y y y y y A n p ′′ n − 1 ; ; w w w w w w w w As indicated in the diagr am, w e obtain B n +1 b y for ming the pull-ba ck ov er p ′ n and p ′′ n . W e complete the induction b y c ho osing an admissible epic p ′ n +1 : P n +1 ։ B n +1 from a pro jective P n +1 , co nstructing p ′′ n +1 as in the ﬁrst pa ragra ph and ﬁnally noticing that p ′′ n +1 d A n +2 = 0. The pro jective complex is giv en b y the P n ’s and th e diﬀeren tial d P n − 1 = i ′ n − 1 p ′ n , whic h satisﬁes ( d P ) 2 = 0 by c onstruction. Let α b e given b y α n = i ′′ n − 1 p ′ n in deg ree n , manifestly a chain map. W e claim that α is a quasi- isomorphism. The mapping cone of α is seen to be exact using Prop osition 2.12 : F or each n there is an exact sequence B n +1 i n = » − i ′ n i ′′ n – − − − − − − − → P n ⊕ A n +1 p n = [ p ′ n p ′′ n ] − − − − − − − − → B n . W e th us obtain an exact complex C with C n = P n ⊕ A n +1 in degree n and diﬀerential d C n − 1 = i n − 1 p n = h − i ′ n − 1 p ′ n − i ′ n − 1 p ′′ n i ′′ n − 1 p ′ n i ′′ n − 1 p ′′ n i = h − d P n − 1 0 α n d A n i which shows that C = cone ( α ). 12.8 Theorem (Horsesho e Lemma) . A horsesho e c an b e ﬁl le d in: Supp ose we ar e given a horsesho e diagr am · · · / / P ′ 2 / / P ′ 1 / / P ′ 0 / / / / A ′     A     · · · / / P ′′ 2 / / P ′′ 1 / / P ′′ 0 / / / / A ′′ , that is to say, the c olumn is short exact and the horizontal r ows ar e pr oje ctive r esolutions of A ′ and A ′′ . Then the dir e ct su m s P n = P ′ n ⊕ P ′′ n assemble to a pr oje ctive r esolution of A in such a way that the horsesh o e c an b e emb e dde d into a c ommut at ive diagr am with 46 exact r ows and c olumn s · · · / / P ′ 2 / /     P ′ 1 / /     P ′ 0     / / / / A ′     · · · / / P 2 / /     P 1 / /     P 0 / / / /     A     · · · / / P ′′ 2 / / P ′′ 1 / / P ′′ 0 / / / / A ′′ . 12.9 Remark. All the columns e xcept the rightmost o ne are split e xact. How ever, the morphisms P n +1 → P n are not the s ums of the morphisms P ′ n +1 → P ′ n and P ′′ n +1 → P ′′ n . This only happens in the trivial case that the sequence A ′ ֌ A ։ A ′′ is a lready s plit exact. Proof. This is an easy a pplication of the ﬁve lemma 3.2 and the 3 × 3-lemma 3.6. By lifting the morphism ε ′′ : P ′′ 0 → A ′′ ov er the a dmissible epic A ։ A ′′ we obtain a morphism ε : P 0 → A and a commutativ e diagram Ker ε ′ / / / /   P ′ 0 ε ′ / / / /   [ 1 0 ]   A ′     Ker ε / / / /   P 0 ε / / [ 0 1 ]     A     Ker ε ′′ / / / / P ′′ 0 ε ′′ / / / / A ′′ . It follows from the ﬁve lemma that ε is actua lly an admissible epic, so its kernel exists. The tw o vertical dotted morphisms exist since the second and the third column ar e short exact. No w the 3 × 3-lemma implies that the do tted column is short exa ct. Finally note that P ′ 1 → P ′ 0 and P ′′ 1 → P ′′ 0 factor ov er admissible e pics to Ker ε ′ and Ker ε ′′ and proceed b y induction. 12.10 Remark. In c oncrete situations it may b e useful to r emember that only the pro jectivity of P ′′ n is used in the pro of. 12.11 Remark (Clas sical Deriv ed F unctors) . Using the results of this section, the the- ory of classical derived functors, see e.g. Cartan-E ilenberg [14], Mac La ne [43], Hilton- Stamm bach [28] or W eib el [59], is easily ada pted to the following situation: Let ( A , E ) b e an e xact catego ry with enough pro jectives and let F : A → B b e an additive functor to an ab elian categor y . By the res olution lemma 1 2.2 a pro jective resolution P • ։ A exists for every ob ject A ∈ A a nd is well-deﬁned up to homotopy equiv alence by the comparison theo rem ( Corolla ry 12.5). It follows that for tw o pro jective resolutions P • ։ A and Q • ։ A the complexes F ( P • ) and F ( Q • ) ar e chain homotopy equiv alent . Therefore it makes sense to deﬁne the left derive d functors of F as L i F ( A ) := H i ( F ( P • )) . 47 Let us indicate wh y L i F ( A ) is a functor . First obse rve that a morphism f : A → A ′ extends uniquely up to c hain ho motopy equiv alence to a chain map f • : P • → P ′ • if P • ։ A and P ′ • ։ A ′ are pr o jective res olutions of A and A ′ . F r om this uniqueness it follows easily that L i F ( f g ) = L i F ( f ) L i F ( g ) a nd L i F (1 A ) = 1 L i F ( A ) as desired. Using the horsesho e lemma 1 2.8 one proves that a s hort exact s equence A ′ ֌ A ։ A ′′ yields a long exact sequence · · · → L i +1 F ( A ′′ ) → L i F ( A ′ ) → L i F ( A ) → L i F ( A ′′ ) → L i − 1 F ( A ′ ) → · · · and that L 0 F sends exact se quences to right exact se quences in B so that the L i F are a universal δ -functor . Mor eov er, L 0 F is characterized by being the b est right exact approximation to F and the L i F measur e the failure of L 0 F to be exact. In pa rticular, if F sends exact sequences to right exact sequences then L 0 F ∼ = F and if F is exact, then in addition L i F = 0 if i > 0 . 12.12 Remark. By the discussion in section 1 0.6, th e assumption that ( A , E ) has enough pro jectives is unnecessarily res trictive. In order for the classical left der ived func- tor of F : A → B to exist, it suﬃces to assume that there is a fully exact subca tegory C ⊂ A satisfying the duals o f the c onditions in Theorem 10.22 with the a dditional prop erty that F restricted to C is ex act (see Lemma 10.26). These conditions ensure that the to tal derived functor L F : D − ( A ) → D − ( B ) ex ists and thus it makes sense to deﬁne L i F ( A ) = H i ( L F ( A )), where the ob ject A ∈ A is considere d as a complex concentrated in degree zero. More explicitly , choose a C -resolution C • → A and let L i F ( A ) := H i ( F ( C • )). It is not diﬃcult to chec k that the L i F are a universal δ -functor: They form a δ -functor as L F is a triangle functor and H ∗ : D − ( B ) → B sends dis- tinguished tria ngles to lo ng exact sequences; this δ - functor is universal b ecause it is eﬀa¸ ca ble, as L i F ( C ) = 0 for i > 0 . 12.13 Exercise (Heller [26, 6.3, 6 .5]) . Let ( A , E ) b e an exact categ ory and consider the exa ct categor y ( E , F ) as describe d in E xercise 3.9. P rov e that an ex act sequence P ′ ֌ P ։ P ′′ is pro jective in ( E , F ) if P ′ and P ′′ (and hence P ) a re pr o jective in ( A , E ). If ( A , E ) has enough pro jectives then so has ( E , F ) and every pro jective is of the form describ ed befor e. 13. Examples and Appli cations It is of course impossible to giv e a n exhaustive list of examples. W e simply list some of the p opular ones. 13.1. A dditive Cate gories Every additive categor y A is exact with resp ect to the cla ss E min of split exact sequences, i.e., the sequences isomorphic to A [ 1 0 ] − − → A ⊕ B [ 0 1 ] − − − → B for A, B ∈ A . Every ob ject A ∈ A is b oth pro jective and injective with resp ect to this exact structure. 48 13.2. Quasi-A b elian Cate gories W e hav e seen in Section 4 that quasi-ab elian ca tegories are exact with r esp ect to the class E max of a ll kernel-cokernel pa irs. Ev iden tly , this class of e xamples includes in par ticular all a belia n catego ries. There is an a bundance of no n-ab elian quasi-a belia n categories arising in functional analysis: 13.1 Example (Cf. e.g. [1 1, IV.2]) . Le t Ban be the catego ry of Banach spaces and bo unded linear ma ps ov er the ﬁeld k of real or complex n um be rs. It ha s kernels and cokernels—the cokernel of a morphism f : A → B is given b y B / Im f . It is an easy consequence of the op en mapping theorem that Ban is qua si-ab elian. Notice that the forgetful functor Ban → Ab is exa ct and reﬂects exactness, it preser ves monics but fails to preser ve epics (morphisms with dense range). The gro und ﬁeld k is pro jective and by Hahn-Bana ch it also is injective. Mo re genera lly , it is easy to se e that for ea ch set S the space ℓ 1 ( S ) is pr o jective and ℓ ∞ ( S ) is injective. Since ev ery Banach space A is iso metrically isomo rphic to a quotient of ℓ 1 ( B ≤ 1 A ) and to a subspace of ℓ ∞ ( B ≤ 1 A ∗ ) there are enoug h of b oth, pro jective a nd injectiv e ob jects in Ban . 13.2 Example. Let F re b e the ca tegory of completely metrizable top olo gical vector spaces (F r´ echet spac es) and contin uous linear maps . Again, F re is quasi-ab elian by the op en mapping theorem (the pro of of Theo rem 2.3.3 in Chapter IV.2 of [11] applies mutatis mutandis ), and there are exact functors Ban → F re and F re → Ab . It is still true that k is pr o jective, but k fails to be injective (Hahn-Bana ch breaks down). 13.3 Example. Co nsider the categor y Pol of p olish abelian groups (i.e., second count - able and completely metrizable top olo gical groups) and con tinuous homomo rphisms. F rom the o pen mapping theorem—which is a s tandard consequence o f Pettis’ theorem (cf. e.g. [35, (9.9), p. 61]) stating that for a non-meager set A in G the set A − 1 A is a neigh bo rho o d of the iden tit y—it follows that P ol is quasi-ab elian (again one easily adapts the pro o f of Theorem 2.3.3 in Chapter IV.2 of [1 1]). F urther functional analy tic exa mples are discussed in detail e.g. in Rump [49] a nd Sch neiders [54]. Rump [52] gives a rather long list of exa mples. 13.3. F ul ly Exact Sub c ate gories Recall from section 10.5 tha t a ful ly exact sub c ate gory B of an exact ca tegory A is a full sub categor y B which is closed under extensions and equipp ed with the e xact structure formed by the sequences which a re exact in A (see Lemma 1 0.20). 13.4 Example. B y the embedding theorem A.1, every sma ll ex act categ ory is a fully exact subcatego ry of an ab elian ca tegory . 13.5 Example. The full sub catego ries of pro jectiv e o r injectiv e ob jects of an ex act category A are fully exact. The induced exa ct structures are the split exact structures. 13.6 Example. Let b ⊗ b e the pro jective tensor pro duct of Banach spaces. A Ba nach space F is ﬂat if F b ⊗ − is exact. It is w ell- known that the ﬂat Banac h spaces are precisely the L 1 -spaces of L indenstrauss-Pe lczy ´ nski. The catego ry o f ﬂat Banach spaces is a fully exact sub category of Ban . The exact structure is the pure exact structure consisting of the short sequences whose Ba nach dua l sequences are s plit exact, s ee [11, Ch. IV.2 ] for further information a nd refer ences. 49 13.4. F r ob enius Cate gories An exact category is said to be F r ob enius provided t hat it has enough pro jectives and injectiv es and, moreover, the classes o f pro jectives and injectives coincide [27]. F rob enius categories A give rise to algebr aic triangulate d c ate gories (see [3 9, 3.6]) b y pa ssing to the stable c ate gory A of A . By deﬁnition, A is the category consisting of the same ob jects as A a nd in which a mo rphism of A is iden tiﬁed with zero if it fa ctors ov er an injectiv e ob ject. It is not hard to prov e that A is a dditiv e and it has the structure of a triangulated category as follows: The tra nslation functor is obtained b y c ho osing for eac h ob ject A a short exact sequence A ֌ I ( A ) ։ Σ( A ) where I ( A ) is injectiv e . The a ssignment A 7→ Σ( A ) induces an auto-equiv alence of A . Giv en a morphism f : A → B in A consider the push-out diagram A / / / / f   PO I ( A )   / / / / Σ( A ) B / / / / C ( f ) / / / / Σ( A ) and call the sequence A → B → C ( f ) → Σ( A ) a standar d triangle . The cla ss ∆ of distinguishe d triangles consists of the tria ngles which are isomor phic in A to (the image of ) a standard triangle. 13.7 Theorem (Happel [24, 2.6, p.16]) . Th e triple ( A , Σ , ∆) is a t riangulate d c ate gory. 13.8 Example. Co nsider the categ ory Ch ( A ) of complexes ov er the additiv e category A equipp ed with the degreewise split exact sequences. It turns out that Ch ( A ) is a F r ob enius c ate gory . The complex I ( A ) in tro duced in t he pro of of Lemma 10.7 is injectiv e. It is not hard to v erify that the stable category Ch ( A ) coincides with the homotopy category K ( A ) and that the triangulated structure provided b y Happel’s theorem 13.7 is the s ame a s the one men tioned in Remark 9.8 (see als o E xercise 1 0.8). The rea der may consult Happ el [24] fo r further infor mation, examples and a pplica- tions. 13.5. F urther Examples 13.9 Exa mple (V ector bundles) . Let X b e a scheme. The category of algebr aic vector bundles ov er X , i.e., the categ ory of lo cally free and coherent O X -mo dules, is an exact category with the exact structure consisting of the loca lly split short exact sequences. 13.10 Example (Chain complexes) . If ( A , E ) is an exact ca tegory then the ca tegory of chain complexes Ch ( A ) is an exac t categor y with resp ect to the exact structure Ch ( E ) of short s equences o f complexes which are exact in each degr ee, see Lemma 9.1. 13.11 Example (Diagram Catego ries) . Let ( A , E ) be a n exact category a nd let D b e a sma ll category . The category A D of functors D → A is an exact catego ry with the exact structure E D . The v er iﬁcation of the a xioms of an ex act catego ry for ( A D , E D ) is straightforw ard, as limits a nd colimits in A D are for med point wise, see e.g. Bor ceux [5, 2.15.1, p. 8 7]. 50 13.12 Example (Filtered Ob jects) . Let ( A , E ) b e an exact category . A (bounded) ﬁlter e d obje ct A in A is a sequence of admissible monics in A A = ( · · · / / / / A n / / i n A / / A n +1 / / / / · · · ) such that A n = 0 for n ≪ 0 and that i n A is a n isomorphism fo r n ≫ 0 . A morphism f from the ﬁltered o b ject A to the ﬁltered ob ject B is a collection of mo rphisms f n : A n → B n in A satisfying f n +1 i n A = i n B f n . Th us there is a ca tegory F A of ﬁltered ob jects. It follows from P rop osition 2 .9 that F A is additiv e. The 3 × 3-lemma 3.6 implies that the class F E consisting o f the pairs of morphisms ( i, p ) of F A such that ( i n , p n ) is in E for each n is an exa ct structure on F A . Notice that for a nonzero ab elian catego ry A the category of ﬁltered ob jects F A is not ab elian. 13.13 E xample. Paul Balmer [2] (following K nebusch) gives the follo wing deﬁnition: An ex act c ate gory with duality is a triple ( A , ∗ ,  ) consisting of a n exact category A , a contra v ar iant and ex act endo functor ∗ o n A tog ether with a natural iso morphism  : id A ⇒ ∗ ◦ ∗ satisfying  ∗ M  M ∗ = id M ∗ for all M ∈ A . There a re na tural no tions of symmetric spaces and is ometries of symmetric spaces, (admissible) La grangia ns of a symmetric s pace and hence of metab olic spaces If A is esse n tially small it makes sense to sp eak o f the set MW ( A , ∗ ,  ) of iso metry classes of symmetric spaces and the subset NW( A , ∗ ,  ) of isometry classes of metabolic spac es and b oth turn out to be ab elian monoids with resp ect to the ortho gonal s u m of symmetric s paces. The Witt gr oup is W( A , ∗ ,  ) = MW( A , ∗ ,  ) / NW( A , ∗ ,  ). In case A is the categor y of vector bundles ov er a scheme ( X, O X ) and ∗ = Hom O X ( − , O X ) is the usual duality fu nctor, o ne obtains the classical Witt gro up of a s c heme. Extending these considera tions to the level of the derived catego ry lea ds to Balmer’s triangular Witt gr oups whic h ha d a num b er o f striking applicatio ns to the theory o f quadratic for ms a nd K -theory , we refer the interested reader to Balmer’s survey [2]. F or a b ea utiful link to algebra ic K -theory we r efer to Schlic hting [53]. 13.6. Higher Algebr aic K -The ory Let ( A , E ) b e a small exact category . The Gr othendie ck gr oup K 0 ( A , E ) is deﬁn ed to be the quo tien t of the free (ab elian) gr oup generated by the i somorphism classes o f ob jects of A mo dulo the relatio ns [ A ] = [ A ′ ][ A ′′ ] for ea ch sho rt exact s equence A ′ ֌ A ։ A ′′ in E . This genera lizes the K -theo ry of a ring, where ( A , E ) is tak en to be the categ ory of ﬁnitely gener ated pro jective mo dules over R with the split exact structure. If ( A , E ) is the categor y of a lgebraic vector bundles ov er a scheme X then by deﬁnition K 0 ( A , E ) is the (na ¨ ıve) Grothendieck gr oup K 0 ( X ) of the scheme (cf. [57, 3.2 , p. 313 ]). Quillen’s la ndmark paper [50] in tro duces today’s deﬁnition of higher algebr aic K - theory and pr ov es its basic prop erties. E xact categ ories enter via the Q -construction, which we outline brieﬂy . Giv en a small exact category ( A , E ) one fo rms a new ca tegory Q A : The ob jects of Q A are the ob jects of A and Hom Q A ( A, A ′ ) is deﬁned to be the set of equiv alence class es of diagrams A B p o o o o / / i / / A ′ , in which p is an admissible epic and i is a n admissible monic, where tw o diagrams ar e considered equiv alent if there is an isomor phism of such diag rams inducing the identit y 51 on A a nd A ′ . The co mpo sition of t wo morphisms ( p, i ), ( p ′ , i ′ ) is g iven by the following construction: form the pull-back over p ′ and i so that by Pr op osition 2.15 there is a diagram B ′′ q v v v v ( ( j ′ ( ( PB B p v v v v n n n n n n ( ( i ( ( R R R R R B ′ p ′ v v v v l l l l l l ( ( i ′ ( ( R R R R R A A ′ A ′′ and put ( p ′ , i ′ ) ◦ ( p, i ) = ( pq , i ′ j ′ ). This is ea sily chec ked to yield a ca tegory and it is not hard to ma ke sense of the sta temen t that the morphisms A → A ′ in Q A cor resp ond to the diﬀeren t wa ys that A ar ises as an admissible sub quotient of A ′ . Now any sma ll category C gives rise to a simplicia l set N C , called the nerve of C whose n s implices ar e g iven by seq uences o f comp osa ble morphisms C 0 → C 1 → · · · → C n , where the i -th face map is obtained by deleting the ob ject C i and the i -th deg eneracy map is obta ined by replacing C i b y 1 C i : C i → C i . The classifying sp ac e B C of C is the geometric realization o f the nerve N C . Quillen proves the fundamental res ult that K 0 ( A , E ) ∼ = π 1 ( B ( Q A ) , 0) which motiv ates the deﬁnition K n ( A , E ) : = π n +1 ( B ( Q A ) , 0) . Obviously , an exact funct or F : ( A , E ) → ( A ′ , E ′ ) yields a functor Q A → Q A ′ and hence a ho momorphism F ∗ : K n ( A , E ) → K n ( A ′ , E ′ ) which is easily seen to depend only o n the iso morphism class of F . W e do not discuss K -theory any further a nd re commend the lecture of Q uillen’s original a rticle [50] a nd Sriniv as’s b o ok [56] expanding on Quillen’s ar ticle. F or a g o o d ov erview over many topics o f curr en t interest we refer to the handb o ok of K - theory [20]. A. The Em b eddi ng Theorem F or a belian categ ories, one has the F rey d-Mitc hell embedding theor em, see [1 7] a nd [46], allowing one to prov e diagram lemmas in ab elian categories “by chasing element s”. In order to prove diag ram lemmas in e xact ca tegories, a similar tec hnique works. More precisely , one ha s: A.1 Theorem ([57, A.7.1, A.7 .16]) . L et ( A , E ) b e a smal l exact c ate gory. (i) T her e is an ab elian c ate gory B and a ful ly faithf u l exact fun ctor i : A → B that r eﬂe cts exactness. Mor e over, A is close d under extensions in B . (ii) The c ate gory B may c anonic al ly b e chosen to b e the c ate gory of left exact funct ors A op → Ab and i t o b e the Y one da emb e dding i ( A ) = Hom A ( − , A ) . (iii) Assume mor e over that A is we akly idemp otent c omplete. If f is a morphism in A and i ( f ) is epic in B then f is an admissible epic. 52 A.2 Remark. In o rder for (iii) to hold it is necessary to assume w ea k idempotent com- pleteness of A . Indeed, if A fails to b e w eakly idempotent co mplete, there m us t b e a retraction r without kernel. By deﬁnition there exists s such that rs = 1, but then i ( r s ) is epic, so i ( r ) is epic as well. A.3 Remark. Let B b e an a be lian ca tegory a nd assume that A is a full sub category which is closed under extensions, i.e., A is fully exact sub category o f B in the sense of Deﬁnition 10.21. Then, by Le mma 10.20, A is an exact category with resp ect to the class E of sho rt sequences in A which ar e exact in B . This is a ba sic reco gnition principle of exact categories, for many examples arise in this wa y . The e m bedding theor em provides a pa rtial con verse to this recognition pr inciple. A.4 Remark. Quillen states in [50, p. “92/ 16/10 0”]: Now supp ose given a n exact catego ry M . Let A b e the additiv e catego ry of additive contrav ariant functors from M to ab elian groups which are left exact, i.e. ca rry [an exact sequence M ′ ֌ M ։ M ′′ ] to an exact sequence 0 → F ( M ′′ ) → F ( M ) → F ( M ′ ) . (Precisely , choose a universe containing M , and let A be the ca tegory of left exact functors whose v a lues are ab elian gro ups in the universe.) F ollowing well-kno wn idea s (e.g. [22]), one can prov e A is an ab elian category , that the Y oneda functor h embeds M a s a full sub categ ory of A closed under extensions, and ﬁna lly that a [short] sequence [. . . ] is in E if and only if h carries it into an exac t s equence in A . The deta ils will b e omitted, as they are not rea lly imp ortant for the sequel. F reyd stated a similar theorem in [16], again without pro o f, a nd with the addi- tional assumption o f idemp otents completeness, since he uses Heller’s axioms . The ﬁrst pro of published is in L aumon [42, 1.0.3], r elying on the Gro thendiec k- V erdier theory of sheafiﬁcation [5 5]. The shea fiﬁcation appro ach was also used and further reﬁned by Thomason [57, App endix A]. A q uite detailed sketc h of the pro of alluded to by Q uillen is given in Keller [36, A.3]. The pro o f given her e is due to Thomason [57, A.7] amalg amated with the pr o of in Laumon [42, 1.0 .3]. W e a lso take the opp ortunity to ﬁx a slight ga p in Tho mason’s argument (our Lemma A.10, compare with the ﬁrst sentence after [57, (A.7.10)]). Since Thomason fails to sp ell out the nice shea f-theoretic interpretations of his construction and since referring to SGA 4 seems rather brutal, w e use the terminology of the more light weigh t Mac Lane-Mo erdijk [45, Chapter I II]. Other go o d in tro ductions to the theory of sheav es may b e found in Artin [1] or Borceux [7], for example. A.1. Sep ar ate d Pr eshe aves and She aves Let ( A , E ) be a small exact category . F o r ea ch ob ject A ∈ A , let C A = { ( p ′ : A ′ ։ A ) : A ′ ∈ A } be the set of a dmissible epics onto A . The elemen ts of C A are the c overings o f A . 53 A.5 Lemma. The family { C A } A ∈ A is a basis for a Grothendieck top olog y J on A : (i) If f : A → B is an isomorphi sm t hen f ∈ C B . (ii) If g : A → B is arbitr ary and ( q ′ : B ′ ։ B ) ∈ C B then the pul l-b ack A ′ / / p ′     PB B ′ q ′     A g / / B yields a m orphism p ′ ∈ C A . (“ Stabilit y under base-change”) (iii) If ( p : B ։ A ) ∈ C A and ( q : C ։ B ) ∈ C B then pq ∈ C A . (“T ra nsitivit y”) In p articular, ( A , J ) is a s ite . Proof. This is o b vious from the deﬁnition, see [45, Deﬁnition 2, p. 111 ]. The Y oneda functor y : A → Ab A op asso ciates to ea ch ob ject A ∈ A the pr esheaf (of ab elian g roups) y ( A ) = Hom A ( − , A ) . In g eneral, a pr eshe af is the sa me thing as a functor G : A op → Ab , which we will assume to b e additiv e except in the next le mma. W e will s ee sho rtly that y ( A ) is in fact a she af on the site ( A , J ). A.6 Lemma. Consider the site ( A , J ) and let G : A op → Ab b e a fun ct or. (i) T he pr eshe af G is s eparated if and only if for e ach admissible epic p t he m orphism G ( p ) is monic. (ii) The pr eshe af G is a sheaf if and only if for e ach admissible epic p : A ։ B the diagr am G ( B ) G ( p ) / / G ( A ) d 0 = G ( p 0 ) / / d 1 = G ( p 1 ) / / G ( A × B A ) is an equalizer (diﬀer enc e kernel), wher e p 0 , p 1 : A × B A ։ A denote the two pr o- je ctions. In other wor ds, the pr eshe af G is a she af if and only if for al l admissible epics p : A ։ B the diagr am G ( B ) G ( p ) / / G ( p )   G ( A ) d 1   G ( A ) d 0 / / G ( A × B A ) is a pul l-b ack. Proof. Again, this is obtained by making the deﬁnitions explicit. Poin t (i) is the deﬁ- nition, [45, p. 129 ], and p oint (ii) is [45, Prop osition 1 [bis], p. 123]. The following lemma shows that the sheav es on the site ( A , J ) are quite familiar gadgets. 54 A.7 Lemma. L et G : A op → Ab b e an additiv e functor. The fol lowing ar e e qu ivalent: (i) T he pr eshe af G is a she af on the site ( A , J ) . (ii) F or e ach admissible epic p : B ։ C the se quenc e 0 − → G ( C ) G ( p ) − − − → G ( B ) d 0 − d 1 − − − − → G ( B × C B ) is exact. (iii) F or e ach short exact se quenc e A ֌ B ։ C in A the se quenc e 0 − → G ( C ) − → G ( B ) − → G ( A ) is exact, i.e., G is left exa ct . Proof. By Lemma A.6 (ii) we hav e tha t G is a sheaf if and only if the sequence 0 − → G ( C ) » G ( p ) G ( p ) – − − − − − → G ( B ) ⊕ G ( B ) [ G ( p 0 ) − G ( p 1 ) ] − − − − − − − − − − − → G ( B × C B ) is exa ct. Since p 1 : B × C B ։ B is a s plit epic with k er nel A , there is a n isomo rphism B × C B → A ⊕ B and it is easy to chec k that the above sequence is isomorphic to 0 − → G ( C ) − → G ( B ) ⊕ G ( B ) − → G ( A ) ⊕ G ( B ) . Because left exa ct sequences are stable under taking direct s ums and passing to di- rect s ummands, (i) is equiv alent to (iii). That (i) is equiv alent to (ii) is obvious by Lemma A.6 (ii). A.8 Corollar y ([57, A.7.6]) . The r epr esente d functor y ( A ) = Hom A ( − , A ) is a she af for every obje ct A of A . Proof. Given an exact s equence B ′ ֌ B ։ B ′′ we need to pr ov e that 0 − → Hom A ( B ′′ , A ) − → Hom A ( B , A ) − → Hom A ( B ′ , A ) is exact. That the seq uence is exact at Hom A ( B , A ) follows from the fa ct that B ։ B ′′ is a cokernel o f B ′ ֌ B . Tha t the sequence is exact at Hom A ( B ′′ , A ) follows from the fact that B ։ B ′′ is epic. A.2. Outline of the Pr o of Let now Y b e the categ ory of a dditiv e functors A op → Ab and let B b e the cat- egory of (additiv e) sheav es on the site ( A , J ). Let j ∗ : B → A b e the inclusion. By Corollar y A.8, the Y o neda functor y factors as A y B B B B B B B B i / / B j ∗   Y 55 via a functor i : A → B . W e will prov e that the category B = Sheav es ( A, J ) is abelia n and we will chec k that the functor i ha s the prop erties asserted in the embedding theo rem. The catego ry Y is a Gr othendieck abelian ca tegory (there is a gener ator, small pro d- ucts and copro ducts exist and ﬁltered colimits a re exact)—as a functor categ ory , these prop erties ar e inherited fro m Ab , as limits and co limits are taken p oint wise. The c rux of the pro of of the embedding theorem is to s how that j ∗ has a left adjoint j ∗ such tha t j ∗ j ∗ = id B , namely sheafiﬁcation. As so o n as this is established, the rest will be relatively painless. A.3. She afiﬁc ation The goal of this sectio n is to construct the shea fiﬁcation functor on the site ( A , J ) and to prove its ba sic pr op erties. W e will constr uct a n endofunctor L : Y → Y which asso ciates to each preshea f a separated presheaf and to ea ch separated pr esheaf a sheaf. The sheafiﬁcation functor will then b e given by j ∗ = LL . W e need one more concept from the theory of sites: A.9 Lemma. L et A ∈ A . A c overing p ′′ : A ′′ ։ A is a reﬁnement of the c overing p ′ : A ′ ։ A if and only if ther e exists a morphism a : A ′′ → A ′ such that p ′ a = p ′′ . Proof. This is the specia lization of a matching family as given in [45, p. 121 ] in the present context. By deﬁnition, re ﬁnemen t gives the structure of a ﬁltered categor y on C A for each A ∈ A . More pr ecisely , let D A be the following category: the ob jects a re the cov erings ( p ′ : A ′ ։ A ) and there exists a t mo st o ne morphism b etw een any tw o ob jects of D A : there exists a morphism ( p ′ : A ′ ։ A ) → ( p ′′ : A ′′ ։ A ) in D A if and only if there exists a : A ′′ → A ′ such that p ′ a = p ′′ . T o se e that D A is ﬁlter ed, let ( p ′ : A ′ ։ A ) and ( p ′′ : A ′′ ։ A ) b e tw o ob jects and put A ′′′ = A ′ × A A ′′ , so there is a pull-back diag ram A ′′′ a     a ′ / / / / PB A ′′ p ′′     A ′ p ′ / / / / A. Put p ′′′ = p ′ a = p ′′ a ′ , so the ob ject ( p ′′′ : A ′′′ ։ A ) of D A is a co mmon reﬁnement of ( p ′ : A ′ ։ A ) and ( p ′′ : A ′′ ։ A ) . A.10 Lemma. L et A 1 , A 2 ∈ A b e any two obje cts. (i) T her e is a functor Q : D A 1 × D A 2 → D A 1 ⊕ A 2 , ( p ′ 1 , p ′ 2 ) 7→ ( p ′ 1 ⊕ p ′ 2 ) . (ii) L et ( p ′ : A ′ ։ A 1 ⊕ A 2 ) b e an obje ct of D A 1 ⊕ A 2 and for i = 1 , 2 let A ′ i / / PB p ′ i     A ′ p ′     A i / / A 1 ⊕ A 2 56 b e a pul l-b ack diagr am in which the b ottom arr ow is the inclusion. This c onst ruction deﬁnes a functor P : D A 1 ⊕ A 2 − → D A 1 × D A 2 , p ′ 7− → ( p ′ 1 , p ′ 2 ) . (iii) T her e ar e a natur al tr ansformation id D A 1 ⊕ A 2 ⇒ P Q and a natur al isomorphi sm QP ∼ = id D A 1 × D A 2 . In p articular, t he images of P and Q ar e c oﬁnal. Proof. That P is a functor follows from its construction and the universal property of pull-back diag rams in conjunction with axiom [E2 op ]. That Q is w ell-deﬁned follows from Prop ositio n 2.9 and that P Q ∼ = id D A 1 × D A 2 is easy to chec k. That there is a natural transformation id D A 1 ⊕ A 2 ⇒ QP follows from the universal prop erty of products. Let ( p ′′ : A ′′ ։ A ) be a r eﬁnemen t of ( p ′ : A ′ ։ A ) , a nd let a : A ′′ → A ′ be such that p ′ a = p ′′ . By the universal prop erty of pull-backs, a yields a unique morphism A ′′ × A A ′′ → A ′ × A A ′ which we denote by a × A a . Hence, for every additive functor G : A op → Ab , w e obtain a comm utative diag ram in Ab : Ker ( d 0 − d 1 ) ∃ !   / / G ( A ′ ) d 0 − d 1 / / G ( a )   G ( A ′ × A A ′ ) G ( a × A a )   Ker ( d 0 − d 1 ) / / G ( A ′′ ) d 0 − d 1 / / G ( A ′′ × A A ′′ ) . The next thing to observe is that the dotted morphism do es not depend on the choice of a . Indeed, if ˜ a is a nother morphism such that p ′ ˜ a = p ′′ , co nsider the diagr am A ′′ a ' ' ˜ a ! ! ∃ ! b $ $ A ′ × A A ′ p ′ 1 / / p ′ 0   PB A ′ p ′   A ′ p ′ / / A and b : A ′′ → A ′ × A A ′ is such that G ( b )( d 0 − d 1 ) = G ( b ) G ( p ′ 0 ) − G ( b ) G ( p ′ 1 ) = G ( a ) − G (˜ a ) , so G ( a ) − G ( ˜ a ) = 0 on Ker ( d 0 − d 1 ). F or G : A op → Ab , we put ℓ G ( p ′ : A ′ ։ A ) := K er ( G ( A ′ ) d 0 − d 1 − − − − → G ( A ′ × A A ′ )) and we hav e just seen that this deﬁnes a functor ℓG : D A → Ab . A.11 Lemma. Deﬁne LG ( A ) = lim − → D A ℓG ( p ′ : A ′ ։ A ) . (i) LG is an additive c ontra variant functor in A . 57 (ii) L is a c ovariant functor in G . Proof. This is immediate from going through the deﬁnitions: T o prov e (i), let f : A → B b e an a rbitrary morphism. Lemma A.5 (ii) shows that by taking pull-bac ks we o btain a functor D B f ∗ − → D A which , by pas sing to the colimit, induces a unique morphism LG ( B ) LG ( f ) − − − − → LG ( A ) compatible with f ∗ . F ro m this uniqueness, we deduce LG ( f g ) = LG ( g ) LG ( f ). The additivit y of LG is a co nsequence o f Lemma A.10. T o prov e (ii), let α : F ⇒ G b e a natural transforma tion betw een tw o (additive) presheav es . Given a n ob ject A ∈ A , we o btain a morphism betw een the colimit diagra ms deﬁning LF ( A ) and LG ( A ) and w e deno te the unique r esulting map b y L ( α ) A . Given a morphism f : A → B , there is a commutativ e dia gram LF ( B ) L ( α ) B   LF ( f ) / / LF ( A ) L ( α ) A   LG ( B ) LG ( f ) / / LG ( A ) , as is easily c heck ed. The uniqueness in the deﬁnition o f L ( α ) A implies that for ea c h A ∈ A the equation L ( α ◦ β ) A = L ( α ) A ◦ L ( β ) A holds. The rea der in need of more details may consult [7, p. 206 f ]. A.12 Lemma ([57, A.7.8]) . The functor L : Y → Y has the fol lowing pr op erties: (i) It is additive and pr eserves ﬁnite limits. (ii) The r e is a natur al t ra nsformation η : id Y ⇒ L . Proof. That L preserves ﬁnite limits follo ws from the fac t that ﬁltered colimits and kernels in Ab co mm ute with ﬁnite limits, as limits in Y a re formed p oint wise , see also [7, Lemma 3.3.1]. Since L prese rves ﬁnite limits, it preserves in particular ﬁnite products, hence it is additive. This settles point (i). F or each ( p ′ : A ′ ։ A ) ∈ D A the morphism G ( p ′ ) : G ( A ) → G ( A ′ ) factors uniquely ov er ˜ η p ′ : G ( A ) → K er ( G ( A ′ ) → G ( A ′ × A A ′ )) . By passing to the colimit ov er D A , this induces a morphism ˜ η A : G ( A ) → L G ( A ) wh ich is clearly natural in A . In other words, the ˜ η A yield a natural transformation η G : G ⇒ L G , i.e., a morphism in Y . W e leave it to the reader to c heck that the construction of η G is compatible with natural tr ansformations α : G ⇒ F so that the η G assemble to yield a natural transformation η : id Y ⇒ L , a s cla imed in p oint (ii). A.13 Lemma ([57, A.7.11, (a), (b), (c)]) . L et G ∈ Y and let A ∈ A . 58 (i) F or al l x ∈ L G ( A ) ther e exists an admissible epic p ′ : A ′ ։ A and y ∈ G ( A ′ ) such that η ( y ) = LG ( p ′ )( x ) in LG ( A ′ ) . (ii) F or al l x ∈ G ( A ) , we h ave η ( x ) = 0 in LG ( A ) if and only if ther e exists an admissible epic p ′ : A ′ ։ A such that G ( p ′ )( x ) = 0 in G ( A ′ ) . (iii) We hav e L G = 0 if and only if for al l A ∈ A and al l x ∈ G ( A ) ther e exists an admissible epic p ′ : A ′ ։ A such that G ( p ′ )( x ) = 0 . Proof. Poin ts (i) and (ii) are immediate from the deﬁnitions. Poin t (iii) follows f rom (i) and (ii). A.14 Lemma ([45, Lemma 2, p. 131], [57, A.7 .11, (d), (e)]) . L et G ∈ Y . (i) T he pr eshe af G is sep ar ate d if and only if η G : G → L G is monic. (ii) The pr eshe af G is a she af if and only if η G : G → L G is an isomorphism. Proof. Poin t (i) f ollows from Lemma A .13 (ii) and p oint (ii) follows from the deﬁnit ions. A.15 Proposition ( [57, A.7.12]) . L et G ∈ Y . (i) T he pr eshe af LG is sep ar ate d. (ii) If G is sep ar ate d then LG is a she af. Proof. Let us pr ov e (i) b y applying Lemma A.6 (i), s o let x ∈ LG ( A ) a nd let b : B ։ A be an admissible epic for whic h LG ( b )( x ) = 0. W e ha ve to pr ov e that then x = 0 in LG ( A ). By the deﬁnition of LG ( A ), w e know that x is repr esent ed by some y ∈ Ker ( G ( A ′ ) d 0 − d 1 − − − − → G ( A ′ × A A ′ )) for some admissible epic ( p ′ : A ′ ։ A ) in D A . Since LG ( b )( x ) = 0 in LG ( B ), we know that the imag e of y in Ker ( G ( A ′ × A B ) d 0 − d 1 − − − − → G (( A ′ × A B ) × B ( A ′ × A B ))) is equiv alent to zero in the ﬁltered colimit ov er D B deﬁning LG ( B ). Ther efore there exists a morphism D → A ′ × A B in A suc h that its comp osite with the pro jection onto B is an admissible epic D ։ B . By Lemma A.13 (ii), it follo ws that y maps to zero in G ( D ). Now the comp osite D ։ B ։ A is in D A and hence y is equiv alent to zero in the ﬁltered colimit o ver D A deﬁning LG ( A ). Thus, x = 0 in LG ( A ) a s r equired. Let us prov e (ii). If G is a sepa rated pres heaf, we have to chec k that for every admis- sible epic B ։ A the diagram LG ( A ) / / LG ( B ) d 0 = G ( p 0 ) / / d 1 = G ( p 1 ) / / LG ( B × A B ) is a diﬀerence kernel. By (i) LG is separ ated, so LG ( A ) → LG ( B ) is monic, and it remains to prove that every element x ∈ L G ( B ) with ( d 0 − d 1 ) x = 0 is in the image of LG ( A ). By Lemma A.13 (i) there is an a dmissible epic q : C ։ B and y ∈ G ( C ) such 59 that η ( y ) = LG ( q )( x ). It follows that η G ( p 0 )( y ) = η G ( p 1 )( y ) in LG ( C × A C ). Now, G is sepa rated, so η : G ⇒ LG is monic by Lemma A.14, and we co nclude from this tha t G ( p 0 )( y ) = G ( p 1 )( y ) in G ( C × A C ). In o ther words, y ∈ Ker ( G ( C ) d 0 − d 1 − − − − → G ( C × A C )) yields a cla ss in LG ( A ) representing x . A.16 Corollar y. F or a pr eshe af G ∈ Y we have LG = 0 if and only if LL G = 0 . Proof. Obviously LG = 0 en tails LLG = 0 as L is additive by Lemma A .12. Conv ers ely , as LG is s eparated by Prop osition A.15, it follows that the morphism η LG : LG → LLG is monic by Lemma A.14 (i), s o if LLG = 0 we must hav e LG = 0. A.17 Definition. The she afiﬁc ation functor is j ∗ = LL : Y → B . A.18 Lemma. The she afiﬁc ation fun ctor j ∗ : Y → B is left adjoi nt to the inclusion functor j ∗ : B → Y and satisﬁes j ∗ j ∗ ∼ = id B . Mor e over, she afiﬁc ation is ex act. Proof. By Lemma A.14 (ii) the morphism η G : G → LG is an iso morphism if and o nly if G is a sheaf, s o it follows that j ∗ j ∗ ∼ = id B . Let Y ∈ Y be a preshea f and let B ∈ B b e a sheaf. The natura l transforma tion η : id Y ⇒ L gives us on the one hand a natural transformation  Y = η LY η Y : Y − → LLY = j ∗ j ∗ Y and on the other hand a natur al iso morphism λ B = ( η LB η B ) − 1 : j ∗ j ∗ B = L LB − → B . Now the compositio ns j ∗ B  j ∗ B − − − → j ∗ j ∗ j ∗ B j ∗ λ B − − − → j ∗ B a nd j ∗ Y j ∗  Y − − − → j ∗ j ∗ j ∗ Y λ j ∗ Y − − − → j ∗ Y are m anifestly equal to id j ∗ B and id j ∗ Y so that j ∗ is ind eed lef t adjoin t t o j ∗ . In particular j ∗ preserves cokernels. That j ∗ preserves kernels follows from the fact that L : Y → Y has this proper t y b y Lemma A.12 (i) and the fact that B is a full subcateg ory of Y . Therefore j ∗ is exact. A.19 Remark. It is an illuminating exer cise to pr ov e exactness of j ∗ directly b y going through the deﬁnitions. A.20 Lemma. The c ate gory B is ab elian. Proof. It is clea r that B is additive. The shea fiﬁcation functor j ∗ = LL preserves kernels by Lemma A.12 (i) and a s a left a djoin t it preserves cokernels. T o prov e B ab elian, it s uﬃces to chec k that e very mor phism f : A → B has an analysis A ) ) ) ) S S S S S S f / / B ) ) ) ) S S S S S S Ker ( f ) 6 6 6 6 l l l l l Coim ( f ) ∼ = / / Im ( f ) 6 6 6 6 m m m m m Coker ( f ) . Since j ∗ preserves kernels and cok er nels and j ∗ j ∗ ∼ = id B such an a nalysis can be obtained b y applying j ∗ to an a nalysis of j ∗ f in Y . 60 A.4. Pr o of of t he Emb e dding The or em Let us recapitulate: one half of t he axioms of an exact s tructure yields that a small ex- act categ ory A b ecomes a site ( A , J ). W e deno ted the Y oneda catego ry of cont rav ar iant functors A → Ab by Y and the Y o neda embedding A 7→ Hom ( − , A ) by y : A → Y . W e hav e shown that the category B o f sheaves on the site ( A, J ) is ab elian, b eing a full reﬂective sub categor y of Y with sheafiﬁcatio n j ∗ : Y → B as reﬂector (left adjoin t). F ollowing Thomas on, we denoted the inclusion B → Y b y j ∗ . Moreover, w e have shown that the Y o neda em b edding takes its image in B , so w e o btained a comm utative diagram of categories A i / / y B B B B B B B B B j ∗   Y , in other words y = j ∗ i . By the Y oneda lemma, y is fully faithful and j ∗ is fully faithful, hence i is fully faithful a s well. This settles the ﬁrst pa rt of the following lemma: A.21 Lemma. The functor i : A → B is ful ly faithful and exact. Proof. By the ab ove discussion, it remains to prov e exa ctness. Clearly , the Y oneda embedding s ends ex act sequences in A to left exact sequences in Y . Sheafiﬁcation j ∗ is exa ct and since j ∗ j ∗ ∼ = id B , we hav e that j ∗ y = j ∗ j ∗ i ∼ = i is left exact as well. It remains to prove that for each admissible e pic p : B ։ C the morphism i ( p ) is epic. B y Corolla ry A.16, it s uﬃces to prov e that G = Co ker y ( p ) satisﬁes LG = 0, beca use Coker i ( p ) = j ∗ Coker y ( p ) = LLG = 0 then implies that i ( p ) is epic. T o this end we us e the cr iterion in Lemma A.13 (iii), so let A ∈ A b e any ob ject and x ∈ G ( A ). W e hav e an exa ct sequence Hom ( A, B ) y ( p ) A − − − − → Hom ( A, C ) q A − − → G ( A ) − → 0 , so x = q A ( f ) for some morphism f : A → C . Now form the pull-back A ′ p ′ / / / / PB f ′   A f   B p / / / / C and observe that G ( p ′ )( x ) = G ( p ′ )( q A ( f )) = q A ′ ( f p ′ ) = q A ′ ( pf ′ ) = 0. A.22 Lemma ([57, A.7.15]) . L et A ∈ A and B ∈ B and supp ose ther e is an epic e : B ։ i ( A ) . Ther e exist A ′ ∈ A and k : i ( A ′ ) → B such that ek : A ′ → A is an admissible epic. Proof. Let G be the c okernel of j ∗ e in Y . Then we ha ve 0 = j ∗ G = LLG because j ∗ j ∗ e ∼ = e is epic. By Corollary A.16 it follows that LG = 0 as w ell. No w observe that G ( A ) ∼ = Hom ( A , A ) / Hom ( i ( A ) , B ) and let x ∈ G ( A ) b e the class of 1 A . F rom Lemma A.13 (iii) we co nclude that there is an admissible epic p ′ : A ′ ։ A such that G ( p ′ )( x ) = 0 in G ( A ′ ) ∼ = Hom ( A ′ , A ) / Hom ( i ( A ′ ) , B ). But this means that the admissi- ble epic p ′ factors as e k for some k ∈ Hom ( i ( A ′ ) , B ) as claimed. A.23 Lemma. The functor i r eﬂe cts exactness. 61 Proof. Suppo se A m − → B e − → C is a se quence in A such that i ( A ) i ( m ) − − − → i ( B ) i ( e ) − − → i ( C ) is s hort exact in B . In pa rticular, i ( m ) is a kernel of i ( e ). Since i is fully faithful, it follows that m is a kernel of e in A , hence we are do ne as so o n as we can s how that e is an admissible epic. Becaus e i ( e ) is epic, Lemma A.22 allows us to ﬁnd A ′ ∈ A and k : i ( A ′ ) → i ( B ) such that ek is an a dmissible epic and since e has a kernel we conclude b y the dual of Prop os ition 2.16. A.24 Lemma. The essential image of i : A → B is close d under ex tensions. Proof. Consider a short exact s equence i ( A ) ֌ G ։ i ( B ) in B , where A, B ∈ A . By Lemma A.22 we ﬁnd an a dmissible epic p : C ։ B such that i ( p ) factors o ver G . Now consider the pull-ba ck dia gram D     / / / / PB G     i ( C ) i ( p ) / / / / i ( B ) and observe that D ։ i ( C ) is a split epic b ecause i ( p ) fa ctors ov er G . Ther efore we have isomorphisms D ∼ = i ( A ) ⊕ i ( C ) ∼ = i ( A ⊕ C ). If K is a kernel o f p then i ( K ) is a k ernel of D ։ G , so we obtain a n exact seq uence i ( K ) / / » i ( a ) i ( c ) – / / i ( A ) ⊕ i ( C ) / / / / G, where c = ker p , which shows that G is the push-out i ( K ) / / i ( c ) / / i ( a )   PO i ( C )   i ( A ) / / / / G. Now i is exact b y Lemma A.21 and hence pres erves push-o uts along a dmissible monics b y P rop osition 5 .2, so i preserves the push-out G ′ = A ∪ K C of a alo ng the admissible monic c and thus G is isomorphic to i ( G ′ ). Proof of the Embedding Theorem A.1. Let us summar ize what we know: the em- bedding i : A → B is fully faithful a nd exact by Lemma A.21. It reﬂects exactness by Lemma A.23 and its imag e is clo sed under e xtensions in B by Le mma A.24. This settles po in t (i) of the theorem. Poin t (ii) is taken care of by L emma A.7 and Corollar y A.8 . It remains to pr ov e (iii). Ass ume that A is weakly idemp otent complete. W e cla im that every mo rphism f : B → C such that i ( f ) is epic is in fact an admissible epic. Indeed, by Lemma A.22 we ﬁnd a morphism k : A → B s uc h that f k : A ։ C is an admissible e pic and we co nclude by Pro po sition 7.6. 62 B. Hell er’s Axio ms B.1 Proposition (Quillen) . L et A b e an additive c ate gory and let E b e a class of kernel- c okernel p airs in A . The p air ( A , E ) is a we akly idemp otent c omplete ex act c ate gory if and only if E satisﬁes Hel ler’s axioms: (i) Id entity morphisms ar e b oth admissible monics and admissible epics; (ii) The class of admi ssible monics and the class of admissi ble epics ar e close d under c omp osition; (iii) L et f and g b e c omp osable morphisms. If g f is an admissible monic then so is f and if g f is an admissible epic t hen so is g ; (iv) A ssume that al l r ows and the se c ond two c olumns of the c ommutative diagr am A ′ / / f ′ / / a   B ′ g ′ / / / /   b   C ′   c   A / / f / / a ′   B g / / / / b ′     C c ′     A ′′ / / f ′′ / / B ′′ g ′′ / / / / C ′′ ar e in E then the ﬁrst c olumn is also in E . Proof. Note that (i) and (ii) are just axioms [E0], [E1] a nd their duals. F or a weakly idemp otent complete exa ct catego ry ( A , E ), p oint (iii) is proved in Prop osition 7.6 a nd p oint (iv) follows from the 3 × 3- lemma 3.6. Conv ersely , assume that E has pr op erties (i)–(iv) and let us c heck tha t E is an exact structure. By prop erties (i) and (iii) an isomor phism is both an admissible monic and a n admissible epic since b y deﬁnition f − 1 f = 1 and f f − 1 = 1. If the short sequence σ = ( A ′ → A → A ′′ ) is isomo rphic to the short exa ct sequence B ′ ֌ B ։ B ′′ then prop erty (iv) tells us that σ is sho rt exact. Thus, E is clo sed under iso morphisms. Heller prov es [26, Prop os ition 4 .1] that (iv) implies its du al, that is: if the comm utative diagram in (iv) has exact rows and b oth ( a, a ′ ) and ( b, b ′ ) be long to E then so do es ( c, c ′ ). 2 It follo ws that Heller’s axioms a re self-dual. Let us prov e that [E 2] holds —the r emaining a xiom [E 2 op ] will follow from the dual argument. Given the diagram A ′ / / f ′ / / a   B ′ A 2 Indeed, by (iii ) c ′ is an admissible epic and so i t has a kernel D . Because c ′ g b = 0, there is a morphism B ′ → D and replacing C ′ b y D in the diagram of (iv) we see that A ′ ֌ B ′ ։ D i s short exact. Therefore C ′ ∼ = D and we conclude by the fact that E is closed under isomorphisms. 63 we w ant to construct its push-out B and prov e that the morphism A → B is an admissible monic. Observe that  a f ′  : A ′ → A ⊕ B ′ is the comp osition A ′ / / [ 0 1 ] / / A ⊕ A ′ ∼ = [ 1 a 0 1 ] / / A ⊕ A ′ / / h 1 0 0 f ′ i / / A ⊕ B ′ . By (iii) split exac t sequences b elong to E , and the pro o f of P rop osition 2.9 shows that the direct s um of tw o sequences in E also belongs to E . Therefore  a f ′  is an admissible monic and it has a co kernel [ − f b ] : A ⊕ B ′ ։ B . It follows that the left hand square in the diagram A ′ / / f ′ / / a   BC B ′ b   g ′ / / / / C ′ A f / / B g / / C ′ is bicartesian. L et g ′ : B ′ ։ C ′ be a cokernel of f ′ and let g b e the morphism B → C ′ such that g f = 0 and g b = g ′ . Now consider the commutativ e diagram A ′ / / [ 0 1 ] / / A ⊕ A ′ [ − 1 0 ] / / / /   h 1 a 0 f ′ i   A f   A ′ / / [ a f ] / / A ⊕ B ′ [ − f b ] / / / / [ 0 g ′ ]     B g   C ′ C ′ in which the r ows are exa ct a nd the ﬁrst tw o columns are exact. It follows that the third column is exa ct and hence f is an admissible monic. Now that we know that ( A , E ) is an exact category , we conclude from (iii) and Prop osition 7.6 that A mu st be weakly idemp otent complete. Ac kno wledgments I would like to thank Paul Balmer for introducing me to exact catego ries and Bernhard Keller for a nswering numerous questions via email or via his excellen t articles. Part o f the pap er was written in Nov em be r 20 08 during a conference at the Er win–Schr¨ odinger – Institut in Vienna. 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