One-sided exact categories

ONE-SIDED EXA CT CA TEGORIES SIL V ANA BAZZONI AND SEPTIMIU CRI VEI Abstra ct. One-sided exact categori es app ear naturally as instances of Grothendiec k pretopolo- gies. In an ad d itive setting they are giv en by considering the one- sided part of Keller’s axioms defining Quillen’s exact ca tegories. W e study one-sided exact additive categories an d a stronger versi on defined by adding the one-sided part of Quillen’s “obscure axiom”. W e show that some homological results, such as the Short Five Lemma and the 3 × 3 Lemma, can b e prov ed in our context. W e also note that the deriv ed category of a one-sided exact additive category can b e constructed. 1. Introduction The framework of exact catego ries has naturally app eared in order to dev el op homolog ical algebra in a categorica l setting more general th an th at of an ab elian category . Sev eral notions of exact categories h av e b een defin ed in the literature, for instance, by Barr [2], Heller [5], Qu illen [12] (simplified by Keller [7]), or Y oneda [20], to men tio n o nly some of the most rep resen tativ e ones. Their imp ortance was un derlined by the b road range of applications in algebraic geometry , algebraic and functional analysis, alg ebraic K -theory etc. Sev eral recen t pap ers giv e rather exhaustiv e accoun ts on exact categories in the s ense of Quillen-Keller, defined by means of a distinguished class of kernel-cok ernel pairs, called conflations, e.g. [3] or [4]. Also, these exact categ ories ha ve b een recen tly shown to provide a suitable setting for dev elopp in g an appro ximation theory [17]. During th e last d ecade the n atur al o ccur r ence of one-sided exact categ ories has b ecome ap- paren t in connection with the n otion of Grothendiec k pretop ology in the sense of [1, E x p os ´ e I I, Definition 1.3]. Su ch a pretop ology in an arbitrary ca tegory C is giv en b y assigning to eac h ob ject U of C a family of arro ws ending in U , ca lled co verings of U , satisfying certain axioms. Rosen b erg [14] recen tly in tr o duced arbitrary righ t exact catego ries, wh ose axioms mean that a distinguished class of strict epimorphisms yields the co verings of a Grothend iec k pretop ol- ogy . He show ed that righ t exac t categ ories offer a s u itable framewo rk for homologica l theories whic h could app ear in non-commutati v e algebraic geometry , whereas left exact categories (that is, catego ries w hose o pp osite c ategories are r igh t exact) giv e th e p ossibilit y to dev elop a more general ve rsion of K -theory . In an additiv e setting, Rump br iefly considered left exact categories [16, Defin ition 4] defi ned by means of a distinguished class of coke rnels in connection with the problem of pr oving the existence of flat co v ers in non-ab elian catego ries. In p articular, on e-sided almost ab elian (also termed qu asi-ab elian) categ ories in th e sense of Rump [15] ha v e a natural structure of one-sided exact cate gory giv en by the class of all k ernel-cok ern el pairs. W e shall consider h ere the context of an additiv e category , and we shall define and study one- sided exact categories. Left exact categ ories in our sense are giv en b y means of a distinguished class of cok ernels, call ed deflations, satisfying certain axioms. Con s idering the deflations ending in an ob ject U of a left exact category as the co v erin gs of U , the axioms of a left exact catego ry are nothing else th an th ose of a Grothendieck pretop ology . Unlike Rosenberg, we shall b e mainly in terested in the con text of additiv e categories, where more r esults ca n b e obta ined. W e shall Date : May 26, 2011. 2000 Mathematics Subje ct C lassific ation. 18E10, 18G50 (primary), 18E30, 18E40 ( secondary ). Key wor ds and phr ases. Additive category , one-sided exact category , w eakly idemp otent complete category . First named author supp orted b y MIUR , PRI N 2007, pro ject “Rings, algebras, modu les and categories” and by Universit` a di Pa dov a (Progetto di Ateneo CPDA10588 5/10 “Differential Graded Categoires”). Second named author ackno wledges the supp ort of t he Romanian grant PN-I I-ID -PCE-2008-2 pro ject ID 2271. H e would like to thank the members of the Department of Mathematics, and especially th e first author, for t he kind hospitalit y during his visits at Un iversit` a di P ad ov a. 1 2 SIL V ANA BAZZON I AND SEPTIMIU CRIVEI also study strongly one-sided exact categories, wh ic h are defined b y adding h alf of Qu illen’s “obscure axiom” to the axioms of a one-sided exact cat egory . It turn s out that many results on exact categories do n ot use this extra axio m, and w e wo uld lik e to emphasize this fact by separating th e t wo notions. Note that our strongly one-sided exact categories are n othing else than th e one-sided v ersion of Quillen’s exact cate gories. Some of our s tatemen ts will b e similar with those giv en for exact categories, as for instance resu lts pro ved in d etail by B ¨ uhler [3], but sev eral times our p r o ofs will b e differen t, since w e shall use a reduced set of axioms. Let us also note that our concept of strongly one-sided exac t ca tegory ge neralizes Rump’s notion of one-sided exact category , and co incides with it pr o vided the categ ory is w eakly idemp oten t complete. W e shall u sually refer to (stron gly) right exact cate gories, b u t the dual results on (strongly) left exact categ ories hold as w ell. The pap er is organized as follo ws. After the preliminaries, we shall define (str on gly) one-sided exact categories in Section 3. In Section 4 we s hall sho w ho w (strongly) one-sided structures may b e transferr ed b etw een catego ries. W e shall pro v e that strongly r igh t exact stru ctures are already exact for a large class of categories, n amely for left quasi-ab elian categories, and in particular for ab elian categories. On the other h and, w e sh all show th at if C is a r igh t quasi-ab elian category , D is a co reflectiv e fu ll sub category of C , and E is the class of k ernel-cok er n el pairs in C with terms in D , then D is a righ t quasi-abelian categ ory and E giv es rise to a strongly righ t exac t structure on D , whic h is exact if and only if D is left q u asi-ab elian. Explicit examples of strongly one-sided exact categories whic h are not exact will b e presented. Section 5 will further d ev elop the theory of right exact cate gories. W e shall study ho w conflations b eha v e w ith resp ect to direct su m s and we giv e sev eral c haracterizations of pushouts in righ t exact categ ories. Also, w e shall sho w that the Short Fi v e Lemma is true in righ t exact cate gories, ge neralizing the same result which w as kno wn to h old in right quasi-ab elian categories [15, Lemma 3]. Moreo ver, we shall pr o ve the 3 × 3 Lemm a in strongly righ t exact categ ories. In Section 6 we shall show that w eakly idemp otent complete strongly righ t exact cate gories are righ t exact in the sense of Rump. S ome further prop erties on direct sums will b e giv en. In the last section we sh all sho w that the derived cate gory of a righ t exa ct category ca n b e constructed similarly to the d eriv ed catego ry of an exact category . 2. Preliminaries In this section w e recall some notions and results that will b e used throughout the pap er. The follo w in g tw o lemmas are str aigh tforward. They are du al to [13, Theorem 5] and [18, Example 3, p. 93] resp ectiv ely , which hold in arb itrary categories. Lemma 2.1. L et C b e a c ate gory. L et i : A → B and f : A → A ′ b e morphisms in C such tha t i has a c okernel d : B → C , and the pushout of i and f exists. Then the push out give s r ise to the c ommutative d iagr am A f   i / / B g   d / / C A ′ i ′ / / B ′ d ′ / / C in which d ′ : B ′ → C ′ is a c okernel of i ′ . Lemma 2.2. L et C b e a c ate gory. Consider a c ommutative dia gr am A i / / B g   d / / C h   A i ′ / / B ′ d ′ / / C ′ in which d is an epimorphism and d ′ : B ′ → C ′ is a c okernel of i ′ . Then the right squar e is a pushout. Definition 2.3. An additiv e catego ry is called pr e-ab elian if it has k ernels and cok ern els. It is calle d right (left) qu asi- ab e lian or right (left) almost ab elian ([15], [16]) if it is p re-ab elian ONE-SIDED EXA CT CA TEGORIES 3 and kernels (cok ernels) are stable und er pushouts (pullbac ks), that is, a pu shout (pullback) of a k ernel (cok ernel) along an arbitrary morphism is a kernel (cok ernel). It is called quasi-ab elian if it is left and right quasi-ab elian. Definition 2.4. L et C b e a category and let D b e a full su b category of C . Then D is cal led r e fle c tive (r esp ectiv ely c or efle ctive ) if the inclusion functor i : D → C has a left (resp ectiv ely righ t) ad j oin t. Definition 2.5 . Let A b e a complete and co complete ab elian category . Recall (e. g. f rom [18, Chapter VI, § 1]) that a pr er adic al on A is a subfu nctor of the id entit y functor on A . A pr eradical r on A is called a r adic al if r ( A/r ( A )) = 0 for ev ery ob ject A of A . A full sub category C of A is called a pr etorsion class if there is a preradical r on A suc h th at C consists of th e ob ject s C of A with r ( C ) = C . Dually , C is c alled a pr etorsion fr e e class if there is a preradical r on A suc h that C consists of the ob jects C of A with r ( C ) = 0. Note th at any pr etorsion class is closed under copro du cts and quotien ts, whereas an y pre- torsion free class is clo sed und er pro d ucts and sub ob jects. Pr etorsion (pr etorsion free) classes are in bijection with idemp oten t prer ad icals (radicals). F or more details we refer to [18, Chap- ter VI]. Every pretorsion class in A with asso ciated preradical r : A → A is a co reflectiv e full sub category of A , and the righ t adjoin t of the inclusion functor i : C → Ab is the fu n ctor r view ed as r : Ab → C . Dually , every pr etorsion free class in A is a reflectiv e full sub category of A (see [18, Chapter X, § 1]). T o set the terminology w e recall the follo wing w ell-kno wn notions. A morph ism s : A → B in a category C is called a se ction if it has a left in verse, that is, there is a m orp hism r : B → A in C suc h that r s = 1 A . A morp hism r : B → A is called a r etr action if it has a righ t inv erse, that is, there is a morph ism s : A → B in C suc h that r s = 1 A . Lemma 2.6. (e.g. [3]) The fol lowing ar e e quivalent in an additive c ate gory: (i) Every se ction has a c okernel. (ii) Every r etr action has a kernel. Definition 2.7. ([3], [11 ]) An additiv e categ ory C is said to have split idemp otents (or b e idemp otent c omplete ) if for an y ob ject A of C , any id emp oten t e = e 2 ∈ End( A ) has a k er n el. Also, C is called we akly idemp otent c omplete if the equiv alen t conditions of Lemm a 2.6 hold. Remark 2.8. (1 ) If C is an add itiv e catego ry w ith s p lit idemp oten ts, then C is weakly idemp oten t complete (e.g. see [3, Remark 6.2]). The con verse do es n ot hold in general (see [3, Exercise 7.11]). (2) Every add itiv e catego ry has an id emp oten t-splitting completion (see [11, p. 7]). The n ext c h aracterizat ion of sections in weakly idemp otent complete additive categories easily follo ws by [3, Remark 7.4]. W e sk etc h a pro of since w e shall use it in Pr op osition 6.1. Lemma 2.9. L et C b e a we akly idemp otent c omplete additive c ate gory. Then a morphism s : A → B in C is a se c tion if and only if s is isomo rphic to [ 1 0 ] : A → A ⊕ C for some obje ct C of C . Pr o of. Assume first that s : A → B is a section in C . Then there is a morphism r : B → A suc h that r s = 1 A . By Lemma 2.6 s h as a cok ernel, say p : B → C . Since (1 B − sr ) s = 0, there is a unique morp hism v : C → B suc h that vp = 1 B − sr . Th en pvp = p − p sr = p , whic h im p lies pv = 1 C . Also, r v p = r − r sr = 0, whence r v = 0, b ecause p is an epimorp hism. No w it is easy to see th at [ r p ] : B → A ⊕ C is an isomorphism with inv erse [ s v ] : A ⊕ C → B . Hence s : A → B ma y b e identified with [ 1 0 ] : A → A ⊕ C . Similarly for the con ve rse.  3. Definition o f one-sided ex act ca tegories W e in tro duce (strongly) one-sided exact categories by considering th e one-sided part of Quillen-Keller’s axioms defi n ing exact categories. Definition 3.1. By a right exact c ate gory we mean an additiv e category C endow ed with a distinguished class of k ernels, which are cal led inflations and are denoted by ֌ , satisfying the follo wing axioms: 4 SIL V ANA BAZZON I AND SEPTIMIU CRIVEI [R0] T he id entit y morph ism 1 0 : 0 → 0 is an inflation. [R1] T he comp osition of an y t wo infl ations is again an inflation. [R2] The push out of an y infl ation along an arbitrary morphism exists and is again an inflation. The pushout of an infl ation i : A ֌ B along A → 0 yields its cok ernel d : B → C , whic h is calle d d eflation and is denoted by d : B ։ C . By [18, Prop osition 3.4 , Chapter IV], ev ery inflation give s rise to a s h ort exact sequence, that is, a kernel-cok ernel pair, A ֌ B ։ C , wh ic h is called c onflation . An add itiv e categ ory C is called left exact if its opp osite cate gory C op is righ t exact and it is called exact if it is b oth left and right exact. In w hat fol lows we shal l r efe r to right exact c ate gories. Obviously, our r esults ar e dualizable for left e xact c ate gories. Definition 3.2. Let C b e a right exact categ ory . W e find it con v enien t to consider a stronger form of axiom [R0], namely: [R0 ∗ ] 0 → A is an inflation for every ob j ect A of C . W e sa y that C is str ongly right exact if it satisfies the f ollo wing axiom: [R3] If i : A → B and p : B → C are morphism s in C suc h that i has a cok ernel and pi is an inflation, then i is an inflation. Remark 3.3. (1) Axiom [R3] is the right part of Qu illen’s “obscur e axiom” . W e p oin t out that b y Keller [7, App endix A], a left and righ t exact category in our sen s e coincides with that of an exact catego ry in th e sense of Quillen [12], that is, it satisfies axiom [R3] and its dual [R3 op ]. Moreo v er, again b y [7, Ap p endix A], an additiv e catego ry is exact if and only if it satisfies [R0], [R1], [R2] and [R2 op ]. (2) Rum p defin ed a righ t exact categ ory as an additiv e catego ry C endo w ed with a distin- guished c lass of k ern els s atisfying the ab ov e axioms [R0]-[R2] and axiom [R3] without asking that i has a cok ernel [16, Defin ition 4]. Prop osition 6.4 b elow sho ws th at a category is strongly righ t exact in our sense if and only if it is right exact in the sense of Ru mp, p ro vid ed the category is weakly id emp oten t complete. (3) A left exact catego ry C in our sense ma y b e see n as a n instance of a Grothendieck pre- top ology [1, Exp os´ e I I, Definition 1.3]. In ge neral, th is pr etop ology is giv en by assigning to eac h ob ject U of a category C a family of arro ws { U i → U } , called c overings of U , satisfying certain axioms. Cons idering the defl ations ending in an ob ject U of a left exact category as the co ve rings of U , the axioms of a left exact category are nothing else than those of a Gr othend iec k pretop ology . (4) Rosen b erg considered the conte xt of not necessarily additiv e categ ories [14, 1.1]. Th e axioms of h is right exact catego ries mean th at a distinguish ed class of strict epimorph ism s yields the co verings of a Grothendiec k pretop ology (note the left-righ t difference b et w een Rosen b erg’s terminology and our s ). (5) By the Ga briel-Quillen theorem (see [19]), every (small) exact ca tegory embed s in to an ab elian ca tegory . Similarly , th er e is a Ga briel-Quillen t yp e emb edding th eorem showing that for every (small) righ t exact category C there exists an exact category C ′ and a fully faithfu l exact functor (in the sense that it pr eserves conflations) from C to C ′ whic h is un iv ers al [14, Prop osition 2.6.1]. W e shall not make use of this emb ed ding for transferring prop erties from exact catego ries to one-sided exact categ ories, preferrin g to giv e a cle arer in sigh t of o ne-sided catego ries thr ough d irect pro ofs fr om the axioms. W e ha ve some immediate consequences of the axioms. Lemma 3.4. L et C b e a right exact c ate g ory. Then: (i) F or e ach obje ct A of C , the identity morphism 1 A is an inflation. (ii) Every iso morphism is an inflation. (iii) If C is str ongly rig ht exact, then C satisfies axiom [R0 ∗ ] . Pr o of. (i) Dual to [7, App end ix A, Step 4]. (ii) S ee [3, Remark 2.3]. ONE-SIDED EXA CT CA TEGORIES 5 (iii) Sin ce 1 A : A → A is a cok ernel of 0 → A and 1 0 : 0 → 0 is the comp osition of 0 → A follo we d by A → 0, the morph ism 0 → A is an inflation b y axioms [R0] and [R3].  Example 3.5. (1) In an y a dditive cat egory t here is an exact structur e whose conflations are the split exact sequences. (2) In any r igh t q u asi-ab elian category there is a strongly right exact structure whose confla- tions are the k ern el-cok ernel pairs (e.g., see [15, Prop osition 2 and Corollary 1]). (3) It is easy to see that th e class of all isomorph isms in a category giv es rise to a righ t exact structure wh ic h in general d o es not satisfy axiom [R0 ∗ ], and so it is not strongly righ t exact b y Lemma 3.4 . W e illustrate the relati ons b et ween different t yp es of categ ories, and (one-sided) exact cate - gories r esp ectiv ely in the follo wing diagrams: ab elian   Quillen exact u u k k k k k k k k k k k k k k k ) ) S S S S S S S S S S S S S S S quasi-ab elian   strongly left exact   strongly right e x act   pre-ab elian   left exact with [R0 ∗ op ]   right exact with [R0 ∗ ]   additive left exact right exact 4. Constructions of one-sided exact structures An extension closed full sub catego ry D of an add itiv e exact category C inh erits the exact structure giv en b y all conflations in C ha vin g terms in D [3, Lemma 10.20]. Th is is also a standard w ay of constructing new one-sided exact s tructures fr om existent ones. Prop osition 4.1. L e t C b e a right exact c ate gory and D an extension close d f u l l sub c ate gory o f C . Then the c lass of al l c onflations in C having terms in D d efines a right e xact structur e on D . Pr o of. See [14, Prop osition 2.4.2 ].  W e con tin ue with a result whic h sho ws ho w one-sided exact structures ma y be transferred b et w een ca tegories, and will be useful for giving exa mples of one-sided exact structure which are not exact. Prop osition 4.2. L et C b e a (str ongly) right exact c ate gory, D a right quasi-ab elian c ate gory, and L : D → C an additive functor which pr eserves c okernels. Then ther e is an induc e d (str ongly) right exact structur e on D define d by the pr op erty that a kernel j in D is an inflation if and only if L ( j ) is an inflation i n C . Pr o of. W e denote by [R0]-[R2] ([R0]-[R3]) the axioms of the (strongly) righ t exact catego ry C . In order to show that the category D is (strongly) righ t exact with resp ect to th e defined stru ctur e, w e c hec k the corresp onding axioms [R0 ′ ]-[R2 ′ ] ([R0 ′ ]-[R3 ′ ]). [R0 ′ ] Consider the ob ject 0 of D . T hen 1 0 is the k ernel of the morphism 0 → 0 and L (1 0 ) = 1 L (0) is an inflation in C by Lemma 3.4, and so 1 0 is an inflation in D . [R1 ′ ] Let j : K → M and j ′ : M → N b e t w o inflations in D . Then j ′ j is a k ernel by [15, Prop osition 2]. Also, L ( j ) : L ( K ) → L ( M ) and L ( j ′ ) : L ( M ) → L ( N ) a re inflations in C , and so L ( j ′ j ) = L ( j ′ ) L ( j ) is an inflation in C by [R1]. Hence j ′ j is an inflation in D . 6 SIL V ANA BAZZON I AND SEPTIMIU CRIVEI [R2 ′ ] Let j : K → M and α : K → K ′ b e morphisms in D with j an in fl ation. Let [ β j ′ ] : M ⊕ K ′ → M ′ b e the cok ern el of  j − α  : K → M ⊕ K ′ in D . Th en the left diagram K α   / / j / / M β   K ′ j ′ / / M ′ L ( K ) L ( α )   / / L ( j ) / / L ( M ) L ( β )   L ( K ′ ) L ( j ′ ) / / L ( M ′ ) is a pushout in D . Since L preserves cok ern els, it also p reserv es pushouts, hence the right diagram is a pushou t in C . Then j ′ is a k ernel b ecause D is righ t quasi-ab elian. Also, L ( j ′ ) is an infl ation in C b y [R2]. Hence j ′ is an inflation in D . [R3 ′ ] Let j : K → M and p : M → N b e morphisms in D su c h that j has a cok ernel a nd pj is an inflation. Th en j is a k ern el b y [15, Prop osition 2]. Also, L ( j ) : L ( K ) → L ( M ) an d L ( p ) : L ( M ) → L ( N ) are morphisms in C su c h that L ( j ) has a cok ernel and L ( p ) L ( j ) = L ( pj ) is an inflation in C . Now [R3] implies th at L ( j ) is an inflation in C , so j is an inflation in D .  In what follo ws we shall present some situations when the h yp otheses of Prop osition 4.2 and its dual hold. But first we give th e follo win g result, w hic h sho ws that str on gly righ t exact structures are already exact for a large class of cat egories, namely for left quasi-abelian categories, and in particular for ab elian categ ories. Prop osition 4.3. If C is a left quasi-ab elian c ate gory, then any str ongly right exact structur e on C is exact. Pr o of. Assume that C is a left quasi-ab elian strongly righ t exact category . W e ha v e recalle d in Remark 3.3 that, in order to pro v e that C is exact, it is en ough to sh o w [R2 op ]. So let d : B → C b e a deflatio n with corresp ondin g inflation i : A → B . Consider the pullbac k of d alo ng an arbitrary morphism h : C ′ → C . By the dual of Lemma 2.1, w e h a ve a comm utativ e diag ram A i ′ / / B ′ g   d ′ / / C ′ h   A i / / B d / / C in whic h the right square is a p ullbac k a nd i ′ : A → B ′ is a kernel of d ′ . Since i ′ has a coke rnel and i is an inflation, so is i ′ b y [R3]. By Example 3.5 (2), the class of all short exact sequences defines a left exact structure in the left quasi-ab elian category C , hence d ′ is a coke rnel. Now b y [18, Prop osition 3.4, Chapter IV] we m ust ha ve d ′ = Cok er( i ′ ), and so d ′ is a defl ation.  Prop osition 4.4. L et D b e a right quasi-ab elian fu l l su b c ate gory of a right quasi- ab e lian c ate gory C such that the inclusion functor i : D → C pr eserves c okernels. Denote by E the class of sho rt exact se quenc e s in C with terms in D . Then E gives rise to a str ongly right exact structur e on D , which is exact if and only if D is left quasi-ab elian. Pr o of. W e start with the strongly righ t exact structure on C given b y all short exact sequences (see Example 3.5 (2)). By Prop osition 4. 2, D inherits from C a strongly righ t exact structure with in flations the ke rnels f : A → B with A and B ob j ects in D ; th us the conflations in D are precisely the sh ort exact sequences in the class E . If the ind uced structure is also left exact, then cok ernels in D are stable und er pu llb ac ks, that is, D is also left quasi-abelian. Con v ers ely , if D is left q u asi-ab elian, then by Prop osition 4.3, E induces an exact structure on D .  Corollary 4.5. L et C b e a right quasi-ab elian c ate gory and let D b e a c or efle ctive ful l sub c ate gory of C . Denote by E the c lass of short exact se quenc es in C with terms in D . Then E gives rise to a str ongly right exact structur e on D , which is exact if and onl y if D is left quasi-ab elian. Pr o of. Since D is a coreflectiv e full sub catego ry of C , the inclusion functor i : D → C has a right adjoin t, sa y b : C → D , an d w e hav e ib ∼ = 1 C . Then D is pre-ab elian (e.g., by dual results of [18, Chapter X, § 1], w h ic h are also v alid in our co n text). More precisely , if g : B → C is a ONE-SIDED EXA CT CA TEGORIES 7 morphism in C with terms in D and k ernel f : A → B in C , then the r estriction f ′ : b ( A ) → B of f to b ( A ) is the ke rnel o f f in D . Also, the cok ernels in D coincide with the co k ernels in C . It is easy to sho w that kernels are stable und er push outs in D . Thus D is r igh t quasi-ab elian. Since i preserves cok ernels, the conclusion follo ws no w b y P rop osition 4.4.  Note that Corollary 4.5 has a dual that uses reflectiv e sub categories. W e now exhibit explicit examples of coreflecti v e and r eflectiv e sub catego ries whic h are strongly r igh t or left exac t, but not exact . Example 4.6. Let Ab b e the ca tegory of ab elian groups and let H b e the subgroup of the group of rational n u m b ers generated b y the elemen ts 1 / p , w here p v aries in the set P of prime n umbers. Consider the id emp oten t preradical r : Ab → Ab giv en b y th e trace of H (recall that the trace of H in an ab elian group G is the sum of all images of morp hisms from H to G ). Let C b e the pretorsion class corresp ond ing to r , that is, the class of ob jec ts C of Ab such th at r ( C ) = C . W e claim th at C admits a strongly righ t exact structure w hic h is not left exact. Moreo ve r, w e shall sho w that the strongly righ t exact stru ctur e ma y b e quite far from b eing left exact; more precisely , none of the axioms [R1 op ], [R2 op ], [R3 op ] holds for C . First n ote that since C is a pr etorsion class, it is a coreflectiv e fu ll sub category of Ab. Then by Corollary 4.5 the monomorphisms 0 → A → B in Ab w ith A and B in C giv e r ise to a strongly righ t exact structure in C . A cok ernel in C is a deflation if and only if its k ernel in Ab b elongs to C . Let P b e the set of all p rimes. Note that w e hav e H/p n Z ∼ = Z ( p n ) for eve ry p ∈ P and every non-zero natural n u m b er n . Th en it follo ws that C cont ains all torsion groups (in the usual sense) and all divisible groups. F or ev ery p ∈ P , let Z ( p ) and Z ( p ∞ ) denote the cyclic group of order p and the divisible Pr ¨ ufer p -group resp ectiv ely . It easy to see that Q p ∈ P Z ( p ) L p ∈ P Z ( p ) is a divisible torsion free group , and that for every group G with L p ∈ P Z ( p ) ( G ≤ Q p ∈ P Z ( p ) w e h av e r ( G ) = L p ∈ P Z ( p ). Note that it is enough that one c hec ks this equalit y for G = Q p ∈ P Z ( p ). The canonical pro jection π : Y p ∈ P Z ( p ∞ ) → Q p ∈ P Z ( p ∞ ) L p ∈ P Z ( p ) is a deflation, as well as the p r o jection ρ : Q p ∈ P Z ( p ∞ ) L p ∈ P Z ( p ) → Q p ∈ P Z ( p ∞ ) Q p ∈ P Z ( p ) , but the k ernel of their co mp osition is Q p ∈ P Z ( p ), whic h is not an ob ject of C . So [R1 op ] do es not hold. Let j : Q → Q p ∈ P Z ( p ∞ ) L p ∈ P Z ( p ) b e the inclusion and consid er the pu llbac k of π along i in Ab, that is: 0 / / L p ∈ P Z ( p ) / / Y   / / Q / / j   0 0 / / L p ∈ P Z ( p ) / / Q p ∈ P Z ( p ∞ ) π / / Q p ∈ P Z ( p ∞ ) L p ∈ P Z ( p ) / / 0 W e hav e r ( Y ) = L p ∈ P Z ( p ), and so L p ∈ P Z ( p )   0 / / Q j   Q p ∈ P Z ( p ∞ ) π / / Q p ∈ P Z ( p ∞ ) L p ∈ P Z ( p ) is a pullbac k in C . Note that the lo wer morphism is a deflation, bu t the upp er morphism is not, so [R2 op ] do es not hold. 8 SIL V ANA BAZZON I AND SEPTIMIU CRIVEI Also, the deflation [ π 0] : Y p ∈ P Z ( p ∞ ) M Q → Q p ∈ P Z ( p ∞ ) ⊕ p ∈ P Z ( p ) is the comp osition of the morphism s Q p ∈ P Z ( p ∞ ) ⊕ Q [ 1 0 0 0 ] / / Q p ∈ P Z ( p ∞ ) ⊕ Q [ π j ] / / Q p ∈ P Z ( p ∞ ) L p ∈ P Z ( p ) , but [ π j ] is not a deflation. This sho ws that neither the axiom [R3 op ] holds. Example 4.7. W e adapt and use prop erties f rom [9]. Let I b e the Isb ell categ ory , that is, the full sub category of the category Ab of ab elian group s consisting of ab elian groups ha ving no el emen t of order p 2 , for a fixed prime n um b er p . I consists of the ab elian g roups G w hose p -primary subgroups are elemen tary group s. W e claim that I admits a strongly left exact structure whic h is not righ t exact and, similarly to Ex amp le 4.6, w e shall sh o w that none of the axioms [R1], [R2], [R3] holds for I . Note that I is a pretorsion free class. In fac t, let r : Ab → Ab b e the preradical d efined b y r ( G ) = pG p , where G p is the p -p r imary subgroup of G . It is not difficult to see that r is a r adical. As a pretorsion free class, I is a reflectiv e full su b category of Ab. By the dual of Corollary 4.5, the e pimorph isms B → C → 0 in Ab with B and C in I giv e rise to a strongly left exact structure. Moreo ver, I is complete and cocomplete: limits in I are formed as in Ab, and coli mits are th e col imits G in Ab factored by pG p . W e shall sho w that I is not righ t exact. A morphism j in I is an in flation if and only if the cok ernel of j in Ab is a morphism in I . No w p : Z → Z is an in fl ation in I , b ut its comp osition with itself is n ot an inflation in I . T his sh o ws that axiom [R1] do es not hold in I . Moreo v er, neither axiom [R2], nor axio m [R3] h old in I . Indeed, th e square Z f   / / p / / Z f   Z ( p ) 0 / / Z ( p ) (with f 6 = 0 and Z ( p ) denoting Z /p Z ) is a pushout and the u pp er morphism is an in flation, bu t the lo w er morphism is not an inflation. Also, the inflation [ p 0 ] : Z → Z ⊕ Z ( p ) is th e comp osition of the morphisms [ 1 0 0 0 ] : Z ⊕ Z ( p ) → Z ⊕ Z ( p ) and [ p π ] : Z → Z ⊕ Z ( p ) (wh ere π : Z → Z ( p ) denotes the canonical pro jection), but the latter is not an in flation. 5. Some homological lemmas In this section w e study ho w conflations b eha v e with resp ect to dir ect sums and pu s houts in righ t exact cate gories, and we emplo y su c h prop erties for pro ving t w o important homologica l results, namely the Sh ort Five Lemma and the 3 × 3 Lemm a, in a (strongly) right exact category . Our results are in s pired b y the analogous statemen ts in [3], but in general we h a ve to fin d new pro ofs, since w e use only the one-sided part of the axioms defi n ing an exact stru cture. F or some results w e ha ve to assume axiom [R0 ∗ ] or ev en [R3]. Lemma 5.1. L e t C b e a right exact c ate gory. Then the class of c onflations is c lose d u nder isomorph isms and dir e ct sums of short exact se quenc es. Pr o of. The first p art follo ws immediately b y [R2]. The second part is similar to [3, Prop osition 2.9] using axiom [R1].  The next result is similar to a part of [3, Prop osition 3.1]. ONE-SIDED EXA CT CA TEGORIES 9 Prop osition 5.2. L et C b e a right exact c ate gory. Every morphism ( f , g , h ) b etwe en two c on- flations A i ֌ B d ։ C and A ′ i ′ ֌ B ′ d ′ ։ C ′ factors thr ough some c onflatio n A ′ ֌ D ։ C A f   / / i / / B g ′   d / / / / C A ′ / / j / / D g ′′   p / / / / C h   A ′ / / i ′ / / B ′ d ′ / / / / C ′ such that th e upp er left squar e and the lower right squar e of the diagr am ar e pushouts. Pr o of. As the pro of of [3, Prop osition 3.1 ] using axiom [R2] and Lemmas 2.1 an d 2.2.  It is known th at th e Short Fiv e Lemma (see b elo w) h olds in an y right quasi-ab elian category [15, Lemma 3], which has the righ t exact str ucture giv en b y all sh ort exact sequences. Also, it holds in an y exact category (see [3, Corollary 3.2]) . W e shall generalize this homological lemma to an arbitrary righ t exact catego ry . The argumen t used in the case of exact catego ries ca nnot b e trasferred in our con text. Our pr o of u ses an idea from [13, Theorem 6]. Lemma 5.3 (Short Fiv e Lemma) . L et C b e a right exact c ate gory. Consider a morphism ( f , g, h ) A f   / / i / / B g   d / / / / C h   A ′ / / i ′ / / B ′ d ′ / / / / C ′ b etwe en two c onflations such that f and h ar e isomo rphisms. Then so is g . Pr o of. W e claim fi r st that g is an epimorphism . Indeed, if v : B ′ → D is a morphism suc h that v g = 0, then v i ′ f = v g i = 0, whence v i ′ = 0. Th en there is a un ique morphism w : C ′ → D suc h that w d ′ = v . W e ha v e w hd = wd ′ g = v g = 0, whence w = 0, and so v = 0. Thus g is an epimorphism. No w consider the pu shout of i ′ and if − 1 and use Lemma 2.1 to obtain the follo wing comm u- tativ e d iagram: A f   / / i / / B g   d / / / / C h   A ′ / / i ′ / / if − 1   B ′ d ′ / / / / g ′   C ′ B / / α / / D β / / / / C ′ Then α is an in flation by [R2] and β is its cok ern el by Lemma 2.1, h ence the last ro w is a conflation. W e h av e ( g ′ g − α ) if − 1 = g ′ i ′ f f − 1 − g ′ i ′ = 0, and so ( g ′ g − α ) i = 0. Since d = Cok er( i ), there is a un ique m orp hism γ : C → D suc h that γ d = g ′ g − α . Then β γ d = β g ′ g − β α = d ′ g = hd , whence it f ollo ws that β γ h − 1 = 1, b ecause d is an epimorphism. Sin ce β (1 D − γ h − 1 β ) = 0 and α = Ker( β ), there is a un ique morphism δ : D → B su c h that αδ = 1 D − γ h − 1 β . Th is implies that αδ α = α and αδ γ = 0, and so δ α = 1 B and δ γ = 0, b ecause α is a monomorphism. No w we ha ve δ g ′ g = δ γ d + δ α = 1 B . On the other hand, the equalit y gδ g ′ g = g implies that gδ g ′ = 1 B ′ , b ecause g is an epimorph ism. This sho ws that g is an isomorphism with inv erse δ g ′ .  Prop osition 5.4. L et C b e a right exact c ate gory. Consider the c ommutative squar e A f   / / i / / B g   A ′ / / i ′ / / B ′ 10 SIL V ANA BAZZON I AND SEPTIMIU CRIVEI wher e i and i ′ ar e inflatio ns. Then the squ ar e is a pushout if and only if it is p art o f a c ommu- tative diagr am A f   / / i / / B g   d / / / / C A ′ / / i ′ / / B ′ d ′ / / / / C wher e the r ows ar e c onflations. Pr o of. Assume fir st th at the square is a pushout. Then the existence of th e r equired comm utative diagram follo ws by Lemma 2.1. Note th at i ′ is an inflation by [R2], and d ′ is the cok ern el of i ′ , so the second ro w is a conflation. Supp ose n o w that the square is part of a commutativ e diagram as ab o ve. By Prop osition 5.2 one obtains a 3 × 3 diagram as in the same prop osition with C ′ = C . No w by L emma 5.3 (Short Fiv e Lemma), g ′′ is an isomorphism , wh ic h implies the conclusion.  Prop osition 5.5. L et C b e a right exact c ate gory and c onsider a c ommutative dia gr am A / / i / / B g   d / / / / C h   / / j / / D f   A / / i ′ / / B ′ d ′ / / / / C ′ / / j ′ / / D ′ in which the morphisms i, i ′ , j, j ′ ar e inflatio ns, d, d ′ ar e defla tions and the right squar e is a pushout. Then the short exact se qu enc e B h g j d i / / B ′ ⊕ D [ j ′ d ′ − f ] / / D ′ is a c onflation. Pr o of. By L emm a 2.2 the square B C B ′ C ′ is a pu shout, whence it foll o w s that the r ectangle B D B ′ D ′ is also a pushout. This is equiv alen t to the fact that [ j ′ d ′ − f ] is a cok ernel of  g j d  . Next let us sho w that the follo wing comm utativ e squ are is a pu shout: A i   / / i ′ / / B ′ [ 1 0 ]   B [ g d ] / / B ′ ⊕ C T o this end, let u : B ′ → E a nd v : B → E b e morphisms su ch that ui ′ = v i . Since ( v − ug ) i = 0 and d = C ok er ( i ), there is a unique morp hism w : C → E suc h that wd = v − ug . Th en it is easy to see that [ u w ] : B ′ ⊕ C → E is the uniqu e morph ism required for the push out pr op ert y of the previous square. It follo ws that [ g d ] is an inflation by [R2]. Since j is an inflation, so is  1 0 0 j  : B ′ ⊕ C → B ′ ⊕ C ′ b y Lemma 5.1. Then  g j d  =  1 0 0 j  [ g d ] is an inflation b y [R1]. Now the conclusion follo ws.  In order to complete Prop osition 5. 4 with other c haracterizations of push outs we need n ow to assu me axiom [R0 ∗ ]. Prop osition 5.6. L e t C b e a right exact c ate gory satisfying also [R0 ∗ ] . F or eve ry obje cts A and B of C , the sho rt exact se quenc e A [ 1 0 ] / / A ⊕ B [ 0 1 ] / / B is a c onflation. Pr o of. By assumption 0 → B is an inflation. The required sequence is the dir ect sum of t w o conflations, n amely A ֌ A ։ 0 and 0 ֌ B ։ B , hence the result follo ws by Lemma 5.1 .  ONE-SIDED EXA CT CA TEGORIES 11 Prop osition 5.7. L et C b e a right exact c ate gory satisfying also [R0 ∗ ] . Consider the c ommuta- tive squar e A f   / / i / / B g   A ′ / / i ′ / / B ′ wher e i and i ′ ar e inflations. The fol lowing a r e e qui valent: (i) The squar e is a pushout. (ii) The short exact se qu e nc e A h i f i / / B ⊕ A ′ [ g − i ′ ] / / B ′ is a c onflation. (iii) The squar e is b oth a p ushout and a pul lb ack. Pr o of. Since we are assum ing axiom [R0 ∗ ], the pr o of is the same as the pro of of th e equiv alence of the first three conditions in [3, Prop osition 2.1 2].  Corollary 5.8. L et C b e a right exact c ate gory satisfying also [R0 ∗ ] , and let f : A → B and f ′ : A → C b e morphisms in C w ith f an infla tion. Then h f f ′ i : A → B ⊕ C is an inflat ion. Pr o of. W e m ay consider the pushout of th e inflation f and the morphism f ′ . T hen h f f ′ i is an inflation b y Prop osition 5.7.  F or the n ext results we need to assume axiom [R3]. Prop osition 5.9. L e t C b e a str ongly right exact c ate gory. L et A i → B d → C an d A ′ i ′ → B ′ d ′ → C ′ b e c omp osable morphisms such that A ⊕ A ′ i ⊕ i ′ ֌ B ⊕ B ′ d ⊕ d ′ ։ C ⊕ C ′ is a c onflation. Then the short exact se qu e nc es A i → B d → C and A ′ i ′ → B ′ d ′ → C ′ ar e also c onflations. Pr o of. See [3, Corollary 2.18].  Lemma 5.10. L et C b e a str ongly right exact c ate gory and c onsider a c ommutative diagr am A / / i / / B g   d / / / / C   h   A / / i ′ / / B ′ d ′ / / / / C ′ in which the r ows ar e c onflatio ns and h is a n inflation. Then g is an inflation. Pr o of. W e claim first that g has a cok ernel. More precisely , we p ro v e th at Coker( g ) = h ′ d ′ , where h ′ = C ok er( h ) : C ′ → C ′′ . Let f : B ′ → D b e a morph ism suc h that f g = 0. The squ are B C B ′ C ′ is a pushout b y Lemma 2.2. Hence there is a un ique morphism γ : C ′ → D such that γ d ′ = f and γ h = 0. S ince h ′ = Cok er( h ), th er e is a un ique morp h ism δ : C ′′ → D suc h that δ h ′ = γ . Then δ h ′ d ′ = f , whence C ok er ( g ) = h ′ d ′ . By Prop osition 5.5, [ g d ] is an infl ation. Since h is an infl ation, so is  1 0 0 h  : B ′ ⊕ C → B ′ ⊕ C ′ b y Lemma 5.1. Note that  1 d ′  g =  1 0 0 h  [ g d ] is an inflation b y [R1]. Since g has a cok ernel, axiom [R3] sho ws that g m u s t b e an inflation.  No w we a re able to pro v e the 3 × 3 Lemma in strongly righ t exact catego ries. O n ce aga in, the proof of the same result in exact categories [3, C orollary 3.6] cannot b e transf erred to our setting. 12 SIL V ANA BAZZON I AND SEPTIMIU CRIVEI Prop osition 5.11 (3 × 3 Lemma) . L et C b e a str ongly right exact c ate gory and c onsider the fol lowing c ommutative diagr am: A   f   / / i / / B   g   d / / / / C   h   A ′ f ′     / / i ′ / / B ′ g ′     d ′ / / / / C ′ h ′     A ′′ i ′′ / / B ′′ d ′′ / / C ′′ in which the c olumns and the first two r ows ar e c onflations. Then the thir d r ow is also a c onflation. Pr o of. By Prop osition 5.2 the morp hism ( f , g , h ) b et ween the conflations A i ֌ B d ։ C and A ′ i ′ ֌ B ′ d ′ ։ C ′ factors th rough some conflation A ′ ֌ D ։ C A   f   / / i / / B u   d / / / / C A ′ / / j / / D v   p / / / / C   h   A ′ / / i ′ / / B ′ d ′ / / / / C ′ suc h th at the upp er left square and the lo w er righ t squ are of the diagram are pushouts. Then u is an in flation, and b y Lemma 5.10, v is a lso an inflation. Since f ′ f = 0 = 0 i , the pushout prop erty of the square AB A ′ D yields the existence of a unique morph ism u ′ : D → A ′′ suc h that u ′ j = f ′ and u ′ u = 0. By Lemma 2.1, u ′ = Cok er ( u ). Denote v ′ = Cok er ( v ). W e claim that th e follo wing diagram is commutativ e: B / / u / / D   v   u ′ / / / / A ′′ i ′′   B / / g / / B ′ v ′     g ′ / / / / B ′′ d ′′   C ′′ C ′′ W e already ha ve v u = g . T he square D C B ′ C ′ is a pushout by Lemma 2.2, w hence we ha v e v ′ = h ′ d ′ = d ′′ g ′ b y Lemma 2.1. F urthermore, n ote that ( i ′′ f ′ ) f = 0 = 0 i . Then i ′′ u ′ , g ′ v : D → B ′′ are b oth solutions of th e p u shout p roblem for the square AB A ′ D , b ecause ( i ′′ u ′ ) j = i ′′ f ′ , ( i ′′ u ′ ) u = 0, and also ( g ′ v ) j = i ′′ f ′ , ( g ′ v ) u = 0. Hence we m ust ha v e i ′′ u ′ = g ′ v . Th e square D A ′′ B ′ B ′′ is a pus h out b y Lemma 2.2. Then i ′′ is an in fl ation b y axiom [R2]. Finally , d ′′ = Cok er ( i ′′ ) by Lemma 2.1 . Hence the sequence A ′′ i ′′ → B ′′ d ′′ → C ′′ is a conflation.  Remark 5.12. The Snak e Lemma also holds in a one-sided exact categ ory by [14, Prop osi- tion C2.2]. 6. Weakl y idempote nt c omplete right exact ca tegories First we shall use the c haracterization of sections in w eakly idemp oten t complete cate gories from Lemma 2.9 to obtain a generaliz ation to right exact categories of a result known for exact catego ries (see [17, Lemma A.2]). Prop osition 6.1. L et C b e a we akly idemp otent c omplete right exact c ate gory and let f : A → B and g : B → C b e morphisms in C . Then [ g g f ] : B ⊕ A → C is a deflation if and only if g is a deflation. ONE-SIDED EXA CT CA TEGORIES 13 Pr o of. Supp ose that [ g g f ] : B ⊕ A → C is a deflation. Th en the k ernel [ u r ] : K → B ⊕ A of [ g g f ] : B ⊕ A → C is an in flation. Consider the isomorphism  1 f 0 1  : B ⊕ A → B ⊕ A and denote  u ′ r ′  =  1 f 0 1  · [ u r ]. Then r ′ = r and w e ha v e an isomorphism of short exact sequences K / / [ u r ] / / B ⊕ A h 1 f 0 1 i   [ g g f ] / / / / C K / / h u ′ r i / / B ⊕ A [ g 0 ] / / / / C whic h implies that the lo wer sequence is a conflation b y Lemma 5.1. The lo w er d eflation is determined as b eing [ g 0 ] b y the commuta tivit y of the right square. Th e c omp osition of the morphisms [ g 0 ] : B ⊕ A → C and [ 0 1 ] : A → B ⊕ A is ze ro, hence th ere is a unique morphism s : A → K suc h that  u ′ r  s = [ 0 1 ]. Since C is we akly idemp otent complete, the sect ion s has a cok ernel, sa y p : K → D . By the pro of of Lemma 2.9, there is a unique morphism v : D → K suc h that v p = 1 K − sr , and [ p r ] : K → D ⊕ A is an isomorphism. W e ha v e u ′ v p = u ′ , b ecause u ′ − u ′ v p = u ′ sr = 0. Thus w e obtain the follo wing comm utativ e diagram: K [ p r ]   / / h u ′ r i / / B ⊕ A [ g 0 ] / / / / C D ⊕ A [ 1 0 ]   / / h u ′ v 0 0 1 i / / B ⊕ A [ 1 0 ]   [ g 0 ] / / / / C D / / u ′ v / / B g / / / / C The second row is a conflation b y Lemma 5.1. The lo w er left squ are is a pu shout, and so u ′ v : D → B is an inflation b y [R2]. Mo reo ver, it may b e completed b y Prop osition 5.4 to the ab o ve commutat iv e diagram, w here the low er row is a confl ation. The lo w er deflation is determined as b eing g b y the commutativit y of the lo we r righ t square. Con v er s ely , sup p ose that g is a deflation. Then it has a k ernel, sa y r : D → B , whic h is an inflation. Th en we ha v e th e follo w ing comm u tativ e d iagram: D [ 1 0 ]   / / r / / B [ 1 0 ]   g / / / / C D ⊕ A / / [ r 0 0 1 ] / / B ⊕ A [ g 0 ] / / / / C This is b ecause the left sq u are is a pu shout, and so [ r 0 0 1 ] : D ⊕ A → B ⊕ A is an inflation b y [R2]. Moreo v er, the square may b e completed by Prop osition 5.4 to the ab o ve comm utativ e diagram, where the lo wer ro w is a conflation. T hen w e ha ve an isomorphism of short exact sequ ences D ⊕ A / / [ r 0 0 1 ] / / B ⊕ A h 1 − f 0 1 i   [ g 0 ] / / / / C D ⊕ A / / h r − f 0 1 i / / B ⊕ A [ g g f ] / / / / C in which the lo wer ro w is a conflation by Lemma 5.1. The lo w er deflation is d etermined as b eing [ g g f ] b y the commutat ivit y of the right square. Hence [ g g f ] : B ⊕ A → C is a deflation.  W e cont in ue with some characte rizations of w eak idemp otent completeness of righ t exa ct catego ries satisfying [R0 ∗ ]. Lemma 6.2. L et C b e a right exact c ate g ory. Then the fol lowing ar e e qu ivalent: (i) C is we akly idemp otent c omplete and satisfies [R0 ∗ ] . 14 SIL V ANA BAZZON I AND SEPTIMIU CRIVEI (ii) Every se ction is an inflation. (iii) Every r etr action is a deflation. Pr o of. Using Lemma 2.9 the pro of is the same as [3, Corollary 7.5].  No w we ha v e the du al of Lemma 3.4 (ii). Corollary 6.3. L et C b e a we akly idemp otent c omplete right exact c ate gory satisfying also [R0 ∗ ] . Then every isomorphism is a deflation. W e sh o w no w that in th e case of w eakly idemp oten t complete categories our n otion of strongly one-sided exact category reduces to the notion of one-sided exact cat egory considered b y Rum p ([16, Defin ition 4]). This amounts to the follo wing prop osition. Prop osition 6.4. L et C b e a str ongly right exact c ate gory. Then the fol lowing a r e e quivalent: (i) C is we akly idemp otent c omplete. (ii) If i : A → B and p : B → C a r e morphisms in C such that p i is an inflation, then i is an inflation. Pr o of. ( i ) ⇒ ( ii ) By [R2], Lemma 2.6 and [R3] the pr o of is dual to that of [3, Prop osition 7.6]. ( ii ) ⇒ ( i ) Let s : A → B b e a section in C . Th en there is r : B → A su c h th at r s = 1 A . Since 1 A is an inflation b y Lemma 3.4, so is s b y h yp othesis. No w C is w eakly idempotent complete b y Lemm a 6.2.  As a consequence, w e h a ve a partial con v erse of Corollary 5.8. It p ro vid es a generalization of the dual of [17, Lemm a 4.2]. Corollary 6.5. L et C b e a we akly idemp otent c omplete str ongly right exact c ate gory and let f : A → B and g : B → C b e morphisms in C . Then h f g f i : A → B ⊕ C is an inflation if and only if f is an infla tion. Pr o of. Note that h f g f i =  1 g  f , and use Prop osition 6.4 and Corollary 5.8.  Let us p oint out that one cannot c haracterize weak idemp oten t completeness of str on gly r igh t exact cat egories by the dual of condition (ii) in Prop osition 6.4. Example 6.6. Let C b e the pretorsion class defined in Examp le 4.6. Then C has a stron gly right, but not left, exact structure. Moreo v er, C is a righ t quasi-ab elian category , and consequently w eakly idemp oten t complete. On the other hand, w e h a ve seen that axiom [R3 op ] do es n ot hold. The Isb ell category fr om Example 4.7 giv es an example for the dual case of a stron gly left exact cat egory . Under ce rtain conditions, th e class of conflatio ns in a righ t exact categ ory is closed under arbitrary direct sums (if th ey do exist) of s h ort exact sequences. In order to see th at, we s h all need th e follo wing notions. Definition 6.7. An ob ject I of a right exact category C is called inje ctive if for eve ry inflation A ֌ B , e v ery morphism A → I extends to a morphism B → I . W e sa y that C has enough inje ctives if for ev ery ob ject A of C there is an inflation i : A ֌ I with I an injectiv e ob ject. Example 6.8. (1) Let C b e a right quasi-ab elian category with enough in j ectiv es and let D b e a coreflectiv e fu ll sub category of C . L et b : C → D be the righ t adjoin t of the inclusion i : D → C . F or an y ob ject D of D one has an in fl ation D → I in A for some injectiv e ob j ect I of C . It is easy to see that b ( I ) is injectiv e in D , and the induced morph ism D → b ( I ) is an inflation in D . Hence D has enough injectiv es. F or in s tance, this applies to a p retorsion class in a Grothendiec k catego ry . Note that D migh t not ha ve an exac t structure (see Example 4.6). (2) Let C b e a finitely accessible additiv e catego ry , that is, an additiv e category with direct limits s u c h that the cla ss of fin itely presente d o b jects is sk eletally small, and ev ery ob ject is a direct limit of finitely presen ted ob jects [11, p. 8]. Th en C has split idemp otent s [11, p. 22], and so it is w eakly idemp otent complete [3, Remark 6.2]. Via the Y oneda fun ctor, C is equiv alent to the full sub category F of flat ob jects in the category A = (fp( C ) op , Ab) of all con trav arian t ONE-SIDED EXA CT CA TEGORIES 15 additiv e functors from the full su b category fp ( C ) of finitely presented ob jects of C to the category Ab of abelian groups, and the pur e exact sequences in C are those whic h b ecome e xact in A through the Y oneda emb edding [11, Theorem 3.4]. Then F is exte nsion closed in the ab elian catego ry A , and so, the class of all pu re exa ct sequences in C giv es rise to an exact s tr ucture on C by [3, Lemma 10 .20]. The injectiv e ob jects in this exa ct category are the pure-injectiv e ob jects, and b y [6, Theorem 6], for ev ery ob ject A of C , there is a pur e monomorph ism A ֌ I for some pure-injectiv e ob ject I of C . Hence C has enough inj ectiv es. Note that C migh t not b e pre-ab elian (see [11, Corollary 3.7]). W e giv e now a generaliz ation of [17, Prop osition A.6], ha ving a similar pro of. Prop osition 6.9. L et C b e a we akly idemp otent c omplete str ongly right exact c ate gory with enough inje ctives. Then: (i) A morphism f : A → B is an inflation if and only if the map Hom C ( f , I ) : Hom C ( B , I ) → Hom C ( A, I ) is an epimorphism of ab elian gr oups for every inje ctive obje ct I of C . (ii) If ( A k i k ֌ B k d k ։ C k ) k ∈ K is a family of c onflations having a c opr o duct, then the sho rt exact se quenc e L k ∈ K A k ⊕ k ∈ K i k / / L k ∈ K B k ⊕ k ∈ K d k / / L k ∈ K C k is a c onflation. Pr o of. (i) The “ only if ” p art is cle ar. C on versely , let i : A ֌ I b e an in flation in C with I an injectiv e ob ject. By h yp othesis, there is a morphism g : B → I su ch that g f = i . By Prop osition 5.7, h f g f i =  f i  : A → B ⊕ I is an inflation. Finally , by Corollary 5.8, f is an inflation. (ii) Clearly , L k ∈ K d k : L k ∈ K B k → L k ∈ K C k is the cok ernel of L k ∈ K i k . Hence it is enough to sh o w that L k ∈ K i k is an inflation. Bu t this follo ws from part (i), since Hom C ( − , I ) tak es copro ducts in C into pr o ducts of ab elian groups, and pro du cts of ab elian groups are exact.  7. Derived ca tegories In this section we sho w that the deriv ed category of a righ t exact category can b e constru cted similarly to the deriv ed categ ory of an exact category (see Neeman [10], Keller [8], B ¨ uhler [3]). The only new th ing to b e c heck ed here is that the m apping cone of a c hain map b et w een acyclic complexes s tays acyc lic. Let u s recall some needed terminology . T hroughout C w ill b e an additiv e category . Denote b y Ch ( C ) the additive category of complexes and chain maps ov er C , and b y K ( C ) the additiv e homotop y category w hose ob jects are th e ob jects of Ch ( C ) and whose morph isms are homotopy classes of morph isms in C h ( C ). Reca ll also th at the mapping c one cone( f ) of a c hain map f : A → B in Ch ( C ) is the complex whose n -th comp onent is cone( f ) n = A n +1 ⊕ B n and n -th differen tial is d n f = h − d n +1 A 0 f n +1 d n B i . Note that the mapping cone d efines an endofunctor of Ch ( C ). The conce pt of acyclic c hain complex o ve r a righ t exact category C can b e introduced in the usual w a y [8]. Definition 7.1. Let C b e a righ t exact catego ry . A chain complex A o ver C is call ed acyclic if eac h differential d n − 1 A factors as A n − 1 p n − 1 # # # # H H H H H H H H H d n − 1 / / A n Z n A < < i n − 1 < < y y y y y y y y where i n − 1 is an inflation that is a k ernel of d n and p n − 1 is a deflation that is a cok ernel of d n − 2 . Ha ving p r epared the needed prop erties in the previous s ections, the f ollo wing lemma ma y b e pr o ved similarly to [3, Lemma 10 .3]. W e include a pro of sin ce we w ould lik e to p oin t out precisely the p laces wh ere we use our results for righ t exact cate gories. 16 SIL V ANA BAZZON I AND SEPTIMIU CRIVEI Lemma 7.2. L et C b e a right exact c ate gory. Then the mapping c one of a chain map f : A → B b etwe en acyclic c omplexes over C is acyclic. Pr o of. It is easy to sho w that the morphisms g n and g n +1 in th e follo wing diagram A n − 1 f n − 1   p n − 1 A # # # # H H H H H H H H H d n − 1 A / / A n f n   p n A # # # # H H H H H H H H H d n A / / A n +1 f n +1   Z n A g n      < < i n A < < y y y y y y y y y Z n +1 A g n +1      : : i n +1 A : : t t t t t t t t t Z n B " " i n B " " E E E E E E E E E Z n +1 B $ $ i n +1 B $ $ J J J J J J J J J B n − 1 p n − 1 B ; ; ; ; v v v v v v v v v d n − 1 B / / B n p n B ; ; ; ; v v v v v v v v v d n B / / B n +1 exist an d they are the uniqu e morphisms making the diagram comm utativ e. Starting with the morphism ( g n , f n , g n +1 ) b et ween the conflations Z n A i n A ֌ A n j n A ։ Z n +1 A and Z n B i n B ֌ B n j n B ։ Z n +1 B , one uses Prop osition 5. 2 to obtain some ob j ect Z n C and a 3 × 3 comm u tative diagram in which the middle ro w is an ind uced conflation Z n B j n ֌ Z n C q n ։ Z n +1 A . Then we hav e the follo wing comm utativ e diagram A n − 1 f n − 1 1   p n − 1 A $ $ $ $ I I I I I I I I I d n − 1 A / / A n f n 1   p n A $ $ $ $ I I I I I I I I I d n A / / A n +1 f n +1 1   Z n A P O g n   ; ; i n A ; ; w w w w w w w w w Z n +1 A P O g n +1   9 9 i n +1 A 9 9 s s s s s s s s s s Z n − 1 C P O f n − 1 2   q n − 1 : : : : u u u u u u u u u Z n C P O f n 2   q n : : : : u u u u u u u u u Z n +1 C f n +1 2   Z n B # # i n B # # G G G G G G G G G ; ; j n ; ; w w w w w w w w w Z n +1 B % % i n +1 B % % K K K K K K K K K K 9 9 j n +1 9 9 s s s s s s s s s s B n − 1 p n − 1 B : : : : u u u u u u u u u d n − 1 B / / B n p n B : : : : u u u u u u u u u d n B / / B n +1 in w hic h we ha v e eac h f n = f n 2 f n 1 and the qu ad r ilaterals marked by PO are push outs. No w by Prop osition 5.5 eac h sequence Z n C / /  − i n +1 A q n f n 2  / / A n +1 ⊕ B n [ f n +1 1 j n +1 p n B ] / / / / Z n +1 C is a conflation. W e also ha v e comm utativ e diagrams A n ⊕ B n − 1 [ f n 1 j n p n − 1 B ] & & & & M M M M M M M M M M  − d n A 0 f n d n − 1 B  / / A n +1 ⊕ B n Z n C 8 8  − i n +1 A q n f n 2  8 8 q q q q q q q q q q It follo ws that the mapping cone of f is acyclic.  Let Ac ( C ) b e the fu ll sub category of the homotop y cat egory K ( C ) consisting of all a cyclic complexes o ver a right exact category C . Note that Ac ( C ) is a fu ll additive s u b category of K ( C ), ONE-SIDED EXA CT CA TEGORIES 17 b ecause Lemm a 5.1 implies that the direct su m of tw o ac yclic complexes is acyclic. Moreo v er , Lemma 7.2 yields the follo wing corollary . Corollary 7.3. L et C b e a right exa ct c ate gory. Then Ac ( C ) is a triangulate d sub c ate gory of K ( C ) . Analogously to the derived categ ory of an exact category , we ma y n o w define the derive d c ate gory of a righ t exact cate gory C as the V erd ier qu otien t D ( C ) = K ( C ) / Ac ( C ) (see Keller [8, § 10, § 11 ]). 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Dip ar timento di Ma tema ti ca Pura e Applica t a, Universit ` a di P adov a, Via Trieste 63, 35121 P adov a, It al y E-mail addr ess : bazzoni@math .unipd.it F acul ty of Ma thema tics and Computer S cience, “Babes ¸ -Bol y a i” Uni versity, Str. Mihail Ko g ˘ alniceanu 1, 400084 Cluj-Napoca, Romania E-mail addr ess : crivei@math. ubbcluj.ro