Relative Homological Algebra and Purity in Triangulated Categories

Transcription

1 Journal of Algera 227, Ž 2000 doi:101006jar , availale online at on Relative Homological Algera and Purity in Triangulated Categories Apostolos Beligiannis Fakultat fur Mathematik, Uniersitat Bielefeld, D Bielefeld, Germany -mail: Communicated y Michel Broué Received June 7, 1999 CONTNTS 1 Introduction 2 Prer classes of triangles and phantom maps 3 The Steenrod and Freyd category of a triangulated category 4 Projectie ojects, resolutions, and deried functors 5 The phantom tower, the cellular tower, homoty colimits, and compact ojects 6 Localization and the relatie deried category 7 The stale triangulated category 8 Projectiity, injectiity, and flatness 9 Phantomless triangulated categories 10 Brown representation theorems 11 Purity 12 Applications to deried and stale categories References 1 INTRODUCTION Triangulated categories were introduced y Grothendieck and Verdier in the early sixties as the prer framework for doing homological algera in an aelian category Since then triangulated categories have found important applications in algeraic geometry, stale homoty theory, and representation theory Our main purpose in this paper is to study a triangulated category, using relative homological algera which is develed inside the triangulated category Relative homological algera has een formulated y Hochschild in categories of modules and later y Heller and Butler and Horrocks in $3500 Cyright 2000 y Academic Press All rights of reproduction in any form reserved 268

2 PURITY IN TRIANGULATD CATGORIS 269 more general categories with a relative aelian structure Its main theme consists of a selection of a class of extensions In triangulated categories there is a natural candidate for extensions, namely the Ž distinguished triangles Let C e a triangulated category with triangulation We devel a homological algera in C which parallels the homological algera in an exact category in the sense of Quillen, y specifying a class of triangles which is closed under translations and satisfies the analogous formal prerties of a prer class of short exact sequences We call such a class of triangles a prer class of triangles Two ig differences occur etween the homological algera in a triangulated category and in an exact category First, the asolute theory in a triangulated category, ie, if we choose as a prer class, is trivial Second, a prer class of triangles is uniquely determined y an ideal Ph Ž C of C, which we call the ideal of -phantom maps This ideal is a homological invariant which almost always is non-trivial and in a sense controls the homological ehavior of the triangulated category However, there are non-trivial examples of phantomless categories To a large extent the relative homological ehavior of C with respect to depends on the knowledge of the structure of -phantom maps The paper is organized as follows Section 2 contains the asic definitions aout prer classes of triangles and phantom maps in a triangulated category C We prove that there exists a ijective correspondence etween prer classes of triangles and special ideals of C, and we discuss riefly an analogue of Baer s theory of extensions In Section 3 we prove that there exists a ijective correspondence etween prer classes of triangles in C and Serre sucategories of the category modž C of finitely presented additive functors C A, which are closed under the suspension functor Using this correspondence, we associate to any prer class of triangles in C a uniquely determined aelian category S Ž C, the -Steenrod category of C The -Steenrod category comes equipped with a homological functor S: C S Ž C, the projectiization functor, which is universal for homological functors out of C which annihilate the ideal of -phantom maps The terminology comes from stale homoty: the Steenrod category of the stale homoty category of spectra with respect to the prer class of triangles induced from the ilenergmaclane spectrum corresponding to Ž p is the module category over the mod-p Steenrod algera We regard S Ž C as an aelian approximation of C and the idea is to use it as a tool for transferring information etween the tological category C and the algeraic category S Ž C This tool is crucial, if the projectivization functor is non-trivial and this happens iff the prer class of triangles generates C in an apprriate sense In Section 4, fixing a prer class of triangles in a triangulated category C, we introduce -projective ojects, -projective resolutions,

3 270 APOSTOLOS BLIGIANNIS -projective and -gloal dimension and their duals, and we prove the asic tools of homological algera ŽSchanuel s lemma, horseshoe lemma, comparison theorem in this setting It follows that we can derive additive functors which ehave well with respect to triangles in We compare the homological invariants of C with respect to with the homological invariants of its -Steenrod category, via the projectivization functor Finally we study riefly the semisimple and hereditary categories In Section 5, we associate with any oject A in C the -phantom and the -cellular tower of A, which are crucial for the study of the homological invariants of A with respect to, eg, the structure of -phantom maps out of A and the -projective dimension of A Using these towers we prove our first main result which asserts that the gloal dimension of C with respect to is less than equal to 1 iff the projectivization functor S: C S Ž C is full and reflects isomorphisms The prer setting for the study of the -phantom and the -cellular tower is that of a compactly generated triangulated category 55 In this case homoty colimits are defined and we prove that under mild assumptions, any oject of C is a homoty colimit of its -cellular tower and the homoty colimit of its -phantom tower is trivial Finally we study the category of extensions of -projective ojects, the phantom tology and the phantom filtration of C induced y the ideal Ph Ž C, and we compute the -phantom maps which live forever, ie maps in Ph n Ž C n1, in terms of the -cellular tower In Section 6 we associate with any triangulated category C with enough -projectives a new triangulated category D Ž C which we call the -deried category Under some mild assumptions D Ž C is realized as a full sucategory of C and is descried as the localizing sucategory of C generated y the -projective ojects Moreover, D Ž C is compactly generated and any compactly generated category is of this form This construction, which generalizes the construction of the usual derived category, allows us to prove the existence of total -deried functors Applying these results to the derived category of a Grothendieck category with projectives, we generalize results of Spaltenstein 67, Boekstedt and Neeman 16, Keller 42, and Weiel 71 concerning resolutions of unounded complexes and the structure of the unounded derived category, proved in the aove papers y using essentially a closed model structure on the category of complexes In Section 7 we prove that the stale category of C modulo -projectives admits a natural left triangulated structure, which in many cases it is useful to study The results of the first seven sections refer to general prer classes of triangles in unspecified categories In Section 8 we study prer classes of triangles Ž X induced y skeletally small sucategories X in a category C with croducts Under a reasonale condition on X we show that the

4 PURITY IN TRIANGULATD CATGORIS 271 Steenrod category of C with respect to the prer class Ž X is the category Mod X of additive functors X A We compare the Ž X - homological prerties of ojects of C with the homological prerties of projective, flat, and injective functors of the category ModŽ X Our first main result in this section shows that if X compactly generates C, then C has enough Ž X -injective ojects in a functorial way and admits Ž X -in- jective enveles This generalizes a result of Krause 50 Our second main result shows that if X compactly generates C or if any oject of C has finite Ž X -projective dimension, then any croduct preserving homological functor from C to a Grothendieck category annihilates the ideal of Ž X -phantom maps This generalizes results otained independently y Christensen and Strickland 21 and Krause 50 In Section 9 we characterize the Ž X -semisimple or Ž X -phantomless categories C y a host of conditions, generalizing work of Neeman 54 Perhaps the most characteristic is the condition that C is a pure-semisimple locally finitely presented category 24 ; in particular, C has filtered colimits Using this and an old result of Heller 34 Žoserved also y Keller and Neeman 44, we give necessary conditions for a skeletally small category D such that the categories of Pro-ojects and Ind-ojects over D admit a triangulated structure These conditions are sufficient for the existence of a Puppe-triangulated or pre-triangulated structure 57 on these categories, in the sense that the octahedral axiom does not necessarily hold In Section 10 we study Brown representation theorems, via the concept of a representation equivalence functor, ie, a functor which is full, surjective on ojects, and reflects isomorphisms We say that a pair Ž C, X consisting of a triangulated category with croducts C and a full skeletally small triangulated sucategory X C satisfies the Brown representaility theorem or that C is an Ž X -Brown category, if the projectivization functor of C with respect to Ž X induces a representation equivalence etween C and the category of cohomological functors over X Our main result shows that this happens iff X compactly generates C and any cohomological functor over X has projective dimension ounded y 1 This generalizes results of Christensen and Strickland 21 and Neeman 56 In Section 11 we study the fundamental concept of purity in a compactly generated triangulated category which provides a link etween representation-theoretic and homological prerties Motivated y the examples of the stale homoty category of spectra, the derived category of a ring, and the stale module category of a quasi-froenius ring, it is natural to regard a compact oject as an analogue of a finitely presented oject in a triangulated category In this spirit the prer class of triangles induced y the compact ojects are the pure triangles and the resulting theory is the pure homological algera of C In this case the pure Steenrod category of

5 272 APOSTOLOS BLIGIANNIS Ž C is the module category Mod C, where C denotes the full sucategory of compact ojects of C We give various formulas for the computation of the pure gloal dimension and we prove that the pure gloal dimension ounds the pure gloal dimension of a smashing or finite localization Our main result characterizes the pure Brown categories as the pure hereditary ones, and also y a host of equivalent conditions, the most notale eing the fullness of the projectivization functor and the conditions concerning the ehavior of weak colimits These results are applied directly to the stale homoty category, which has pure gloal dimension 1 By the results of Section 9, a compactly generated triangulated category is pure semisimple iff its pure Steenrod category is locally Noetherian Motivated y the pure homological theory of module categories, we define a category to e of finite type if its pure Steenrod category is locally finite We prove that the phantomless, Brown, or finite type prerty is preserved under smashing or finite localization, generalizing results of Hovey et al 38 In Section 12 we apply the results of the previous sections Žmainly aout purity to derive categories of rings and stale categories of quasi-froenius rings We compute the pure gloal dimension for many classes of rings and we prove that the derived category DŽ of a ring is of finite type iff Ž Ž D is of finite type iff D and D are pure semisimple, so finite type is a symmetric condition However, we show that the Brown prerty is not symmetric in general Motivated y the pure homological theory of module categories we formulate the deried pure semisimple conjecture, DPSC for short, which asserts that if DŽ is pure semisimple, then DŽ is of finite type DPSC implies the still en pure semisimple conjecture asserting that a right pure semisimple ring is of finite representation type We prove DPSC for many classes of rings, including Artin algeras Indeed we characterize the Artin algeras with pure semisimple derived categories as the iterated tilted algeras of Dynkin type We prove that the analogue of DPSC holds for the stale category of a quasi-froenius ring, showing that is representation-finite iff the stale category of is pure semisimple Ž of finite type The analogue of DPSC in a general compactly generated triangulated category fails We prove this y answering in the negative a seemingly unrelated prolem of Roos 64 concerning the structure of quasi-froenius Grothendieck categories Our results indicate that finite type is the apprriate notion of representation-finiteness, at least for stale or derived categories We close the paper y applying the results of Sections 6 and 7 to derived categories In particular we otain structure results for the unounded derived category and we generalize results of Wheeler 72, 73 concerning the structure of the stale derived category

6 PURITY IN TRIANGULATD CATGORIS 273 A general convention used in the paper is that composition of morphisms in a category is meant diagramatically: if f: A B, g: B C are morphisms, their composition is denoted y f g However, we compose functors in the usual Ž anti-diagrammatic order Our additive categories admit finite direct sums 2 PROPR CLASSS OF TRIANGLS AND PHANTOM MAPS 21 Triangulated Categories Let C e an additive category and : C C an additive functor We define the category DiagŽ C, as follows An oject of DiagŽ C, is a diagram in C of the form A B C Ž A A morphism in DiagŽ C, f i g i h etween A B C i Ž A i i i i, i 1, 2, is a triple of morphisms : A A, : B B, : C C, such that the diagram Ž commutes: f1 g1 h1 A1 B1 C1 A1 f2 g2 h A B C Ž A A triangulated category 70 is a triple Ž C,,, where C is an additive category, : C C is an autoequivalence of C, and is a full sucategory of DiagŽ C, which is closed under isomorphisms and satisfies the following axioms: Ž T 1 For any morphism f: A B in C, there exists an oject f g h A B C Ž A in For any oject A C, the diagram 0 A 1 A A 0isin f g h g h T If A B C A is in, then B C Ž A 2 Ž f Ž B is in f i g i h T If A B C i Ž A 3 i i i i, i 1, 2 are in, and if there are morphisms : A1 A2 and : B1 B2 such that f2 f1, then there exists a morphism : C C such that the diagram Ž 1 2 is a morphism in Ž T 4 The Octahedral Axiom For the formulation of this we refer to Prosition 21 PROPOSITION 21 Žsee also 51 Let C e an additie category equipped with an autoequialence : C C and a class of diagrams DiagŽ C, Suppose that the triple Ž C,, satisfies all the axioms of a triangulated category except possily of the octahedral axiom Then the following are Ž

7 274 APOSTOLOS BLIGIANNIS equialent: Ž i Base Change f g h For any triangle A B C Ž A and any morphism : C, there exists a commutatie diagram 0 M M 0 f g h A G Ž A f g h A B C Ž A 0 Ž M Ž M 0 in which all horizontal and ertical diagrams are triangles in f g h ii Coase Change For any triangle A B C Ž A and any morphism : A D, there exists a commutatie diagram 0 N N 0 1 Ž h f g 1 Ž C A B C 1 Ž h f g 1 Ž C D F C 0 Ž N Ž N 0 in which all horizontal and ertical diagrams are triangles in Ž iii Octahedral Axiom For any two morphisms f 1: A B, f 2: B C there exists a commutatie diagram f1 g1 h1 A B X Ž A f2 f1 f2 g3 h3 A C Y Ž A f1 Ž f1 f2 g2 h2 B C Z Ž B 0 h2 Ž g1 0 Ž X Ž X 0 in which all horizontal and the third ertical diagrams are triangles in

8 PURITY IN TRIANGULATD CATGORIS 275 Ž f 1 f Proof i iii Let A B 2 C e a diagram in C By Ž T 1 1 f 2 1 f there are triangles Z B C Z and X A 1 B X in Applying ase change for the last triangle along, it is easy to see that if we arrange prerly the resulting diagram, then we otain an octahedral diagram as in iii iii i If M C Ž M is a triangle, then applying the octahedral axiom to the composition g, we otain a diagram which if we apply 1 and we arrange it prerly, we have a diagram as in Ž i The proof of Ž ii Ž iii is similar Hence for any triangulated category we may use the aove equivalent forms instead of the octahedral axiom, when it is more convenient Throughout the paper we fix a triangulated category C Ž C,, ; is the suspension functor, is the triangulation, and the elements of are f g h the triangles of C A triangle T : A B C Ž A is called split if h 0 Trivially if Ž T is split, then the morphisms f, g induce a direct sum decomposition B A C We denote y 0 the full sucategory of consisting of the split triangles We call a triangle semi-split if it is isomorphic to a croduct of suspensions of triangles of the form A A 0 Ž A Any split triangle is semi-split ut the converse is not true 22 Prer Classes of Triangles A class of triangles is closed under ase change if for any triangle f g h A B C Ž A and any morphism : C as in Ž i of f g h Prosition 21 the triangle A G Ž A elongs to Dually a class of triangles is closed under coase change if for any f g h triangle A B C Ž A and any morphism : A D as in f g h ii of Prosition 21 the triangle D F C Ž D elongs to A class of triangles is closed under suspensions if for any triangle f g h A B C Ž A and for any i the triangle i Ž A Ž1 i i f i Ž1 i i g i Ž1 i i h i1 B C Ž A Finally a class of triangles is called saturated if the following condition holds: if in the situation of ase change in Prosition 21 the third vertical and the second horizontal triangle is in, then the triangle A f B g C h Ž A is in An easy consequence of the octahedral axiom is that is saturated iff in the situation of coase change in Prosition 21, if the second vertical and the third horizontal triangle is in, then the triangle f g h A B C Ž A is in The following concept is inspired from the definition of an exact category 61

9 276 APOSTOLOS BLIGIANNIS DFINITION 22 A full sucategory Diag C, is called a prer class of triangles if the following conditions are true: Ž i Ž ii Ž iii is closed under isomorphisms, finite croducts, and 0 is closed under suspensions and is saturated is closed under ase and coase change XAMPL 23 Ž 1 The class of split triangles 0 and the class of all triangles in C are prer classes of triangles If is a prer class of triangles in C then is a prer class of triangles in C If; i I4 i is a family of prer classes of triangles, then i I i is a prer class of triangles If ; i I4 i is an increasing chain, then the class of triangles i I i is prer Ž 2 Let U: C D e an exact functor of triangulated categories and 1 let e a prer class of triangles in D Let F U Ž e the class of triangles T in C such that UT Then F is a prer class of triangles in C Ž 3 Let F: C U e a Ž co- homological functor from C to an aelian category U Then we otain a prer class of triangles Ž F in C as follows: A triangle A B C Ž A is in Ž F iff i, the induced sequence 0 F i Ž A F i Ž B F i Ž C 0 is exact in U, where F i F i Ž 4 If X C is a class of ojects satisfying Ž X X, then we otain a prer class of triangles X, resp Ž X, inc as follows: A triangle A B C Ž A is in Ž X, resp Ž X, iff X X, the induced sequence 0 CŽ X, A CŽ X, B CŽ X, C 0, resp 0 CŽ C, X CŽ B, X CŽ A, X 0, is exact in A The family of prer classes of triangles in C is easily seen to e a Ž ig poset with 0 Ž the class and 1 Ž the class, defining Phantom Maps In general it is difficult to distinguish the morphisms occurring in a triangle y a characteristic prerty To solve this prolem we proceed as follows DFINITION 24 Let e a class of triangles in C Žnot necessarily f g h prer and let A B C Ž A e a triangle in Ž i The morphism f: A B is called an -prer monic

10 PURITY IN TRIANGULATD CATGORIS 277 Ž ii The morphism g: B C is called an -prer epic Ž iii The morphism h: C Ž A is called an -phantom map The class of -phantom maps is denoted y Ph C The concept of phantom maps has homoty-theoretic origin and the tological terminology is due to A Heller Žsee 52 For a justification of the definition see Section 3 and 52 Let Ph Ž A, B e the set of all -phantom maps from A to B and let Ph n Ž A, B e the set of all maps from A to B which can e written as a composition of n -phantom maps We set Ph n Ž C Ph n Ž A, B A, B C We recall that an ideal of C is an additive sufunctor of CŽ, An ideal I of C is called -stale if f I n Ž f I, n If I is a -stale ideal of C, then define a class of triangles I in C as follows f g h The triangle A B C Ž A if h IŽC, Ž A I A -stale ideal I in C is called saturated, if the following condition is true: if f is I -prer epic and f I then I By the octahedral axiom this is equivalent to the condition that if g is an I -prer monic and g I then I Trivially if is a prer class of triangles, then Ph n Ž C is a -stale saturated ideal of C, n 1 Conversely if I is a -stale saturated ideal, then the class of triangles I is prer and we have the relations: Ph Ž C I and Ph Ž C We collect these oser- I vations in the following PROPOSITION 25 The map : I I is a poset isomorphism etween the poset of -stale saturated ideals of C and the poset of prer classes of triangles in C, with inerse gien y : Ph Ž C, e g, CŽ, and 0 0 It follows that is a prer class of triangles in C iff Ph Ž C is a -stale saturated ideal of C This indicates an analogy etween the homological theory in C ased on prer classes of triangles and the formulation y Butler and Horrocks 18 of relative homology in an aelian category, ased on sufunctors of xt 24 Baer s Theory f g Let e a prer class of triangles in C and let T : A B C h Ž A e a triangle in We call h: C Ž A the characteristic class of Ž T, and usually we denote it y chž T h Let A, C e two ojects of f g h C and consider the class * C, A of all triangles A B C Ž A f i g in We define a relation in * C, A as follows If T : A B i i i C h i Ž A, i 1, 2, are elements of * Ž C, A, then we define Ž T Ž T 1 2

11 278 APOSTOLOS BLIGIANNIS if there exists a morphism of triangles f1 g1 h1 1 A B C Ž A f2 g2 h2 2 A B C Ž A Oviously is an isomorphism and is an equivalence relation on the class * Ž C, A Using ase and coase change, it is easy to see that we can define Ž as in the case of the classical Baer s theory in an aelian category a sum in the class Ž C, A * Ž C, A in such a way that Ž C, A ecomes an aelian group and, : C C A an additive ifunctor Trivially we have the following COROLLARY 26 ch is an isomorphism of ifunctors: Ž, Ph Ž, 3 TH FRYD AND STNROD CATGORY OF A TRIANGULATD CATGORY 31 The Freyd Category Let C e an additive category We recall that an additive functor F: C A is called finitely presented if there exists an exact sequence CŽ, A CŽ, B F 0 We denote y AŽ C or modž C the category of finitely presented functors and we call it the Freyd category of C The Yoneda emedding is denoted y : C AŽ C We define also the Ž Ž Freyd category B C to e A C,so B C mod C is the dual of the category of finitely presented covariant functors over C The corresponding Yoneda emedding is denoted y : C BŽ C The category AŽ C, resp BŽ C, has cokernels, resp kernels, and the functor, resp, has the universal prerty that any additive functor F: C M where M has cokernels, resp kernels, admits a unique cokernel, resp kernel, preserving extension through, resp We recall that a morphism f: A B is a weak kernel of g: B C, if the sequence of Ž f Ž g functors A B Ž C is exact The notion of a weak cokernel is defined dually By results of Freyd 29, the category AŽ C, resp BŽ C, is aelian iff any morphism in C has a weak kernel, resp weak cokernel In this case, resp, emeds C as a class of projective, resp injective, ojects in AŽ C, resp BŽ C, containing enough projective, resp injective, ojects

12 PURITY IN TRIANGULATD CATGORIS 279 We recall that an aelian category A is called Froenius if A has enough projectives, enough injectives, and the projectives coincide with the injectives By results of Freyd 29, for any triangulated category C, the category AŽ C is Froenius aelian and the Yoneda emedding : C AŽ C realizes C as a class of projectiveinjective ojects in AŽ C, containing oth enough projectives and injectives Moreover there exists a unique equivalence : A C B C such that It follows that any oject F in AŽ C has a description as a kernel, image, or cokernel of a morphism etween representale functors The Freyd category is the uniersal homological category of C in the sense that is a homological functor and if F: C K is a homological functor to an aelian category K, then there exists a unique exact functor F*: AŽ C K such that F* F Note that if idempotents split in C, then induces an equivalence C ProjŽ AŽ C InjŽ AŽ C 32 The Steenrod Category Let C e a triangulated category and let I e a class of morphisms in C We denote y AŽ I the full sucategory of AŽ C consisting of all functors of the form Im Ž h, for h I Similarly let BŽ I e the full sucategory of B C consisting of all functors of the form Im Ž h Trivially the suspension of C extends to an automorphism of AŽ C and BŽ C, denoted also y A full sucategory U of AŽ C or BŽ C is called -stale if U is closed under THORM 31 The following statements are equialent Ž i I is a -stale saturated ideal of C Ž ii AŽ I is a -stale Serre sucategory of AŽ C Ž iii BŽ I is a -stale Serre sucategory of BŽ C If one of the aoe is true, then there is an equialence AŽ I BŽ I which induces an equialence AŽ C AŽ I BŽ C BŽ I in the localizations Proof Ž ii Ž i Let AŽ C AŽ I e the quotient category and let Q: AŽ C AŽ C AŽ I e the exact quotient functor Then the composition S I Q is a homological functor and it is easy to see that the kernel ideal ker S I I Then oviously I is a -stale saturated ideal of C Ž i Ž ii Assume that I is a -stale saturated ideal of C Then AŽ I is a -stale full sucategory of AŽ C Let : G H e an epimorphism in AŽ C with G AŽ I Then there exists a morphism : B K in I such that Im Ž G and a morphism : D L such that Im Ž H Let Ž and Ž e the canonical factor-

13 280 APOSTOLOS BLIGIANNIS izations of Ž and Ž Since is epic and Ž D is projective, there exists : Ž D G such that Similarly since is epic, there exists a morphism Ž : Ž D Ž B such that Ž Hence Ž On the other hand, since Ž L is injective, there exists a morphism Ž : Ž K Ž L such that Ž Then Ž Ž Ž Ž Ž Hence Since I, it follows that I Hence H AŽ I and AŽ I is closed under quotient ojects The proof that AŽ I is closed under suojects is dual and is left to the reader It remains to prove that AŽ I is closed under extensions Let 0 F1 F2 F3 0 e a short exact se- f i g quence in A C with F, F A I Let A B i C Ž A 1 3 i i i i e triangles in C such that F Im Ž g and g, g I, i 1, 3 Let Ž g i i 1 3 i e the canonical factorizations Since Ž B i i 3 is projective, there exists : Ž B F such that Since Ž C is injective there exists : F Ž C 2 1 such that 1 It is easy to see that F2 is the image of the morphism Ž g, where 2 g1 0 g : B B C C ž g3 / and : B C is the unique morphism such that Ž 3 1 Hence it suffices to show that g or equivalently is in I Since Ž f Ž f 0, there exists a morphism : Ž A 3 3 F1 such that Ž f Since Ž A is projective, there exists : Ž A Ž B such that Then Ž f ; hence Ž f Ž g Since is of the form Ž , it follows that Ž f Ž f Ž Ž g f3 g 1 Since g1 is in I, it follows that f3 I But f3 is I -epic Since I is saturated, we have I We conclude that F2 A Ž I and A Ž I is closed under extensions The proof of Ž i Ž iii is similar and is left to the reader Finally it is easy to see that the equivalence : AŽ C BŽ C defined y Ž Coker f Ker Ž f sends AŽ I to BŽ I, so it induces an equivalence AŽ C AŽ I BŽ C BŽ I COROLLARY 32 The map AŽPh Ž C gies a ijection etween prer classes of triangles in C and -stale Serre sucategories of AŽ C, with the inerse the map U Ž F U, where FU is the homological functor C AŽ C AŽ C U DFINITION 33 Let e a prer class of triangles in C The Steenrod category, resp dual Steenrod category, of C with respect to is

14 PURITY IN TRIANGULATD CATGORIS 281 defined y S Ž C AŽ C A Ph Ž C, resp S Ž C BŽ C B Ph Ž C where Ph Ž C is the -stale saturated ideal of -phantom maps The canonical functor S: C S C, resp T: C S Ž C, given y the composition of the Yoneda emedding : C A C, resp : C BŽ C, and the quotient functor AŽ C AŽ C AŽPh Ž C, resp BŽ C BŽ C BŽPh Ž C, is called the projectiization functor, resp injectiization functor We refer to the next section for the justification of the aove definition The following result summarizes the aove oservations and explains the terminology for the phantom maps, since these maps are precisely the maps which are invisile in the Steenrod category THORM 34 Let e a prer class of triangles in C Then the Steenrod category S Ž C is aelian and the projectiization functor S: C S Ž C is a homological functor haing the prerty that SŽ 0, Ph Ž C Moreoer the pair ŽS, S Ž C has the following uniersal prerty If H: C M is a homological functor to an aelian category M such that HŽ 0, Ph Ž C, then there exists a unique exact functor H*: S Ž C M such that H*S H Proof It suffices to prove only the universal prerty Let H: AŽ C M e the unique exact functor which extends uniquely H through Since H kills -phantom maps, H kills ojects of the Serre sucategory AŽPh Ž C, so there exists a unique exact functor H*: S Ž C M such that H*Q H Then H*S H*Q H H and H* is the unique exact functor with this prerty Oserve that S Ž C 0 and S Ž C AŽ C 0 By the aove theorem, there exists an equivalence D: S C S Ž C such that DS T We leave to the reader to formulate the aove result for the dual Steenrod category S Ž C and the injectivization functor T We regard the Steenrod category as an aelian approximation of C So it is useful to know when the projectivization functor S: C S Ž C is non-trivial DFINITION 35 The prer class of triangles generates C if the projectivization functor S: C S Ž C reflects zero ojects, ie, SŽ A 0 A 0

15 282 APOSTOLOS BLIGIANNIS We recall that the Jacoson radical JacŽ C of an additive category C is the ideal in C defined y JacŽ CŽ A, B f: A B; g: B A, the morphism 1 f g is invertile 4 A If F is an additive functor, we denote y kerž F, resp KerŽ F, the ideal of morphisms, resp the full sucategory of ojects, annihilated y F If I is a -stale ideal of C, let S I : C AŽ C AŽ I e the canonical functor, ie, the composition of : C AŽ C and the quotient functor AŽ C AŽ C AŽ I LMMA 36 kerž S I Moreoer I JacŽ C iff KerŽ S 0 I Proof The first assertion is trivial If I JacŽ C and S Ž A I 0, then oviously 1 I JacŽ C ; hence A 0 Conversely if KerŽ S A I 0 and f: A B is in I, let g: B A e any morphism and let 1A f g Then trivially S S 1 If A A C Ž A I I A is a triangle in C, then S Ž C I 0; hence C 0 It follows that is invertile and this implies that f JacŽ C COROLLARY 37 generates C iff Ph Ž C JacŽ C I 4 PROJCTIV OBJCTS, RSOLUTIONS, AND DRIVD FUNCTORS We fix throughout a prer class of triangles category C in the triangulated 41 Projectie Ojects and Gloal Dimension DFINITION 41 An oject P C, resp I C, is called -projectie, resp -injectie, if for any triangle A B C Ž A in, the induced sequence 0 CŽ P, A CŽ P, B CŽ P, C 0, resp 0 CŽ C, I CŽ B, I CŽ A, I 0, is exact in A We denote y PŽ Ž IŽ the full sucategory of -projective Ž-in- jective ojects of C As an immediate consequence of the aove definition we have that the categories PŽ and IŽ are full, additive, closed under isomorphisms, direct summands, and -stale, ie, Ž PŽ PŽ and Ž IŽ IŽ We say that C has enough -projecties if for any oject A C there h exists a triangle K P A Ž K in with P PŽ Oserve that in this case h is a weakly uniersal -phantom map out of A in the sense that any -phantom map h: A B factors through h in a not necessarily unique way Dually one defines when C has enough -injecties We study only the case of -projectives since the study of the -injectives is dual However, we use freely the dual results when it is necessary Note that an

16 PURITY IN TRIANGULATD CATGORIS 283 equivalent formulation of the setup of a prer class of triangles in C such that C has enough -projectives is that of a projective class of morphisms in C in the sense of ilenerg and Moore 26 ; see 19 We recall that a full sucategory X C is called contraariantly finite 6 if for any A C there exists a morphism f: X A with X X, such that any morphism g: X A with X X factors through f The following result is a direct consequence of the definitions and its proof is left to the reader LMMA 42 Ž i P PŽ :Ph Ž P, 0 and I IŽ :Ph Ž, I 0 Ž ii If C has enough -projecties, then generates C iff A C and CŽ P, A 0, P PŽ, implies A 0 Ž iii C has enough -projecties iff PŽ is contraariantly finite In this case, Ž PŽ ; ie, a triangle D F C Ž D is in iff P PŽ the induced sequence 0 CŽ P, D CŽ P, F CŽ P, C 0 is exact Moreoer a morphism f: A Bis -phantom iff CŽ P, f 0, P PŽ Further A C : A PŽ iff Ph Ž A, 0 Ž iv Let P e a full additie contraariantly finite sucategory of C, closed under direct summands such that Ž P P Then P PŽŽ P COROLLARY 43 The maps P Ž P, PŽ are inerse ijections etween contraariantly finite -stale additie sucategories of C closed under direct summands and prer classes of triangles in C such that C has enough -projecties f i g i h PROPOSITION 44 Schanuel s lemma If K P A i Ž K i i i are triangles in with P PŽ, i 1, 2, then K P K P Proof g1 h 2: i Consider the octahedral diagram induced from the composition g1 h1 Ž f P A Ž K Ž P h2 g1 h P Ž K X Ž P g1 h2 Ž f2 Ž g2 2 2 A Ž K Ž P Ž A 0 Ž g2 Ž h Ž K Ž K 0

17 284 APOSTOLOS BLIGIANNIS Since h, h are -phantoms and P, Ž P are -projectives, we have g1 h2 0 and g2 h1 0 Hence the second horizontal and the third vertical triangles are split So X Ž K Ž P Ž K Ž P K2 P1 K1 P 2 If K P A Ž K is in with P in PŽ, then we call the oject K a first -syzygy of A Annth -syzygy of A is defined as usual y induction By Schanuel s lemma any two -syzygies of A are isomorphic modulo -projectives We define inductively the -projectie dimension -pd A of an oject A C as follows If A PŽ then -pd A 0 Next if -pd A 0 define -pd A n if there exists a triangle K P A Ž K in with P PŽ and -pd K n 1 Finally define -pd A n if - pd A n and -pd A n 1 Of course we set -pd A, if - pd A n, n 0 The -gloal dimension -gldim C of C is defined y -gldim C sup-pd A; A C 4 XAMPL 45 C 0:-pd C, soif0 C : -gldim C On the other extreme 0-gldim C 0 The following is proved using standard arguments PROPOSITION 46 Ž i A, B C : -pd A B max -pd A, - pd B 4 Ž ii A C, n : -pd A -pd Ý n Ž A Ž iii Let A P B Ž A e a triangle in with P PŽ Then either B PŽ or else -pd A -pd B 1 42 Resolutions and Deried Functors DFINITION 47 An -exact complex X A over A C is a diagram X n1 d n 1 X n X 1 d1 X 0 d 0 A 0 in C, such that n 0: Ž n1 g n n f n n h n Ž n1 i There are triangles K X K K in, where K 0 A n n n1 0 0 ii The differential d f g, n 1 and d f An -projectie resolution of A C is an -exact complex P A as n aove such that P P, n 0 The next result gives a useful characterization of an -projective resolution PROPOSITION 48 Assume that C has enough -projecties and consider 1 d 1 0 d 0 n a complex P : P P A 0 in C with P PŽ, n 0

18 PURITY IN TRIANGULATD CATGORIS 285 Then P is an -projectie resolution of A iff the induced complex CŽQ, P is exact, Q PŽ Proof Assume that the induced complex CŽQ, P is exact in A, g 0 0 f 0 h Q P Set f d and let T : K P A 0 Ž K 1 0 e a triangle in C Since CŽQ, f 0 is epic, we have CŽQ, h 0 0, Q PŽ 0 By Lemma 42 it follows that h is -phantom; hence Ž T 0 is in Since d 1 f 0 0, there exists a morphism f 1 : P 1 K 1 such that f 1 g 0 d 1 Ž 1 2 g 1 1 f 1 1 h It follows easily that C Q, f is epic Let T : K P K 1 1 Ž K 2 e a triangle in C As aove we have that Ž T 1 is in Since Ž 2 d d d f g 0, we have C P, d f 1 CŽ PŽ, g 0 0 Since CŽ PŽ, g 0 Ž 2 is monic, we have C P, d f 1 0; hence y 2 1 Lemma 42, d f 2 is -phantom Since P PŽ, y Lemma 42, we have d 2 f 1 0 So there exists a morphism f 2 : P 2 K 2 such that d 2 f 2 g 1 Continuing inductively in this way we see that P is an -projective resolution of A The converse is trivial COROLLARY 49 Let 0 Pi n Pi n1 Pi 0 Ai 0 e - projectie resolutions of A i, i 1, 2 If A1 A 2, then P1 0 P2 1 P1 2 P2 0 P1 1 P2 2 From now on we assume that C has enough -projectives For any oject A C we fix an -projective resolution of A A n1 A 1 fa 0 n1 n 1 0 A A A A A P P P P P A 0 which, y definition, is the Yoneda composition of triangles n n1 g A n n fa n n h n A n1 A A A A A T K P K Ý K n 0 with PA P, n 0 and where K A A Using standard arguments from relative homological algera, one can prove a version of the comparison theorem 68 for -projective resolutions, the ovious formulation of which is left to the reader It follows that any two -projective resolutions of an oject are homoty equivalent Using Schanuel s lemma and the aove oservations we have the following consequence COROLLARY 410 -pd A n iff there exists an -projectie resolution of A of the form 0 P n P 1 P 0 A 0 PROPOSITION 411 horseshoe lemma Let T : A B C Ž A e a triangle in Then there are -projectie resolutions P Ž A,

19 286 APOSTOLOS BLIGIANNIS P B, and P C of A, B, and C, respectiely, and a commutatie diagram p q 0 Ž P A P B P C P A A B C Ž A n p n n q n n 0 such that P A P B P C ŽP n Ž A is a split triangle, n 0 Such a diagram is called an -projective resolution of the triangle Ž T Proof Let fa 0 : PA 0 A and fc 0 : PC 0 C e -prer epics with P A, PC P Since is -phantom, fc 0 Using that is an automorphism and a result of Verdier Žsee 71, p 378, the commutative square on the t left corner elow is emedded in a diagram Ž p 0 Ž q 0 C A B C P Ž P Ž P Ž P 0 Ž 0 Ž 0 Ž 0 fc f A fb fc Ž Ž C Ž A Ž B Ž C 0 Ž 0 Ž 0 Ž 0 hc h A h B hc 1 Ž Ž Ž 2 1 C A B C Ž K Ž K Ž K Ž K Ž 0 2Ž 0 2Ž 0 2Ž 0 gc g A gb gc p Ž q C A B C Ž P Ž P Ž P Ž P which is commutative except the lower right square which anticommutes and where the rows and columns are triangles Then we have the following commutative diagram in which the first three vertical and horizontal diagrams are triangles: A B C A K K K Ž K Ž 0 ga gb gc g A p q PA PB PC PA Ž 0 fa fb fc f A A B C Ž A Ž 0 ha hb hc h A 1 Ž 1 Ž 1 Ž 1 A B C A Ž K Ž K Ž K Ž K

20 PURITY IN TRIANGULATD CATGORIS Since the second horizontal triangle is split, we have that P is in PŽ B Applying to the aove diagram the homological functor CŽ P,, P PŽ, a simple diagram chasing argument shows that 0 CŽP, K 1 A CŽ P, K 1 CŽP, K 1 B C 0 is exact By Lemma 42, the first horizontal triangle is in Similarly the second vertical triangle is in Inductively the aove procedure completes the proof Let F: C A and G: C A e additive functors, where A is an aelian category Then we define the -deried functors L n F : C A and R n G: C A, n 0 of F and G with respect to, as follows Let P C e an -projective resolution of C C Then define L FC n to e the nth-homology of the Ž n induced complex F P and R GC to e the nth-cohomology of the induced complex GP Ž In particular C C, we define the -extension functors xt n Ž, C y xt n Ž, C R n C Ž, C : C A, n 0 By the comparison theorem the aove -derived functors are well defined COROLLARY 412 Let A B C Ž A e in If F: C A and G: C A are additie functors where A is aelian, then we hae exact sequences L F C L F A L F B L FŽ C 0 0 R 0 GŽ C R 0 GŽ B R 0 GŽ A R 1 GŽ C In particular for any X C we hae a long exact sequence 0 xt 0 Ž C, X xt 0 Ž B, X xt 0 Ž A, X xt 1 Ž C, X Proof Applying F and G to the -projective resolution of the triangle as in Prosition 411 and taking Ž co- homology, we get the desired exact sequences The easy proof of the following is left to the reader COROLLARY 413 Let F: C A, resp G: C A, e a homological, resp cohomological, functor where A is aelian Then the natural morphism L 0 F F, resp G R 0 G, is an isomorphism iff F, resp G, kills -phantom maps

21 288 APOSTOLOS BLIGIANNIS 43 Projectiely Generating Prer Classes DFINITION 414 A prer class of triangles in C projectiely generates C, if generates C and C has enough -projectives A full sucategory G C is called a generating sucategory of C, if G is -stale and A C: CŽ G, A 0 A 0 By Lemma 42, we have the following COROLLARY 415 The map PŽ is a ijection etween projectiely generating prer classes of triangles in C and contraariantly finite generating sucategories of C closed under direct summands The inerse is gien y P Ž P Remark 416 Oviously C has enough -projectives and PŽ 0 Conversely if C has enough -projectives and PŽ 0, then Ž PŽ Ž 0 If C 0, then is not projectively generating Similarly C has enough -projectives and PŽ 0 0 C Conversely if C has enough -projectives and PŽ C, then Ž PŽ Ž C is always projectively generating 0 0 The next result, proved independently y Christensen 19, shows that if projectively generates C, then the -projective dimension is controlled y the vanishing of the -extension functors PROPOSITION 417 If projectiely generates C, then A C, n 0: -pd A n xt n1 Ž A, 0 n Proof It suffices to prove the direction Consider the triangles T A and the resolution PA after Corollary 49 Since the cohomology of the Ž n Ž n1 Ž n2 n1 short complex C P, C P, C P, is xt Ž A, A A A 0, the morphism fa n1 : PA n1 K A n1 factors through A n1 Hence A n1 fa n1 g A n fa n1, for some : PA n K A n1 Applying the Ž n1 functor C P,, P P, to this relation and using that C P, f A is epic, we have that CŽ P, g n CŽ P, 1 Ž Ž n n 1, so C P,, f A C Ž P, K A : A Ž n Ž n1 n n n1 n C P, PA C P, K A K A Since generates C, PA K A K AIt n follows that K PŽ or equivalently -pd A n A 44 The Cartan Morphism Assume that C has split idempotents and is skeletally small; ie, the collection of isoclasses of ojects of C, denoted henceforth y IsoŽ C, isa set The Grothendieck group K Ž C, 0 of C with respect to is defined as the quotient of the free aelian group on IsoŽ C, modulo the sugroup generated y all elements Ž A Ž B Ž C, where A B C Ž A

22 PURITY IN TRIANGULATD CATGORIS 289 is a triangle in If, then K Ž C, 0 is the usual group defined y Grothendieck If, then K Ž C, is the group K Ž C, of the monoidal category Ž C, Oserve that we have canonical epimorphisms K Ž C, K Ž C, and K Ž C, K Ž C, In case C has enough -projectives, then we define the Cartan morphism c : K Ž PŽ, 0 K Ž C, y c ŽP P 0 PROPOSITION 418 If C C : -pd C, then c is an isomorphism Proof We construct an inverse of c as follows Let 0 Pn P 0 A 0ean -projective resolution of A C Using Schanuel s and horseshoe lemma it follows directly that the assignment A n Ý Ž 1 i P i is a well defined morphism d : K Ž C, K Ž PŽ, i0 0 0 Clearly d is the inverse of c 45 The Steenrod Categories Let F: C K e a homological functor, where K is aelian Let Ž F e the prer class of triangles in C induced y F Following Street 68, an oject P C is called F-projectie if FŽ P is projective in K and A C, the canonical map CŽ P, A K FŽ P, FŽ A is an isomorphism Let PŽ F e the full sucategory of F-projectives C has enough F-projecties if A C there exists a triangle K P A Ž K Ž F with P PŽ F It is trivial to see that if C has enough F-projectives, then C has enough Ž F -projectives and PŽŽ F PŽ F The next result shows that if C has enough -projectives for a prer class of triangles, then Ž S and PŽŽ S PŽ S, where S is the projectivization functor It follows that in this case, our formulation of relative homology in C is equivalent to the theory develed y Street in 68 PROPOSITION 419 P PŽ, SŽ P Proj S Ž C and A C, the canonical map S : CŽ P, A S Ž CSŽ P, SŽ A P, A is an isomorphism Hence S induces a full emedding P Ž Proj S Ž C If C has enough -projecties, then: Ž i S Ž C is equialent to AŽ PŽ and S is isomorphic to the restriction A CŽ, A PŽ In particular S Ž C has enough projecties and Ž S ii If idempotents split in P, then S: P Proj S Ž C Ž iii A complex P Aoer A is an -projectie resolution of A iff SŽ P SŽ A is a projectie resolution of SŽ A in S Ž C Proof Since the kernel ideal of the functor S is Ph Ž C, it follows that S is injective, since Ph Ž P, A 0, P PŽ Now let : SŽ P P, A SŽ A e a morphism in S Ž C Since S Q, where Q: AŽ C

23 290 APOSTOLOS BLIGIANNIS AŽ C AŽPh Ž C is the quotient functor, is represented y a fraction s f P F Ž A, where s has kernel and cokernel in AŽPh Ž C Let : Ž P G e the cokernel of s Then G is the image of a morphism Ž h : Ž C Ž D, where h is an -phantom map; let Ž h e the canonical factorization Then there exists a morphism Ž : Ž P Ž C such that Ž Then Ž Ž h Since h is -phantom and P is -projective, h 0 Hence 0 and s is split epic Let s: Ž P F e a morphism such that s s 1 Ž P Then s f is of the form Ž for : P A Trivially SŽ QŽ s f Hence S is surjective The proof that SŽ P P, A is projective is similar and is left to the reader Ž i One can prove this y using the universal prerty of S Ž C Here is a quick proof Since C has enough -projectives, the sucategory PŽ is contravariantly finite in C By 13, there exists a short exact Ž R sequence of aelian categories 0 A CP A C AŽ PŽ 0, where CPŽ is the stale category of C modulo the full sucategory PŽ and R is the restriction functor F F PŽ It is easy to see that AŽCPŽ is equivalent to AŽPh Ž C so y Definition 33, S Ž C is equivalent to AŽ PŽ In particular S Ž C has enough projectives Part Ž ii follows from 28 and part Ž iii follows from Ž i The aove result explains the terminology for the projectivization functor The next example explains the terminology for the Steenrod category XAMPL 420 Let C e the stale homoty category of spectra 51 and consider the full sucategory X n Ž KŽŽ p ; n 4, where KŽŽ p is the ilenergmaclane spectrum corresponding to Ž p Then the Steenrod category S Ž C Ž X of C with respect to the prer class of triangles Ž X is equivalent to the category of modules over the mod-p Steenrod algera; see 74 for details The description of the Steenrod category in Prosition 419 allows us to give a formula for the derived functors of an additive functor F: C M, where M is aelian Let G F PŽ : PŽ M Since S Ž C AŽ PŽ, G has a unique right exact extension G*: S Ž C M through the Steenrod category The easy proof of the following and its contravariant analogue is left to the reader COROLLARY 421 L F Ž L G* S, n 0 n n Ž COROLLARY 422 i For any B C we hae natural isomorphisms n n xt, B xt S Ž C S, S B, n 0

24 PURITY IN TRIANGULATD CATGORIS The natural map : C, B R CŽ, B xt 0 Ž, B, B coincides with the map S : CŽ, B S Ž CSŽ, SŽ B, B Hence Ker, B is the right ideal Ph Ž, B Ž ii A C : -pd A pd SŽ A ; if projectiely generates C, then -pd A pd SŽ A In particular -gldim C gldim S Ž C Ž iii Assume that C is skeletally small with split idempotents If C is -projectiely generated and gldim S Ž C, then the projectiization functor S: C S C induces an isomorphism S: K0 C, K Ž S Ž C, and an isomorphism KŽ AŽ C KŽ S Ž C 0 KŽ AŽCPŽ in Quillen s higher K-theory Proof The assertions Ž i and Ž ii follow trivially from Prosition 419 The first part of Ž iii follows from Prosition 418 and the commutativity of the diagram K Ž PŽ, S K ŽProjŽ S Ž C, 0 0 c c S Ž C S K Ž C, K Ž S Ž C 0 0 The second part of iii follows from a result of Auslander and Reiten 5 The dual of Prosition 419 is also true We state only the following PROPOSITION 423 If C has enough -injecties then the dual Steenrod category of C with respect to is equialent to BŽ I Ž modž I Ž and the injectiization functor T: C S Ž C is isomorphic to the restriction A CŽ A, I Ž If I Ž has split idempotents, then T induces an equialence: I Inj S Ž C If X C is a class of ojects, then addž X denotes the full sucategory of C consisting of all direct summands of finite croducts of ojects of X An -injectie enele of A C is an -prer monic : A with I Ž such that if, then is an automorphism of Without assuming the existence of enough -injectives we have the following THORM 424 The projectiization functor S induces a full emedding S: I Ž Inj S Ž C and Inj S Ž C addž Im S In particular S: I Inj S Ž C if Im S is closed under direct summands If this is the case, then C has enough -injecties iff S Ž C has

25 292 APOSTOLOS BLIGIANNIS enough injecties In particular if S Ž C has injectie eneles, then C has -injectie eneles Proof Let P 1 A 1 P 0 f A 0 A A A 0 e the start of an -projective resolution of A C If I is -injective then oviously the map S : CŽ A, I A, I S Ž CSŽ A, SŽ I is injective Let : SŽ A SŽ I e any morphism Ž 0 Ž 0 0 Consider the morphism S fa : S PA S I Since PA is -projective, Ž 0 0 S f S, for some morphism : P I Then SŽ 1 SŽ A A A Ž 1 Ž S A S fa 0, so A : PA I is -phantom Since PA is -projective, 1 0 Since A 1 fa 1 g A 0, we have that the morphism g A Ž factors through h A: K A K A So g A ha, for some morphism Ž : K A I But since ha is -phantom, it follows that g A : K A I is -phantom Since I is -injective, g A 0 0 Hence there exists : 0 A I such that f Then SŽ f 0 SŽ SŽ Since SŽf 0 A A A is epic, SŽ ; hence S is an isomorphism It remains to show that SŽ I A, I SŽ Inj S C Let F S C and let S P SŽ P 1 0 F 0 e the start of a projective resolution of F Let T : A P P Ž A 1 0 e f a triangle in C and let K P A Ž K 2 e an -projective presen- SŽ SŽ tation of A in C Since the sequence S A S P SŽ P 1 0 F 0 is exact, if G KerŽSŽ, then we have an epimorphism SŽ f S f coim S : S P G Then S P SŽ P SŽ P F 0 is part of a projective resolution of F Applying S Ž C, SŽ I to this sequence and using that the map SC, I is an isomorphism C C, it follows easily that the resulting sequence is exact This means that 1 xt F, SŽ I 0 Hence SŽ I S Ž C is injective From the triangle Ž T it follows that we have an inclusion : F SŽŽ A If F is injective then F is a direct summand of SŽŽ A ; hence if in addition Im S is closed under direct summands, then F SŽ I for some oviously -injective I If C has enough -injectives, let A I B 2 Ž A e a triangle in with I I Ž Then SŽ : F SŽ I is monic and SŽ I is injective; hence S Ž C has enough injectives Conversely if S Ž C has enough injectives, C C, let : SŽ C e monic with Inj S Ž C Then SŽ I for some I I Ž and S for a morphism : C I in C Since the triangle C I B Ž C is in, it follows that C has enough -injectives Clearly if SŽ : SŽ C SŽ I is an injective envele, then : C I is an -injective envele 46 Semisimple and Hereditary Categories The following characterization of -phantomless categories follows easily from our previous results