Relative Homological Algebra and Purity in Triangulated Categories

Journal of Algebra 227, 268᎐361Ž 2000. doi:10.1006rjabr.1999.8237, available online at http:rrwww.idealibrary.com on

Apostolos Beligiannis

Fakultat¨¨ fur Mathematik, Uni¨ersitat ¨ Bielefeld, D-33501 Bielefeld, Germany E-mail: [email protected], [email protected]

Communicated by Michel Broue´

Received June 7, 1999

CONTENTS

1. Introduction. 2. Proper classes of triangles and phantom maps. 3. The Steenrod and Freyd category of a triangulated category. 4. Projectië objects, resolutions, and deriëd functors. 5. The phantom tower, the cellular tower, homotopy colimits, and compact objects. 6. Localization and the relatië deriëd category. 7. The stable triangulated category. 8. Projectiïty, injectiïty, and flatness. 9. Phantomless triangulated categories. 10. Brown representation theorems. 11. Purity. 12. Applications to deriëd and stable categories. References.

1. INTRODUCTION

Triangulated categories were introduced by Grothendieck and Verdier in the early sixties as the proper framework for doing homological algebra in an abelian category. Since then triangulated categories have found important applications in algebraic geometry, stable homotopy theory, and representation theory. Our main purpose in this paper is to study a triangulated category, using relative homological algebra which is developed inside the triangulated category. Relative homological algebra has been formulated by Hochschild in categories of modules and later by Heller and Butler and Horrocks in

268 0021-8693r00 $35.00 Copyright ᮊ 2000 by Academic Press All rights of reproduction in any form reserved. PURITY IN TRIANGULATED CATEGORIES 269 more general categories with a relative abelian structure. Its main theme consists of a selection of a class of extensions. In triangulated categories there is a natural candidate for extensions, namely theŽ distinguished. triangles. Let C be a triangulated category with triangulation ⌬. We develop a homological algebra in C which parallels the homological algebra in an exact category in the sense of Quillen, by specifying a class of triangles E : ⌬ which is closed under translations and satisfies the analo- gous formal properties of a proper class of short exact sequences. We call such a class of triangles E a proper class of triangles. Two big differences occur between the homological algebra in a triangulated category and in an exact category. First, the absolute theory in a triangulated category, i.e., if we choose ⌬ as a proper class, is trivial. Second, a proper class of triangles is uniquely determined by an ideal Ph E ŽC.of C, which we call the ideal of E-phantom maps. This ideal is a homological invariant which almost always is non-trivial and in a sense controls the homological behavior of the triangulated category. However, there are non-trivial examples of phantomless categories. To a large extent the relative homological behavior of C with respect to E depends on the knowledge of the structure of E-phantom maps. The paper is organized as follows. Section 2 contains the basic definitions about proper classes of triangles and phantom maps in a triangulated category C. We prove that there exists a bijective correspondence between proper classes of triangles and special ideals of C, and we discuss briefly an analogue of Baer’s theory of extensions. In Section 3 we prove that there exists a bijective correspondence between proper classes of triangles in C and Serre subcategories of the category modŽC.of finitely presented op additive functors C ª Ab, which are closed under the suspension functor. Using this correspondence, we associate to any proper class of triangles E in C a uniquely determined abelian category SE ŽC., the E-Steen- rod category of C. The E-Steenrod category comes equipped with a homological functor S: C ª SE ŽC., the projecti¨ization functor, which is universal for homological functors out of C which annihilate the ideal of E-phantom maps. The terminology comes from stable homotopy: the Steenrod category of the stable homotopy category of spectra with respect to the proper class of triangles induced from the Eilenberg᎐MacLane spectrum corresponding to ޚrŽp.is the module category over the mod-p Steenrod algebra. We regard SE ŽC.as an abelian approximation of C and the idea is to use it as a tool for transferring information between the topological category C and the algebraic category SE ŽC.. This tool is crucial, if the projectivization functor is non-trivial and this happens iff the proper class of triangles E generates C in an appropriate sense. In Section 4, fixing a proper class of triangles E in a triangulated category C, we introduce E-projective objects, E-projective resolutions, 270 APOSTOLOS BELIGIANNIS

E-projective and E-global dimension and their duals, and we prove the basic tools of homological algebraŽ Schanuel’s lemma, horseshoe lemma, comparison theorem. in this setting. It follows that we can derive additive functors which behave well with respect to triangles in E. We compare the homological invariants of C with respect to E with the homological invariants of its E-Steenrod category, via the projectivization functor. Finally we study briefly the semisimple and hereditary categories. In Section 5, we associate with any object A in C the E-phantom and the E-cellular tower of A, which are crucial for the study of the homological invariants of A with respect to E, e.g., the structure of E-phantom maps out of A and the E-projective dimension of A. Using these towers we prove our first main result which asserts that the global dimension of C with respect to E is less than equal to 1 iff the projectivization functor S: C ª SE ŽC.is full and reflects isomorphisms. The proper setting for the study of the E-phantom and the E-cellular tower is that of a compactly generated triangulated categorywx 55 . In this case homotopy colimits are defined and we prove that under mild assumptions, any object of C is a homotopy colimit of its E-cellular tower and the homotopy colimit of its E-phantom tower is trivial. Finally we study the category of extensions of E-projective objects, the phantom topology and the phantom filtration of C induced by the ideal Ph E ŽC., and we compute the E-phantom maps which n Ž. ‘‘live forever,’’ i.e. maps in FnG1 Ph E C , in terms of the E-cellular tower. In Section 6 we associate with any triangulated category C with enough

E-projectives a new triangulated category DE ŽC.which we call the E-deri¨ed category. Under some mild assumptions DE ŽC.is realized as a full subcategory of C and is described as the localizing subcategory of C generated by the E-projective objects. Moreover, DE ŽC.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 E-deri¨ed functors. Applying these results to the derived category of a Grothendieck category with projectives, we generalize results of Spaltensteinwx 67 , Boekstedt and Neemanwx 16 , Keller wx 42 , and Weibel wx 71 concerning resolutions of unbounded complexes and the structure of the unbounded derived category, proved in the above papers by using essentially a closed model structure on the category of complexes. In Section 7 we prove that the stable category of C modulo E-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 proper classes of triangles in unspecified categories. In Section 8 we study proper classes of triangles EŽX .induced by skeletally small subcategories X in a category C with coproducts. Under a reasonable condition on X we show that the PURITY IN TRIANGULATED CATEGORIES 271

Steenrod category of C with respect to the proper class EŽX .is the op category ModŽX .of additive functors X ª Ab. We compare the EŽX .- homological properties of objects of C with the homological properties of projective, flat, and injective functors of the category ModŽX .. Our first main result in this section shows that if X compactly generates C, then C has enough EŽX .-injective objects in a functorial way and admits EŽX .-injective envelopes. This generalizes a result of Krausewx 50 . Our second main result shows that if X compactly generates C or if any object of C has finite EŽX .-projective dimension, then any coproduct preserving homological functor from C to a Grothendieck category annihilates the ideal of EŽX .-phantom maps. This generalizes results obtained independently by Christensen and Stricklandwx 21 and Krause wx 50 . In Section 9 we characterize the EŽX .-semisimple or EŽX .-phantomless categories C by a host of conditions, generalizing work of Neemanwx 54 . Perhaps the most characteristic is the condition that C is a pure-semisimple locally finitely presented categorywx 24 ; in particular, C has filtered colimits. Using this and an old result of Hellerwx 34Ž observed also by Keller and Neemanwx 44. , we give necessary conditions for a skeletally small category D such that the categories of Pro-objects and Ind-objects over D admit a triangulated structure. These conditions are sufficient for the existence of a Puppe-triangulated or ‘‘pre-triangulated’’ structurewx 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, i.e., a functor which is full, surjective on objects, and reflects isomorphisms. We say that a pair ŽC, X . consisting of a triangulated category with coproducts C and a full skeletally small triangulated subcategory X : C satisfies the Brown representability theorem or that C is an EŽX .-Brown category, if the projectivization functor of C with respect to EŽX .induces a representation equivalence between 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 bounded by 1. This generalizes results of Christensen and Stricklandwx 21 and Neeman wx 56 . In Section 11 we study the fundamental concept of purity in a compactly generated triangulated category which provides a link between representation-theoretic and homological properties. Motivated by the examples of the stable homotopy category of spectra, the derived category of a ring, and the stable module category of a quasi-Frobenius ring, it is natural to regard a compact object as an analogue of a finitely presented object in a triangulated category. In this spirit the proper class of triangles induced by the compact objects are the pure triangles and the resulting theory is the pure homological algebra of C. In this case the pure Steenrod category of 272 APOSTOLOS BELIGIANNIS

C is the module category ModŽC b., where C b denotes the full subcategory of compact objects of C. We give various formulas for the computa- tion of the pure global dimension and we prove that the pure global dimension bounds the pure global dimension of a smashing or finite localization. Our main result characterizes the pure Brown categories as the pure hereditary ones, and also by a host of equivalent conditions, the most notable being the fullness of the projectivization functor and the conditions concerning the behavior of weak colimits. These results are applied directly to the stable homotopy category, which has pure global 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 by the pure homological theory of module categories, we define a category to be of finite type if its pure Steenrod category is locally finite. We prove that the phantomless, Brown, or finite type property is preserved under smashing or finite localization, generalizing results of Hovey et al.wx 38 . In Section 12 we apply the results of the previous sectionsŽ mainly about purity. to derive categories of rings and stable categories of quasi-Frobenius rings. We compute the pure global dimension for many classes of rings and we prove that the derived category DŽ⌳.of a ring ⌳ is of finite type iff DŽ⌳op .is of finite type iff DŽ⌳.and DŽ⌳op . are pure semisimple, so finite type is a symmetric condition. However, we show that the Brown property is not symmetric in general. Motivated by the pure homological theory of module categories we formulate the deri¨ed pure semisimple conjecture, DPSC for short, which asserts that if DŽ⌳.is pure semisimple, then DŽ⌳. is of finite type. DPSC implies the still open 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 algebras. Indeed we characterize the Artin algebras with pure semisimple derived categories as the iterated tilted algebras of Dynkin type. We prove that the analogue of DPSC holds for the stable category of a quasi-Frobenius ring ⌳, showing that ⌳ is representation-finite iff the stable category of ⌳ is pure semisimpleŽ of finite type. . The analogue of DPSC in a general compactly generated triangulated category fails. We prove this by answering in the negative a seemingly unrelated problem of Rooswx 64 concerning the structure of quasi-Frobenius Grothendieck categories. Our results indicate that finite type is the appropriate notion of ‘‘representation-finiteness,’’ at least for stable or derived categories. We close the paper by applying the results of Sections 6 and 7 to derived categories. In particular we obtain structure results for the unbounded derived category and we generalize results of Wheelerwx 72, 73 concerning the structure of the stable derived category. PURITY IN TRIANGULATED CATEGORIES 273

A general convention used in the paper is that composition of morphisms in a category is meant diagramatically: if f: A ª B, g: B ª C are morphisms, their composition is denoted by f ( g. However, we compose functors in the usualŽ anti-diagrammatic. order. Our additive categories admit finite direct sums.

2. PROPER CLASSES OF TRIANGLES AND PHANTOM MAPS

2.1. Triangulated Categories

Let C be an additive category and ⌺: C ª C an additive functor. We define the category DiagŽC, ⌺.as follows. An object of DiagŽC, ⌺.is a diagram in C of the form A ª B ª C ª ⌺ŽA.. A morphism in DiagŽC, ⌺. f i g i h i between Aiiª B ª C iª ⌺ŽA i., i s 1, 2, is a triple of morphisms ␣: A12ª A , ␤: B 12ª B , ␥ : C 12ª C , such that the diagramŽ †. commutes:

fgh1116 6 6 ABC111⌺ŽA 1.

Ž. 6 6 6 ␣ 6 ␤ ␥ ⌺Ž␣ . †

fgh2226 6 6 ABC222⌺ŽA 2..

A triangulated categorywx 70 is a triple ŽC, ⌺, ⌬., where C is an additive category, ⌺: C ª C is an autoequivalence of C, and ⌬ is a full subcategory of DiagŽC, ⌺.which is closed under isomorphisms and satisfies the following axioms:

ŽT1. For any morphism f: A ª B in C, there exists an object fgh A ª B ª C ª ⌺ŽA.in ⌬. For any object A g C, the diagram 0 ª A 1 ª A A ª 0 is in ⌬. fgh gh ŽT2 . If A ª B ª C ª ⌺ŽA.is in ⌬, then B ª C ª ⌺ŽA. ⌺ Ž f . ªy ⌺ŽB.is in ⌬. f i g i h i ŽT3. If Aiiª B ª C iª ⌺ŽA i., i s 1, 2 are in ⌬, and if there are morphisms ␣: A12ª A and ␤: B 12ª B such that ␣ ( f 21s f ( ␤, then there exists a morphism ␥ : C12ª C such that the diagramŽ †. is a morphism in ⌬.

ŽT4 . The Octahedral Axiom. For the formulation of this we refer to Proposition 2.1.

PROPOSITION 2.1Ž see alsowx 51. . Let C be an additië category equipped with an autoequiälence ⌺: C ª C and a class of diagrams ⌬ : DiagŽC, ⌺.. Suppose that the triple ŽC, ⌺, ⌬.satisfies all the axioms of a triangulated category except possibly of the octahedral axiom. Then the following are 274 APOSTOLOS BELIGIANNIS equiälent: fgh Ži. Base Change. For any triangle A ª B ª C ª ⌺ŽA.g ⌬ and any morphism ␧: E ª C, there exists a commutatië diagram

0 6 MMs 6 6 0

6 6 6 6 ␣ ␦

f Ј 6 g Ј 6 hЈ 6

AG E⌺ŽA.

6 6 6 5 6 ␤ ␧ 5

fg6 6 h 6

AB C⌺ŽA.

6 6 6 ␥ 6 ␨

0 6 ⌺ŽM .s 6 ⌺ŽM . 6 0 in which all horizontal and ¨ertical diagrams are triangles in ⌬. fgh Žii. Cobase Change. For any triangle A ª B ª C ª ⌺ŽA.g ⌬ and any morphism ␣: A ª D, there exists a commutati¨e diagram

0 6 NNs 6 6 0

6 6 6 6 ␨ ␦

y1 ⌺ Žh.6 fg6 6

⌺y1 ŽCABC.y

6 6 6 5 6 ␣ ␤ 5

y1 ⌺ ŽhЈ.6 f Ј 6 g Ј 6

⌺y1 ŽCDFC.y

6 6 6 6 ␩ ␽ 0 6 ⌺ŽN .s 6 ⌺ŽN . 6 0 in which all horizontal and ¨ertical diagrams are triangles in ⌬.

Žiii. Octahedral Axiom. For any two morphisms f12: A ª B, f : B ª C there exists a commutati¨e diagram

fgh6 6 6

ABX111⌺ŽA.

6 6 6 556 f2 ␣

f12( fgh6 36 36

ACY⌺ŽA.

6 6 6 f1 6 5 ␤ ⌺Žf1.

fgh6 6 6

BCZ222⌺ŽB.

6 6 6 6 0 h21( ⌺Žg . 0 6 ⌺ŽX .s 6 ⌺ŽX . 6 0 in which all horizontal and the third ¨ertical diagrams are triangles in ⌬. PURITY IN TRIANGULATED CATEGORIES 275

f1 f 2 Proof. Ži.« Žiii. Let A ª B ª C be a diagram in C. ByŽ T1. 1 ␣ f 1 f there are triangles ⌺y ŽZ.ª B ª 2 C ª Z and ⌺y ŽX .ª A ª 1 B ª X in ⌬. Applying base change for the last triangle along ␣, it is easy to see that if we arrange properly the resulting diagram, then we obtain an ␧ ␨ octahedral diagram as inŽ iii. .Ž iii.« Ži. If M ª E ª C ª ⌺ŽM .is a triangle, then applying the octahedral axiom to the composition g (␨, we obtain a diagram which if we apply ⌺y1 and we arrange it properly, we have a diagram as inŽ i. . The proof ofŽ ii.m Žiii. is similar.

Hence for any triangulated category we may use the above equivalent forms instead of the octahedral axiom, when it is more convenient. Throughout the paper we fix a triangulated category C s ŽC, ⌺, ⌬.; ⌺ is the suspension functor, ⌬ is the triangulation, and the elements of ⌬ are fgh the triangles of C. A triangle ŽT .: A ª B ª C ª ⌺ŽA.g ⌬ is called split if h s 0. Trivially if ŽT .is split, then the morphisms f, g induce a direct sum decomposition B ( A [ C. We denote by ⌬ 0 the full subcategory of ⌬ consisting of the split triangles. We call a triangle semi-split if it is isomorphic to a coproduct of suspensions of triangles of the form A ªs A ª 0 ª ⌺ŽA.. Any split triangle is semi-split but the converse is not true.

2.2. Proper Classes of Triangles

A class of triangles E is closed under base change if for any triangle fgh A ª B ª C ª ⌺ŽA.g E and any morphism ␧: E ª C as inŽ i. of f Ј g Ј hЈ Proposition 2.1 the triangle A ª G ª E ª ⌺ŽA.belongs to E. Dually a class of triangles E is closed under cobase change if for any fgh triangle A ª B ª C ª ⌺ŽA.g E and any morphism ␣: A ª D as in f Ј g Ј hЈ Žii. of Proposition 2.1 the triangle D ª F ª C ª ⌺ŽD.belongs to E. A class of triangles E is closed under suspensions if for any triangle fgh i A ª B ª C ª ⌺ŽA.g E and for any i g ޚ the triangle ⌺ ŽA. Ž 1.i ⌺ i fi Ž 1.i ⌺ i gi Ž 1.i ⌺ i hi1 ª y ⌺ ŽB.ª y ⌺ ŽC.ª y ⌺ q ŽA.g E. Finally a class of triangles E is called saturated if the following condition holds: if in the situation of base change in Proposition 2.1 the third vertical and the f g h second horizontal triangle is in E, then the triangle A ª B ª C ª ⌺ŽA.is in E. An easy consequence of the octahedral axiom is that E is saturated iff in the situation of cobase change in Proposition 2.1, if the second vertical and the third horizontal triangle is in E, then the triangle fgh A ª B ª C ª ⌺ŽA.is in E. The following concept is inspired from the definition of an exact categorywx 61 . 276 APOSTOLOS BELIGIANNIS

DEFINITION 2.2. A full subcategory E : DiagŽC, ⌺.is called a proper class of triangles if the following conditions are true:

Ži. E is closed under isomorphisms, finite coproducts, and ⌬ 0 : E : ⌬. Žii. E is closed under suspensions and is saturated. Žiii. E is closed under base and cobase change.

EXAMPLE 2.3.Ž 1. The class of split triangles ⌬ 0 and the class of all triangles ⌬ in C are proper classes of triangles. If E is a proper class of op op triangles in C then E is a proper class of triangles in C . If ÄEi; i g I4 is a family of proper classes of triangles, then Fig IiE is a proper class of triangles. If ÄEi; i g I4is an increasing chain, then the class of triangles Dig IiE is proper. Ž2. Let U: C ª D be an exact functor of triangulated categories and let E be a proper class of triangles in D. Let F [ Uy1 ŽE.be the class of triangles T in C such that UTŽ.g E. Then F is a proper class of triangles in C. Ž3. Let F: C ª U be aŽ co-. homological functor from C to an abelian category U. Then we obtain a proper class of triangles EŽF.in C as follows: ⅷ A triangle A ª B ª C ª ⌺ŽA.is in EŽF.iff ᭙ i g ޚ, the i i i induced sequence 0 ª FAŽ.ª FBŽ.ª FCŽ.ª 0 is exact in U, where i i F s F ⌺ . Ž4. If X : C is a class of objects satisfying ⌺ŽX .s X, then we obtain a proper class of triangles EŽX ., resp. E opŽX ., in C as follows: op ⅷ A triangle A ª B ª C ª ⌺ŽA.is in EŽX ., resp. E ŽX ., iff ᭙ X g X, the induced sequence 0 ª CŽX, A.ª CŽX, B.ª CŽX, C.ª 0, resp. 0 ª CŽC, X .ª CŽB, X .ª CŽA, X .ª 0, is exact in Ab. The family of proper classes of triangles in C is easily seen to be aŽ big. poset with 0Ž the class ⌬ 01212.and 1Ž the class ⌬., defining E U E m E : E .

2.3. Phantom Maps In general it is difficult to distinguish the morphisms occurring in a triangle by a characteristic property. To solve this problem we proceed as follows.

DEFINITION 2.4. Let E be a class of triangles in C Žnot necessarily fgh proper. and let A ª B ª C ª ⌺ŽA.be a triangle in E. Ži. The morphism f: A ª B is called an E-proper monic. PURITY IN TRIANGULATED CATEGORIES 277

Žii. The morphism g: B ª C is called an E-proper epic. Žiii. The morphism h: C ª ⌺ŽA.is called an E-phantom map.

The class of E-phantom maps is denoted by Ph E ŽC..

The concept of phantom maps has homotopy-theoretic origin and the topological terminology is due to A. HellerŽ seewx 52. . For a justification of the definition see Section 3 andwx 52 . Let Ph E ŽA, B.be the set of all n E-phantom maps from A to B and let Ph E ŽA, B.be the set of all maps from A to B which can be written as a composition of nE-phantom maps. n Ž. n Ž. We set Ph E C s DA, B g C Ph E A, B . We recall that an ideal of C is an additive subfunctor of CŽy, y.. An n ideal I of C is called ⌺-stable if f g I m ⌺ Žf .g I, ᭙ n g ޚ. If I is a ⌺-stable ideal of C, then define a class of triangles EI in C as follows. fgh The triangle A ª B ª C ª ⌺ŽA.g EI if h g IŽŽC, ⌺ A...A ⌺-stable ideal I in C is called saturated, if the following condition is true: if f is EI -proper epic and f (␴ g I then ␴ g I. By the octahedral axiom this is equivalent to the condition that if g is an EI -proper monic and ␳ ( g g I then ␳ g I. Trivially if E is a proper class of triangles, then n Ph E ŽC.is a ⌺-stable saturated ideal of C, ᭙ n G 1. Conversely if I is a ⌺-stable saturated ideal, then the class of triangles EI is proper and we have the relations: Ph ŽC. I and E E. We collect these obser- EI s Ph E ŽC . s vations in the following.

PROPOSITION 2.5. The map ⌽: I ¬ EI is a poset isomorphism between the poset of ⌺-stable saturated ideals of C and the poset of proper classes of triangles in C, with in¨erse gi¨en by ⌿: E ¬ Ph E ŽC., e. g., ⌿⌬Ž.s CŽy, y. and ⌿⌬Ž0 .s 0.

It follows that E is a proper class of triangles in C iff Ph E ŽC.is a ⌺-stable saturated ideal of C. This indicates an analogy between the homological theory in C based on proper classes of triangles and the formulation by Butler and Horrockswx 18 of relative homology in an abelian category, based on subfunctors of E xt.

2.4. Baer’s Theory fg Let E be a proper class of triangles in C and let ŽT .: A ª B ª C h ª ⌺ŽA.be a triangle in E. We call h: C ª ⌺ŽA.the characteristic class of ŽT ., and usually we denote it by chŽT .s h. Let A, C be two objects of fgh C and consider the class E*ŽC, A.of all triangles A ª B ª C ª ⌺ŽA. f i g i in E. We define a relation in E*ŽC, A.as follows. If ŽTii.: A ª B ª C h i ª ⌺ŽA., i s 1, 2, are elements of E*ŽC, A., then we define ŽT12.;ŽT . 278 APOSTOLOS BELIGIANNIS if there exists a morphism of triangles

fgh1116 6 6

AB1 C⌺ŽA.

6 6 6 5 6 ␤ 55 6 6 fgh2226 AB2 C⌺ŽA..

Obviously ␤ is an isomorphism and ; is an equivalence relation on the class E*ŽC, A.. Using base and cobase change, it is easy to see that we can defineŽ as in the case of the classical Baer’s theory in an abelian category. a sum in the class EŽC, A.[ E*ŽC, A.r; in such a way that EŽC, A. op becomes an abelian group and EŽy, y.: C =C ª Ab an additive bifunctor. Trivially we have the following.

COROLLARY 2.6. ch is an isomorphism of bifunctors: EŽy, y.ª Ph E Žy, ⌺y..

3. THE FREYD AND STEENROD CATEGORY OF A TRIANGULATED CATEGORY

3.1. The Freyd Category Let C be an additive category. We recall that an additive functor F: op C ª Ab is called finitely presented if there exists an exact sequence CŽy, A.ª CŽy, B.ª F ª 0. We denote by AŽC.or modŽC.the category of finitely presented functors and we call it the Freyd category of C. The Yoneda embedding is denoted by ޙ: C ¨ AŽC.. We define also the op op op op Freyd category BŽC.to be AŽC . , so BŽC.s modŽC . is the dual of the category of finitely presented covariant functors over C. The op corresponding Yoneda embedding is denoted by ޙ : C ¨ BŽC.. The category AŽC., resp. BŽC., has cokernels, resp. kernels, and the functor ޙ, resp. ޙ op, has the universal property that any additive functor F: C ª M where M has cokernels, resp. kernels, admits a unique cokernel, resp. kernel, preserving extension through ޙ, resp. ޙ op. We recall that a morphism f: A ª B is a weak kernel of g: B ª C, if the sequence of ޙŽ f . ޙŽ g . functors ޙŽA.ª ޙŽB.ª ޙŽC.is exact. The notion of a weak cokernel is defined dually. By results of Freydwx 29 , the category AŽC., resp. BŽC., is abelian iff any morphism in C has a weak kernel, resp. weak cokernel. In this case ޙ, resp. ޙ op, embeds C as a class of projective, resp. injective, objects in AŽC., resp. BŽC., containing enough projective, resp. injective, objects. PURITY IN TRIANGULATED CATEGORIES 279

We recall that an abelian category A is called Frobenius if A has enough projectives, enough injectives, and the projectives coincide with the injectives. By results of Freydwx 29 , for any triangulated category C, the category AŽC.is Frobenius abelian and the Yoneda embedding ޙ: C ¨ AŽC.realizes C as a class of projective᎐injective objects in AŽC., containing both enough projectives and injectives. Moreover there exists a unique op equivalence ބ: AŽC.ªf BŽC.such that ބޙ s ޙ . It follows that any object F in AŽC.has a description as a kernel, image, or cokernel of a morphism between representable functors. The Freyd category is the uni¨ersal homological category of C in the sense that ޙ is a homological functor and if F: C ª K is a homological functor to an abelian category K, then there exists a unique exact functor F*: AŽC.ª K such that F*ޙ s F. Note that if idempotents split in C, then ޙ induces an equivalence C f ProjŽŽA C..s InjŽŽA C...

3.2. The Steenrod Category Let C be a triangulated category and let I be a class of morphisms in C. We denote by AŽI .the full subcategory of AŽC.consisting of all functors of the form Im ޙŽh., for h g I. Similarly let BŽI .be the full subcategory of BŽC.consisting of all functors of the form Im ޙ opŽh.. Trivially the suspension ⌺ of C extends to an automorphism of AŽC.and BŽC., denoted also by ⌺. A full subcategory U of AŽC.or BŽC.is called ⌺-stable if U is closed under ⌺.

THEOREM 3.1. The following statements are equi¨alent. Ži. I is a ⌺-stable saturated ideal of C. Žii.AŽI .is a ⌺-stable Serre subcategory of AŽC.. Žiii.BŽI .is a ⌺-stable Serre subcategory of BŽC..

If one of the aboë is true, then there is an equiälence AŽI .ªf BŽI . which induces an equiälence AŽC.rAŽI .ªf BŽC.rBŽI .in the localizations.

Proof. Žii.« Ži. Let AŽC.rAŽI .be the quotient category and let Q: AŽC.ª AŽC.rAŽI .be the exact quotient functor. Then the composition S I s Qޙ is a homological functor and it is easy to see that the kernel ideal ker S I s I. Then obviously I is a ⌺-stable saturated ideal of C. Ži.« Žii. Assume that I is a ⌺-stable saturated ideal of C. Then AŽI .is a ⌺-stable full subcategory of AŽC.. Let ␤: G ª H be an epimorphism in AŽC.with G g AŽI .. Then there exists a morphism ␮: B ª K in I such that Im ޙŽ␮.s G and a morphism ␯ : D ª L such that Im ޙŽ␯ .s H. Let ޙŽ␮.s ␳ (␬ and ޙŽ␯ .s ␴ ( ␭ be the canonical factor- 280 APOSTOLOS BELIGIANNIS izations of ޙŽ␮.and ޙŽ␯ .. Since ␤ is epic and ޙŽD.is projective, there exists ␻ : ޙŽD.ª G such that ␻ ( ␤ s ␴ . Similarly since ␳ is epic, there exists a morphism ޙŽ⑀ .: ޙŽD.ª ޙŽB.such that ޙŽ⑀ .( ␳ s ␻ . Hence ޙŽ⑀ .( ␳ ( ␤ s ␴ . On the other hand, since ޙŽL.is injective, there exists a morphism ޙŽ␦ .: ޙŽK .ª ޙŽL.such that ␬ (ޙŽ␦ .s ␤ ( ␭.Then ޙŽ⑀ .(ޙŽ␮.(ޙŽ␦ .s ޙŽ⑀ .( ␳ ( ␤ ( ␭ s ␴ ( ␭ s ޙŽ␯ .. Hence ⑀ (␮(␦ s ␯. Since ␮ g I, it follows that ␯ g I. Hence H g AŽI .and AŽI .is closed under quotient objects. The proof that AŽI .is closed under subobjects is dual and is left to the reader. It remains to prove that AŽI .is closed ␣ ␤ under extensions. Let 0 ª F123ª F ª F ª 0 be a short exact se- f i g i quence in AŽC.with F13, F g AŽI .. Let Aiiª B ª C iª ⌺ŽA i.be triangles in C such that Fiis Im ޙŽg .and g13, g g I, i s 1, 3. Let ޙŽgi. s ␳ii(␴ be the canonical factorizations. Since ޙŽB3.is projective, there exists ␰ : ޙŽB32.ª F such that ␰ ( ␤ s ␳ 3. Since ޙŽC 1.is injective there exists ␻ : F21ª ޙŽC .such that ␣ ( ␻ s ␴ 1. It is easy to see that F 2is the image of the morphism ޙŽg2 ., where

g 0 g 1 B B C C 21313s ␶ g : [ ª [ ž3 /

and ␶ : B31ª C is the unique morphism such that ޙŽ␶ .s ␰ ( ␻ . Hence it suffices to show that g233or equivalently ␶ is in I. Since ޙŽf .( ␳ s ޙŽf3.(␰ ( ␤ s 0, there exists a morphism ␬: ޙŽA31.ª F such that ޙŽf33.(␰ s ␬ ( ␣. Since ޙŽA .is projective, there exists ␭: ޙŽA 31.ª ޙŽB . such that ␭( ␳1313s ␬. Then ޙŽf .(␰ s ␭( ␳ ( ␣; hence ޙŽf .(␰ ( ␻ s ␭( ␳1111( ␣ ( ␻ s ␭( ␳ (␴ s ␭(ޙŽg .. Since ␭ is of the form ޙŽ␮., it follows that ޙŽf33.(␰ ( ␻ s ޙŽf .(ޙŽ␶ .s ޙŽ␮( g 131.« f (␶ s ␮( g . Since g133is in I, it follows that f (␶ g I. But f is EI -epic. Since I is saturated, we have ␶ g I. We conclude that F2 g AŽI .and AŽI .is closed under extensions. The proof ofŽ i.m Žiii. is similar and is left to the reader. Finally it is easy to see that the equivalence ބ: AŽC.ª BŽC.defined by op ބŽŽCoker ޙ f .. s Ker ޙ Žf .sends AŽI .to BŽI ., so it induces an equivalence AŽC.rAŽI .f BŽC.rBŽI ..

COROLLARY 3.2. The map E ¬ AŽŽPh E C.. gi¨es a bijection between proper classes of triangles in C and ⌺-stable Serre subcategories of AŽC., with the in¨erse the map U ¬ EŽFU ., where FU is the homological functor C ª AŽC.ª AŽC.rU.

DEFINITION 3.3. Let E be a proper class of triangles in C. The Steenrod category, resp. dual Steenrod category, of C with respect to E is PURITY IN TRIANGULATED CATEGORIES 281 defined by

op SE ŽC .[ AŽC .rAŽPh E ŽC .., resp. SE ŽC .[ BŽC .rBŽPh E ŽC ..

where Ph E ŽC.is the ⌺-stable saturated ideal of E-phantom maps. op The canonical functor S: C ª SE ŽC., resp. T: C ª SE ŽC., given by op the composition of the Yoneda embedding ޙ: C ª AŽC., resp. ޙ : C ª BŽC., and the quotient functor AŽC.ª AŽC.rAŽŽPh E C.., resp. BŽC.ª BŽC.rBŽŽPh E C.., is called the projecti¨ization functor, resp. injecti¨ization functor. We refer to the next section for the justification of the above definition. The following result summarizes the above observations and explains the terminology for the phantom maps, since these maps are precisely the maps which are invisible in the Steenrod category.

THEOREM 3.4. Let E be a proper class of triangles in C. Then the

Steenrod category SE ŽC.is abelian and the projectiïzation functor S: C ª SE ŽC.is a homological functor haïng the property that SŽ␾.s 0, ᭙␾ g Ph E ŽC.. Moreoër the pair ŽŽS, SE C.. has the following uniërsal property.

ⅷ If H: C ª M is a homological functor to an abelian category M such that HŽ␾.s 0, ᭙␾ g Ph E ŽC., then there exists a unique exact functor H*: SE ŽC.ª M such that H*S s H. Proof. It suffices to prove only the universal property. Let HЈ: AŽC. ª M be the unique exact functor which extends uniquely H through ޙ. Since H kills E-phantom maps, HЈ kills objects of the Serre subcategory AŽŽPh E C.., so there exists a unique exact functor H*: SE ŽC. ª M such that H*Q s HЈ. Then H*S s H*Qޙ s HЈޙ s H and H* is the unique exact functor with this property. Observe that S ŽC.0 and S ŽC.AŽC.. By the above theorem, ⌬⌬s 0 s f op there exists an equivalence D: SE ŽC.ª SE ŽC.such that DS s T. We leave to the reader to formulate the above result for the dual Steenrod op category SE ŽC.and the injectivization functor T. We regard the Steenrod category as an abelian approximation of C. So it is useful to know when the projectivization functor S: C ª SE ŽC.is non-trivial.

DEFINITION 3.5. The proper class of triangles E generates C if the projectivization functor S: C ª SE ŽC.reflects zero objects, i.e., SŽA.s 0 « A s 0. 282 APOSTOLOS BELIGIANNIS

We recall that the Jacobson radical JacŽC.of an additive category C is the ideal in C defined by JacŽC.ŽA, B. s Ä f: A ª B; ᭙ g: B ª A, the morphism 1A y f ( g is invertible4 . If F is an additive functor, we denote by kerŽF., resp. KerŽF., the ideal of morphisms, resp. the full subcategory of objects, annihilated by F. If I is a ⌺-stable ideal of C, let S I : C ª AŽC.rAŽI .be the canonical functor, i.e., the composition of ޙ: C ¨ AŽC.and the quotient functor AŽC.ª AŽC.rAŽI ..

LEMMA 3.6. kerŽS I .s I. Moreo¨er I : JacŽC.iff KerŽS I .s 0.

Proof. The first assertion is trivial. If I : JacŽC.and S I ŽA.s 0, then obviously 1A g I : JacŽC.; hence A s 0. Conversely if KerŽS I .s 0 and f: A ª B is in I, let g: B ª A be any morphism and let ␣ [ 1A y f ( g. ␣ Then trivially S I Ž␣ .s S I Ž1A. . If A ª A ª C ª ⌺ŽA.is a triangle in C, then S I ŽC.s 0; hence C s 0. It follows that ␣ is invertible and this implies that f g JacŽC..

COROLLARY 3.7. E generates C iff Ph E ŽC.: JacŽC..

4. PROJECTIVE OBJECTS, RESOLUTIONS, AND DERIVED FUNCTORS

We fix throughout a proper class of triangles E in the triangulated category C.

4.1. Projectië Objects and Global Dimension DEFINITION 4.1. An object P g C, resp. I g C, is called E-projectië, resp. E-injectië, if for any triangle A ª B ª C ª ⌺ŽA.in E, the induced sequence 0 ª CŽP, A.ª CŽP, B.ª CŽP, C.ª 0, resp. 0 ª CŽC, I.ª CŽB, I.ª CŽA, I.ª 0, is exact in Ab. We denote by PŽE.ŽŽI E..the full subcategory of E-projective ŽE-injective. objects of C. As an immediate consequence of the above definition we have that the categories PŽE.and IŽE.are full, additive, closed under isomorphisms, direct summands, and ⌺-stable, i.e., ⌺ŽŽP E..s PŽE. and ⌺ŽŽI E..s IŽE.. We say that C has enough E-projectiës if for any object A g C there h exists a triangle K ª P ª A ª ⌺ŽK .in E with P g PŽE.. Observe that in this case h is a weakly uniërsal E-phantom map out of A in the sense that any E-phantom map hЈ: A ª B factors through h in a not necessarily unique way. Dually one defines when C has enough E-injectiës. We study only the case of E-projectives since the study of the E-injectives is dual. However, we use freely the dual results when it is necessary. Note that an PURITY IN TRIANGULATED CATEGORIES 283 equivalent formulation of the setup of a proper class of triangles in C such that C has enough E-projectives is that of a projective class of morphisms in C in the sense of Eilenberg and Moorewx 26 ; see wx 19 . We recall that a full subcategory X : C is called contraäriantly finite wx6 if for any A g C there exists a morphism f: X ª A with X g X, such that any morphism g: XЈ ª A with XЈ g X factors through f. The following result is a direct consequence of the definitions and its proof is left to the reader.

LEMMA 4.2.Ž i.᭙ P g P ŽE .:PhE ŽP , ys. 0 and ᭙ I g IŽE.: Ph E Žy, I.s 0. Žii. If C has enough E-projectiës, then E generates C iff A g C and CŽP, A.s 0, ᭙P g PŽE., implies A s 0. Žiii.C has enough E-projectiës iff PŽE.is contraäriantly finite. In this case, E s EŽŽP E..; i.e., a triangle D ª F ª C ª ⌺ŽD. is in E iff ᭙P g PŽE.the induced sequence 0 ª CŽP, D.ª CŽP, F.ª CŽP, C.ª 0 is exact. Moreoër a morphism f: A ª B is E-phantom iff CŽP, f .s 0, ᭙P g PŽE.. Further ᭙A g C : A g PŽE.iff Ph E ŽA, ys. 0. Živ. Let P be a full additië contraäriantly finite subcategory of C, closed under direct summands such that ⌺ŽP.s P. Then P s PŽŽE P... COROLLARY 4.3. The maps P ¬ EŽP., E ¬ PŽE.are inërse bijec- tions between contraäriantly finite ⌺-stable additië subcategories of C closed under direct summands and proper classes of triangles E in C such that C has enough E-projectiës.

f i g i h i PROPOSITION 4.4Ž Schanuel’s lemma. . If K iiª P ª A ª ⌺ŽK i. are triangles in E with Pi g PŽE., i s 1, 2, then K12[ P ( K 21[ P . Proof. Consider the octahedral diagram induced from the composition g12( h :

gh1116 6 y⌺Žf .6

PA111⌺ŽK .⌺ŽP .

6 6 6 556 h2

g12( h 6 6 6

P12⌺ŽKX.⌺ŽP 1.

6 6 6 g1 5 6

h2226 y⌺Žf .6 y⌺Žg .6

A ⌺ŽK22.⌺ŽP .⌺ŽA.

6 6 6 6 0 y⌺Žg21.( ⌺Žh .

6 22s 6 6 0 ⌺ ŽK11.⌺ ŽK . 0. 284 APOSTOLOS BELIGIANNIS

Since h12, h are E-phantoms and P 1, ⌺ŽP 2.are E-projectives, we have g12( h s 0 and g 21( h s 0. Hence the second horizontal and the third vertical triangles are split. So X ( ⌺ŽK21.[ ⌺ŽP .( ⌺ŽK 12.[ ⌺ŽP .« K21[ P ( K 12[ P . If K ª P ª A ª ⌺ŽK .is in E with P in PŽE., then we call the object K a first E-syzygy of A. An nth E-syzygy of A is defined as usual by induction. By Schanuel’s lemma any two E-syzygies of A are isomorphic modulo E-projectives. We define inductively the E-projecti¨e dimension E-p.d A of an object A g C as follows. If A g PŽE.then E-p.d A s 0. Next if E-p.d A )0 define E-p.d A F n if there exists a triangle K ª P ª A ª ⌺ŽK .in E with P g PŽE.and E-p.d K F n y 1. Finally define E-p.d A s n if E- p.d A F n and E-p.d A g n y 1. Of course we set E-p.d A s ϱ, if E- p.d A /n, ᭙ n G 0. The E-global dimension E-gl.dim C of C is defined by E-gl.dim C s supÄE-p.d A; A g C4. EXAMPLE 4.5. ᭙ C /0:⌬-p.d C s ϱ, so if 0 /C : ⌬-gl.dim C s ϱ. On the other extreme ⌬ 0-gl.dim C s 0. The following is proved using standard arguments.

PROPOSITION 4.6.Ž i. ᭙A, B g C : E-p.d A [ B s maxÄ E-p.d A, E- p.d B4. n Žii. ᭙A g C, ᭙ n g ޚ : E-p.d A s E-p.d Ý ŽA.. Žiii.Let A ª P ª B ª ⌺ŽA. be a triangle in E with P g PŽE.. Then either B g PŽE.or else E-p.d A s E-p.d B y 1.

4.2. Resolutions and Deri¨ed Functors DEFINITION 4.7. An E-exact complex X ⅷ ª A over A g C is a diagram n 1 d nq 1 n 1 d1 0 d 0 иии ª X q ª X ª иии ª X ª X ª A ª 0 in C, such that ᭙ n G 0: n 1 g n nfn nhn n 1 Ži. There are triangles K q ª X ª K ª ⌺Ž K q . in E, where K 0 [ A. nnn100 Žii. The differential d s f ( g y , ᭙ n G 1 and d s f . An E-projecti¨e resolution of A g C is an E-exact complex P ⅷ ª A as n above such that P g PŽE., ᭙ n G 0. The next result gives a useful characterization of an E-projective resolution.

PROPOSITION 4.8. Assume that C has enough E-projecti es and consider 1 0 ¨ ⅷ 1 d 0 d n a complex P : иии ª P ª P ª A ª 0 in C with P g PŽE., ᭙ n G 0. PURITY IN TRIANGULATED CATEGORIES 285

Then Pⅷ is an E-projecti¨e resolution of A iff the induced complex CŽQ, P ⅷ . is exact, ᭙ Q g PŽE.. Proof. Assume that the induced complex CŽQ, P ⅷ . is exact in Ab, 00 1 g 0 0 f 0 h0 1 ᭙ Q g PŽE.. Set f [ d and let ŽT0 .: K ª P ª A ª ⌺Ž K . be a 0 0 triangle in C. Since CŽQ, f .is epic, we have CŽQ, h .s 0, ᭙ Q g PŽE.. 0 By Lemma 4.2 it follows that h is E-phantom; hence ŽT0 .is in E. Since 1 0 1 1 1 1 0 1 d ( f s 0, there exists a morphism f : P ª K such that f ( g s d . 1 2 g 1 1 f 1 1 h1 It follows easily that CŽQ, f .is epic. Let ŽT1.: K ª P ª K ª 2 ⌺Ž K .be a triangle in C. As above we have that ŽT1.is in E. Since 21 210 21 0 d ( d s d ( f ( g s 0, we have CŽŽP E., d ( f .(CŽŽP E., g . s 0. 0 21 Since CŽŽP E., g .is monic, we have CŽŽP E., d ( f . s 0; hence by 21 2 Lemma 4.2, d ( f is E-phantom. Since P g PŽE., by Lemma 4.2, we 2 1 2 2 2 have d ( f s 0. So there exists a morphism f : P ª K such that 2 2 1 ⅷ d s f ( g . Continuing inductively in this way we see that P is an E-projective resolution of A. The converse is trivial.

n ny1 0 COROLLARY 4.9. Let 0 ª Piiª P ª иии ª P iª A iª 0 be E- 0 1 2 projecti¨e resolutions of Ai, i s 1, 2. If A12( A , then P 121[ P [ P [ иии 0 1 2 ( P212[ P [ P иии . From now on we assume that C has enough E-projectives. For any object A g C we fix an E-projective resolution of A

nq 1 1 0 ␧AAA6 ␧ 6 f 6 ⅷ nq 1 n 10 PAAAAAиии ª PPª иии ª PPAª 0 which, by definition, is the ‘‘Yoneda composition’’ of triangles

n n n

g 6 f 6 h 6 nnq1 AAnn A nq1 TAAAAAKPKÝŽK .g E

n 0 with PAAg PŽE., ᭙ n G 0 and where K [ A. Using standard arguments from relative homological algebra, one can prove a version of the comparison theoremwx 68 for E-projective resolutions, the obvious formulation of which is left to the reader. It follows that any two E-projective resolutions of an object are homotopy equivalent. Using Schanuel’s lemma and the above observations we have the following consequence.

COROLLARY 4.10. E-p.d A F n iff there exists an E-projecti¨e resolution n 1 0 of A of the form 0 ª P ª иии ª P ª P ª A ª 0. ␣ ␤␥ PROPOSITION 4.11Ž horseshoe lemma. . LetŽ T .: A ª B ª C ª ⌺ŽA. be a triangle in E. Then there are E-projecti¨e resolutions P ⅷ ŽA., 286 APOSTOLOS BELIGIANNIS

P ⅷ ŽB., and P ⅷ ŽC. of A, B, and C, respecti¨ely, and a commutati¨e diagram

ⅷ ⅷ

p 6 q 6 0 6

P ⅷⅷⅷŽA.P ŽB.P ŽC.⌺ŽŽP ⅷA..

6 6 6 6

␣ 6 ␤ 6 ␥ 6 ABC⌺ŽA.

n p n n q n n 0 n such that P ŽA.ª P ŽB.ª P ŽC.ª ⌺ŽP ŽA.. is a split triangle, ᭙ n G 0. Such a diagram is called an E-projective resolution of the triangle ŽT ..

0 0 0 0 Proof. Let fAA: P ª A and f CC: P ª C be E-proper epics with 00 0 PAC, P g PŽE.. Since ␥ is E-phantom, f C(␥ s 0. Using that ⌺ is an automorphism and a result of VerdierŽ seewx 71, p. 378. , the commutative square on the top left corner below is embedded in a diagram

6 ⌺Žp.6 ⌺Žq. 6 000 y 00 PCA⌺ŽP .⌺ŽP BC.⌺ŽP .

00 0 0 6 6 6 fCA6 y⌺Žf .y⌺Žf B.⌺Žf C.

␥ 6 ⌺Ž␣ .6 y⌺Ž␤ .6 C ⌺ŽA.y ⌺ŽB.⌺ŽC.

00 0 0 6 6 6 hCA6 y⌺Žh .y⌺Žh B.⌺Žh C. 2 2

⌺Ž␾ .6 ⌺ Ž␺ .6 6 1212121y ⌺ Ž␻ . ⌺ŽKCA.⌺ ŽK .⌺ ŽK B.⌺ ŽK C.

0202020 6 6 6 y⌺ŽgCA.6 ⌺ Žg .⌺ Žg B.y⌺ Žg C. 22

6 ⌺ Žp.6 ⌺ Žq.6 02020200 y ⌺ŽPCA.⌺ ŽP .⌺ ŽP B.⌺ ŽP C. 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:

␺ 6 6 ␾ 6 111y ␻ y 1 KKKABC⌺ŽK A.

0000 6 6 6 gggABCA6 ⌺Žg .

pq6 6 6 0000 0 PPPABCA⌺ŽP .

0000 6 6 6 fffABCA6 ⌺Žf .

␣ 6 ␤ 6 ␥ 6 ABC⌺ŽA.

0000 6 6 6 hhhABCA6 ⌺Žh .

⌺Ž␺ .6 6 ⌺Ž␾ .6 1111y ⌺Ž␻ . y ⌺ŽK ABCA.⌺ŽK .⌺ŽK .⌺ŽK .. PURITY IN TRIANGULATED CATEGORIES 287

0 Since the second horizontal triangle is split, we have that PB is in PŽE.. Applying to the above diagram the homological functor CŽP, y., ᭙P g 1 PŽE., a simple diagram chasing argument shows that 0 ª CŽP, K A. ª 1 1 CŽ P, K BC.ª CŽP, K . ª 0 is exact. By Lemma 4.2, the first horizontal triangle is in E. Similarly the second vertical triangle is in E. Inductively the above procedure completes the proof.

op Let F: C ª A and G: C ª A be additive functors, where A is an abelian category. Then we define the E-deri¨ed functors

E n op Ln F : C ª A and RE G: C ª A, n G 0 of F and G with respect to E, as follows. Let P ⅷ ª C be an E-projective E resolution of C g C. Then define Ln FCŽ.to be the nth-homology of the ⅷ n induced complex FPŽ . and RE GCŽ.to be the nth-cohomology of the induced complex GPŽ ⅷ .. In particular ᭙ C g C, we define the E-extension n functors E xtE Žy, C.by

n n op E xtE Žy, C.[ RE C Žy, C.: C ª Ab, ᭙ n G 0.

By the comparison theorem the above E-derived functors are well defined.

COROLLARY 4.12. Let A ª B ª C ª ⌺ŽA. be in E. If F: C ª A and op G: C ª A are additi¨e functors where A is abelian, then we ha¨e exact sequences

E E E E иии ª L10FCŽ.ª L FAŽ.ª L 00FBŽ.ª L FCŽ.ª 0 0 0 0 1 0 ª RE GCŽ.ª RE GBŽ.ª RE GAŽ.ª RE GCŽ.ª иии

In particular for any X g C we ha¨e a long exact sequence

0 0 0 1 0 ª E xtE ŽC, X .ª E xtE ŽB, X .ª E xtE ŽA, X .ª E xtE ŽC, X .ª иии .

Proof. Applying F and G to the E-projective resolution of the triangle as in Proposition 4.11 and takingŽ co-. homology, we get the desired exact sequences. The easy proof of the following is left to the reader.

op COROLLARY 4.13. Let F: C ª A, resp. G: C ª A, be a homological, resp. cohomological, functor where A is abelian. Then the natural morphism E 0 L0 F ª F, resp. G ª RE G, is an isomorphism iff F, resp. G, kills E-phantom maps. 288 APOSTOLOS BELIGIANNIS

4.3. Projecti¨ely Generating Proper Classes DEFINITION 4.14. A proper class of triangles E in C projecti¨ely generates C, if E generates C and C has enough E-projectives. A full subcategory G : C is called a generating subcategory of C, if G is ⌺-stable and ᭙A g C: CŽG, A.s 0 « A s 0. By Lemma 4.2, we have the following.

COROLLARY 4.15. The map E ¬ PŽE.is a bijection between projectiëly generating proper classes of triangles in C and contraäriantly finite generating subcategories of C closed under direct summands. The inërse is giën by P ¬ EŽP.. Remark 4.16. Obviously C has enough ⌬-projectives and PŽ⌬.s 0. Conversely if C has enough E-projectives and PŽE.s 0, then E s EŽŽP E..s EŽ0. s ⌬. If C /0, then ⌬ is not projectively generating. Similarly C has enough ⌬ 00-projectives and PŽ⌬ .s C. Conversely if C has enough E-projectives and PŽE.s C, then E s EŽŽP E..s EŽC. s ⌬ 00. ⌬ is always projectively generating. The next result, proved independently by Christensenwx 19 , shows that if E projectively generates C, then the E-projective dimension is controlled by the vanishing of the E-extension functors.

PROPOSITION 4.17. If E projecti¨ely generates C, then ᭙A g C, ᭙ n G 0:

nq 1 E-p.d A F n m E xtE ŽA, ys.0.

n Proof. It suffices to prove the Ž¥ .direction. Consider the triangles TA ⅷ and the resolution PA after Corollary 4.9. Since the cohomology of the n nq 1 nq 2 nq 1 short complex CŽ PAA, y.ª CŽP , y.ª CŽP A, y. is E xtE Ž A, y. nq 1 nq 1 nq 1 nq 1 s 0, the morphism fAA: P ª K Afactors through ␧ A. Hence nq 1 nq 1 n nq 1 n nq 1 ␧AAAA( ␣ s f ( g ( ␣ s f , for some ␣: P AAª K . Applying the nq 1 functor CŽP, y., ᭙P g PŽE., to this relation and using that CŽP, fA . is n n epic, we have that CŽ P, g .CŽP, ␣ .1nq 1 , so CŽŽP, ␣, f ..: A ( s C Ž P, K A . A n nq 1 n nnq1 n CŽ P, PAAA.( CŽP, K [ K .. Since E generates C, P AAA( K [ K . It n follows that K A g PŽE.or equivalently E-p.d A F n.

4.4. The Cartan Morphism Assume that C has split idempotents and is skeletally small; i.e., the collection of isoclasses of objects of C, denoted henceforth by IsoŽC., is a set. The Grothendieck group K0ŽC, E.of C with respect to E is defined as the quotient of the free abelian group on IsoŽC., modulo the subgroup generated by all elements ŽA.y ŽB.q ŽC., where A ª B ª C ª ⌺ŽA. PURITY IN TRIANGULATED CATEGORIES 289 is a triangle in E. If E s ⌬, then K0ŽC, ⌬.is the usual group defined by Grothendieck. If E s ⌬ 000, then K ŽC, ⌬ .is the group K 0ŽC, [.of the monoidal category ŽC, [.. Observe that we have canonical epimorphisms K00ŽC, ⌬ .ª K 0ŽC, E.and K 0ŽC, E.ª K 0ŽC, ⌬.. In case C has enough E-projectives, then we define the Cartan morphism cE : K0ŽŽP E.., [ ª K0ŽC, E.by cPE Žwx. s wxP .

PROPOSITION 4.18. If ᭙ C g C : E-p.d C -ϱ, then cE is an isomorphism.

Proof. We construct an inverse of cE as follows. Let 0 ª Pn ª иии ª 0 P ª A ª 0 be an E-projective resolution of A g C. Using Schanuel’s and horseshoe lemma it follows directly that the assignment wxA ª n Ž.i i Ž.ŽŽ.. Ýis0 y1 w P x is a well defined morphism dE : K00C, E ª K P E , [ . Clearly dE is the inverse of cE .

4.5. The Steenrod Categories

Let F: C ª K be a homological functor, where K is abelian. Let EŽF. be the proper class of triangles in C induced by F. Following Streetwx 68 , an object P g C is called F-projecti¨e if FPŽ.is projective in K and ᭙A g C, the canonical map CŽP, A.ª K w FPŽ., FAŽ.x is an isomorphism. Let PŽF.be the full subcategory of F-projectives. C has enough F-projecti¨es if ᭙A g C there exists a triangle K ª P ª A ª ⌺ŽK .g EŽF.with P g PŽF.. It is trivial to see that if C has enough F-projectives, then C has enough EŽF.-projectives and PŽŽE F..s PŽF.. The next result shows that if C has enough E-projectives for a proper class of triangles E, then E s EŽS.and PŽŽE S..s PŽS., where S is the projectivization functor. It follows that in this case, our formulation of relative homology in C is equivalent to the theory developed by Street inwx 68 .

PROPOSITION 4.19. ᭙P g PŽE., SŽP.g Proj SE ŽC.and ᭙A g C, the canonical map SP, A: CŽP, A.ª SE ŽC.wSŽP., SŽA.x is an isomorphism. Hence S induces a full embedding PŽE.¨ Proj SE ŽC.. If C has enough E-projecti¨es, then:

Ži.SE ŽC.is equiälent to AŽŽP E.. and S is isomorphic to the restriction A ¬ CŽy, A.< PŽ E .. In particular SE ŽC.has enough projectiës and E s EŽS.. f Žii.If idempotents split in PŽE., then S: PŽE.ª Proj SE ŽC.. Žiii. A complex P ⅷ ª Aoër A is an E-projectië resolution of A iff ⅷ SŽ P .ª SŽA. is a projectië resolution of SŽA. in SE ŽC..

Proof. Since the kernel ideal of the functor S is Ph E ŽC., it follows that SP, A is injective, since Ph E ŽP, A.s 0, ᭙P g PŽE.. Now let ␣: SŽP.ª SŽA.be a morphism in SE ŽC.. Since S s Qޙ, where Q: AŽC.ª 290 APOSTOLOS BELIGIANNIS

AŽC.rAŽŽPh E C.. is the quotient functor, ␣ is represented by a fraction sf ޙŽP.¤ F ª ޙŽA., where s has kernel and cokernel in AŽŽPh E C... Let ␰ : ޙŽP.ª G be the cokernel of s. Then G is the image of a morphism ޙŽh.: ޙŽC.ª ޙŽD., where h is an E-phantom map; let ޙŽh.s ⑀ (␫ be the canonical factorization. Then there exists a morphism ޙŽ␾.: ޙŽP.ª ޙŽC.such that ޙŽ␾.(⑀ s ␰. Then ␰ (␫ s ޙŽ␾.(⑀ (␫ s ޙŽ␾ ( h.. Since h is E-phantom and P is E-projective, ␾ ( h s 0. Hence ␰ s 0 and s is split epic. Let sЈ: ޙŽP.ª F be a morphism such that sЈ( s s 1ޙŽ P . . Then sЈ( f is of the form ޙŽ␣ Ј.for ␣ Ј: P ª A. Trivially SŽ␣ Ј.s QŽsЈ( f .s ␣. Hence SP, A is surjective. The proof that SŽP.is projective is similar and is left to the reader.

Ži. One can prove this by using the universal property of SE ŽC.. Here is a quick proof. Since C has enough E-projectives, the subcategory PŽE.is contravariantly finite in C. Bywx 13 , there exists a short exact R sequence of abelian categories 0 ª AŽŽCrP E..ª AŽC. ª AŽŽP E.. ª 0, where CrPŽE.is the stable category of C modulo the full subcategory PŽE.and R is the restriction functor F ¬ F < PŽ E .. It is easy to see that AŽŽCrP E..is equivalent to AŽŽPh E C..so by Definition 3.3, SE ŽC. is equivalent to AŽŽP E... In particular SE ŽC. has enough projectives. Part Žii. follows fromwx 28 and partŽ iii. follows fromŽ i. . The above result explains the terminology for the projectivization functor. The next example explains the terminology for the Steenrod category.

EXAMPLE 4.20. Let C be the stable homotopy category of spectrawx 51 n and consider the full subcategory X s Ä⌺ ŽŽŽK ޚr p...; n g ޚ4, where KŽŽޚr p..is the Eilenberg᎐MacLane spectrum corresponding to ޚrŽp.. Then the Steenrod category SE op ŽX .ŽC.of C with respect to the proper class of triangles E opŽX .is equivalent to the category of modules over the mod-p Steenrod algebra; seewx 74 for details. The description of the Steenrod category in Proposition 4.19 allows us to give a formula for the derived functors of an additive functor F: C ª M, where M is abelian. Let G [ F < PŽ E .: PŽE.ª M. Since SE ŽC.s AŽŽP E.., G has a unique right exact extension G*: SE ŽC. ª M through the Steenrod category. The easy proof of the following and its contravariant analogue is left to the reader.

E COROLLARY 4.21. LnnF ( ŽL G*.S, ᭙ n G 0.

COROLLARY 4.22.Ž i. For any B g C we ha¨e natural isomorphisms

E xt nn, B E xt S , S B , ᭙ n 0. E Žy .( S E ŽC . Žy.Ž. G PURITY IN TRIANGULATED CATEGORIES 291

␮ Ž. 0 Ž. 0 Ž. The natural map y, B: C y, B ª RE C y, B s E xtE y, B coincides Ž.Ž.Ž.Ž. ␮ with the map Sy, B: C y, B ª SE C wS y , S B x. Hence Ker y, B is the right ideal Ph E Žy, B.. Žii.᭙A g C : E-p.d A G p.d SŽA.; if E projecti¨ely generates C, then E-p.d A s p.d SŽA.. In particular E-gl.dim C F gl.dim SE ŽC.. Žiii. Assume that C is skeletally small with split idempotents. If C is

E-projecti¨ely generated and gl.dim SE ŽC.-ϱ, then the projecti¨ization ( functor S: C ª SE ŽC.induces an isomorphism S#: K0ŽC, E.ª K 0 ŽŽS E C .., and an isomorphism K# ŽŽA C ..( K# ŽŽS E C ..[ K#ŽŽŽA CrP E... in Quillen’s higher K-theory. Proof. The assertionsŽ i. andŽ ii. follow trivially from Proposition 4.19. The first part ofŽ iii. follows from Proposition 4.18 and the commutativity of the diagram

S ( 6

K00ŽŽP E.., [ K ŽŽŽProj SE C.., [ .

6 6 c cE S EŽC .

S# 6 K00ŽC, E.K ŽŽSE C...

The second part ofŽ iii. follows from a result of Auslander and Reitenwx 5 .

The dual of Proposition 4.19 is also true. We state only the following.

PROPOSITION 4.23. If C has enough E-injectiës then the dual Steenrod op op category of C with respect to E is equiälent to BŽŽI E..s modŽŽI E. . op and the injectiïzation functor T: C ª SE ŽC.is isomorphic to the restriction A ¬ CŽA, y.< I Ž E .. If I ŽE.has split idempotents, then T induces an op equiälence: I ŽE.f Inj SE ŽC.. If X : C is a class of objects, then addŽX .denotes the full subcategory of C consisting of all direct summands of finite coproducts of objects of X. An E-injectië enëlope of A g C is an E-proper monic ␮: A ª E with E g I ŽE.such that if ␮( ␣ s ␮, then ␣ is an automorphism of E. Without assuming the existence of enough E-injectives we have the following.

THEOREM 4.24. The projecti¨ization functor S induces a full embedding

S: I ŽE .¨ Inj SE ŽC .and Inj SE ŽC .: addŽ Im S..

f In particular S: I ŽE.ª Inj SE ŽC.if Im S is closed under direct summands. If this is the case, then C has enough E-injecti¨es iff SE ŽC.has 292 APOSTOLOS BELIGIANNIS

enough injectiës. In particular if SE ŽC.has injectië enëlopes, then C has E-injectië enëlopes.

1 0 1 ⑀A 0 f A Proof. Let PAAª P ª A ª 0 be the start of an E-projective resolution of A g C. If I is E-injective then obviously the map SA, I : CŽA, I. ª SE ŽC.wSŽA., SŽI.x is injective. Let ␮: SŽA.ª SŽI.be any morphism. 0 0 0 Consider the morphism SŽ fAA.(␮: SŽP .ª SŽI.. Since P Ais E-projective, 0 0 1 SŽ fAAA.(␮ s SŽ␣ ., for some morphism ␣: P ª I. Then SŽ⑀ .(SŽ␣ .s 1 0 11 1 SŽ⑀AA.(SŽf .(␮ s 0, so ⑀ A( ␣: P Aª I is E-phantom. Since P Ais E-pro- 1 1 1 0 0 jective, ⑀ ( ␣ s 0. Since ⑀AAAs f ( g , we have that the morphism g A( ␣ 11 2 01 factors through hAA: K ª ⌺Ž K A.. So g A( ␣ s h A( ␤, for some morphism 2 101 ␤: ⌺Ž K AA. ª I. But since h is E-phantom, it follows that g AA( ␣: K ª I 0 is E-phantom. Since I is E-injective, g A ( ␣ s 0. Hence there exists ␥ : 0 0 0 A ª I such that fAA(␥ s ␤. Then SŽ f .(SŽ␥ .s SŽ␤ .. Since SŽf A. is epic, ␮ s SŽ␥ .; hence SA, I is an isomorphism. It remains to show that SŽI.g SŽ ␤ . Inj SE ŽC.. Let F g SE ŽC.and let SŽP10.ª SŽP .ª F ª 0 be the ␣ ␤ start of a projective resolution of F. Let ŽT .: A ª P10ª P ª ⌺ŽA.be f a triangle in C and let K ª P2 ª A ª ⌺ŽK .be an E-projective presen- SŽ ␣ . SŽ ␤ . tation of A in C. Since the sequence SŽA.ª SŽP10.ª SŽP .ª F ª 0 is exact, if G s KerŽŽS ␤ .., then we have an epimorphism SŽ f ( ␣ . SŽf .(coim SŽ␤ .: SŽP2210.ª G. Then SŽP .ª SŽP .ª SŽP .ª F ª 0 is part of a projective resolution of F. Applying SE ŽC.wy, SŽI.x to this sequence and using that the map SC, I is an isomorphism ᭙ C g C, it follows easily that the resulting sequence is exact. This means that E xt1 F, SŽI. 0. Hence SŽI.is injective. S E ŽC .w x s From the triangle ŽT .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 ( SŽI.for some ␶ Žobviously E-injective.I. If C has enough E-injectives, let ⌺ŽA.ª I ª 2 B ª ⌺ ŽA.be a triangle in E with I g I ŽE.. Then ␰ (SŽ␶ .: F ª SŽI.is monic and SŽI.is injective; hence SE ŽC.has enough injectives. Con- versely if SE ŽC.has enough injectives, ᭙ C g C, let ␨ : SŽC.¨ E be monic with E g Inj SE ŽC.. Then E s SŽI.for some I g I ŽE.and ␨ s ␯ SŽ␯ .for a morphism ␯ : C ª I in C. Since the triangle C ª I ª B ª ⌺ŽC.is in E, it follows that C has enough E-injectives. Clearly if SŽ␮.: SŽC.ª SŽI.is an injective envelope, then ␮: C ª I is an E-injective envelope.

4.6. Semisimple and Hereditary Categories The following characterization of E-phantomless categories follows easily from our previous results. PURITY IN TRIANGULATED CATEGORIES 293

THEOREM 4.25. The following are equi¨alent: Ži. E-gl.dim C s 0. Žii.C s PŽE..

Žiii. E s ⌬ 0.

Živ. Ph E ŽC.s 0.

Žv.The projectiïzation functor S: C ª SE ŽC.is faithful. f Žvi.The projectiïzation functor S induces an equiälence SE ŽC.ª AŽC.. op LEMMA 4.26. Let G: C ª M be a homological and F: C ª M a cohomological functor, where M is abelian. If E-p.d A F 1, there are exact sequences E E 0 ª L01GAŽ.ª GAŽ.ª L GŽ⌺ŽA..ª 0, 1 0 0 ª RE F Ž⌺ŽA..ª FAŽ.ª RE FAŽ.ª 0. Proof. Since E-p.d A 1, an E-projective resolution of A is simply a 0 0 F 0 1 g A 0 f A h A 1 10 triangle PAAP A ⌺Ž P A. in E, with P AA, P PŽE.. From ª ª ª 0 g 1 GŽ g A. 0 1 the exact sequence иии GPŽ AA. GPŽ .GAŽ.G⌺ŽP A. 0 ª ª ª ª G ⌺Ž g A. 0 0 ª G⌺Ž PA .ª иии , by construction we have CokerŽŽGgA.. s E 0 E L0 GAŽ.and KerŽŽG⌺ g A.. s L1 G⌺ŽA.and the result follows. The claim for F is similar.

The next result computes the ideal Ph E ŽC.in case E-gl.dim C F 1 and can be regarded as a universal coefficient theorem.

THEOREM 4.27. Let A g C with E-p.d A F 1. Then we ha¨e the following. Ži. For any B g C there exists a short exact sequence

␮ 6 10A, B 0 ª E xtE Ž⌺ŽA., B.ª C ŽA, B.E xtE Ž A, B.ª 0. 1 1 Hence Ph E ŽA, B.s E xtE ŽŽ⌺ A.., B and A is E-projecti¨e iff E xtE Ž A, y. s 0. 2 Žii. Ph E ŽA, ys.0.

If E-gl.dim C F 1, then the projecti¨ization functor S: C ª SE ŽC.is full and reflects isomorphisms, E projecti¨ely generates C, and the following relations hold: 2 Ph E ŽC .s 0, Ph E ŽC .: JacŽC .,

y1 1 Ph E Ž⌺ y , y(.E xtE Žy, y.. Proof. PartŽ i. follows from Lemma 4.26 setting F s CŽy, B.. To ␣ ␤ proveŽ ii. let A ª B ª C be E-phantom maps and consider the E-pro- 294 APOSTOLOS BELIGIANNIS jective resolution of A as in the proof of Lemma 4.26. Since ␣ is 0 1 E-phantom, the composition fAA( ␣ s 0. So there exists a map y: ⌺Ž P . ª 011 B with hAAA( y s ␣. But with P , also ⌺Ž P . is E-projective. Since ␤ 0 is E-phantom, the composition y( ␤ s 0. Then ␣ ( ␤ s hA ( y( ␤ s 0. 2 0 Hence Ph E ŽA, ys.0. If E-gl.dim C F 1, then since E xtE Ž A, B.( SE ŽC.wSŽA., SŽB.x, ᭙A, B g C, the functor S is full. Since ker S s Ph E ŽC. is a square zero ideal, it is contained in the Jacobson radical of C. This implies that S reflects isomorphisms. The last claim follows fromŽ i. and Žii. . Remark 4.28. If C is skeletally small and A is abelian, let AddŽC, A. be the category of additive functors from C to A and let HomŽC, A.be the full subcategory of homological functors. If E-gl.dim C F 1, it is easy E E to see that the kernel-ideal kerŽ L00. of the functor L : HomŽC, A.ª E 2 AddŽC, A., satisfies kerŽL0 . s 0. Hence if ␾: F ª G is a map of E homological functors, then ␾ is invertible iff L0 Ž␾.is also. op COROLLARY 4.29. Assume that E-gl.dim C F 1 and let F: C ª Ab be a cohomological functor. Then the following are equi¨alent. Ži. F is representable. ( Žii. There exists ⍀ g C and a natural isomorphism F < PŽ E . ª CŽy, ⍀.< PŽ E .. ( Proof. Žii.« Ži. Let ␻ : CŽy, ⍀.< PŽ E . ª F < PŽ E . be a natural isomorphism. Using ␻ , it follows easily that the short exact sequence of Lemma 4.26 is isomorphic to

␤ ␣ 10 0 ª E xtE wx⌺y , ⍀ ª F ª E xtE wxy, ⍀ ª 0. ␮ Ž⍀. 0 ⍀ ␣ Let y, ⍀ : C y, ª E xtE wxy, be the canonical morphism. Since is an epimorphism and CŽy, ⍀.is a projective functor, there exists a ␾ Ž⍀. ␾ ␣ ␮ ␾ morphism : C y, ª F such that ( s y, ⍀ . Obviously P is an isomorphism, ᭙P g PŽE., since ␮P, ⍀ and ␣P are. Using the natural morphism ␾ and the projective resolution of A, using that F is cohomological and finally using 5-Lemma, we see directly that ␾A: CŽA, ⍀.ª FAŽ.is an isomorphism. Hence ␾: CŽy, ⍀.ª( F.

5. THE E-PHANTOM TOWER, THE E-CELLULAR TOWER, HOMOTOPY COLIMITS, AND COMPACT OBJECTS

Throughout this section we assume that E is a proper class of triangles in the triangulated category C and C has enough E-projectives. Our PURITY IN TRIANGULATED CATEGORIES 295 purpose in this section is to construct for any object A of C the phantom tower and the cellular tower of A, generalizing topological constructions fromwx 51 . These towers are crucial in the study of the homological behavior of A with respect to E.

5.1. The Phantom Tower We fix an object A g C and let

nq 1 1 0 ␧AAA6 ␧ 6 f 6 ⅷ nq 1 n 10 PAA: иии ª PP Aª иии ª PPA AAª 0 be the E-projective resolution of A as in Section 4. We recall that the

ⅷ resolution PA is obtained by splicing triangles

n n n n nq 1 g A n f A n h A nq 1 TAA: K ª P Aª K Aª ⌺ŽK A.g E

n 0 nnny1 nny1 where PAAAAAAAg PŽE., ᭙ n G 0, K [ A and ␧ s f ( g : P ª P , ᭙ n G 1. ny1 ny1 n Consider the morphisms hAA: K ª ⌺Ž K A., ᭙ n G 1. By con- ny1 struction all these maps are E-phantoms; i.e., we have hA g ny1 n ny1 ny1 ny1 ny1 n n Ph E Ž K AA, ⌺Ž K ... Hence the maps ⌺ Žh A.: ⌺ Ž K A. ª ⌺ Ž K A. are also E-phantom maps. In this way we obtain a tower of objects and E-phantom maps

0 1

h 6 ⌺Žh . 6 AA122 A ⌺ŽK AA.⌺ ŽK . ny1 ny1

⌺ Žh . 6 ny1 ny1 A nn ª иии ª ⌺ ŽK AA. ⌺ ŽK .ª иии which we call the E-phantom tower of A, with respect to the proper class E.

PROPOSITION 5.1. For any morphism ␣: A ª B, the following are equi¨- alent: n Ži. ␣ g Ph E ŽA, B.. n n Žii. There exists a morphism ␴ : ⌺ Ž K A. ª B such that

nq 1 0122 ny1 ny1 Žy1.hAA(⌺Žh .(⌺ Žh A.( иии (⌺ Žh A.(␴ s ␣ .

000 Proof. Ži.« Žii. If ␣ g Ph E ŽA, B., then fAAA( ␣ s 0, since f : P ª A 1 0 and ␣ is E-phantom. Hence there exists ␴ : ⌺Ž K AA. ª B with h (␴ s ␣. 2 If ␣ g Ph E ŽA, B., then ␣ s ␣12( ␣ where ␣ 1: A ª X and ␣ 2: X ª B 1 are E-phantom maps. By the above argument, there exists ␴1: ⌺Ž K A. ª X 0 1 with hA (␴11s ␣ . Since the composition ␴ 12( ␣ : ⌺Ž K A. ª B is E- 1 1 phantom and ⌺Ž PAA.g PŽE., we have y⌺Žf .(␴12( ␣ s 0. Then as 296 APOSTOLOS BELIGIANNIS

2 2 1 above there exists ␴ : ⌺ Ž K AA.ª B with y⌺Žh .(␴ s ␴12( ␣ . Then 0 1 0 hAA(ŽŽy⌺ h ..(␴ s h A(␴12( ␣ s ␣ 12( ␣ s ␣. A simple induction argument completes the proof. The converse is trivial.

5.2. The Cellular Tower In this subsection we use the phantom tower to construct the cellular tower of an object. We fix an object A in the triangulated category C and consider the E-projective resolution of A as above. For simplicity we set n n 0 1 2 2 n n nq 1 nq 1 ␻ AAAAA[ Ž1.h ⌺Žh . ⌺ Žh . иии ⌺ Žh .: A ⌺ Ž K A., A0 [ 0, y ( ( ( ( 0 ª 0 0 01g A 0 f A h A 1 and A1 [ PAAAA. Consider the triangle K ª P ª A ª ⌺Ž K . and 1 form the cobase change of this triangle along the phantom map yhA: 1 2 K AAª ⌺Ž K .:

0 0 0

g 6 f 6 h 6 10AAA 1 KPAAA⌺ŽK A.

11 6 6 6 6 ␣ 5 Ž. yh AA1 y⌺ h Ž. 1 q

␤ 6 ␥␻6 6 22222A ⌺ŽKAAA.2 ⌺ ŽK A..

1 2 ␤2 ␥ 2 ␻ A Next consider the second horizontal triangle ⌺Ž K A. ª A2 ª A ª 2 2 ⌺ Ž K A. in the above diagram, which by construction is in E, and form the 2 2 cobase change of this triangle along the phantom map y⌺ŽhAA.: ⌺ŽK . ª 2 3 ⌺ Ž K A.:

␤ 6 ␥␻6 6 22222A ⌺ŽKAAA.2 ⌺ ŽK A.

222 6 6 6 6 ␣ y⌺Žh AA.2 5 y⌺ Žh . 2

␤ 6 ␥␻6 6 2333A 33 ⌺ ŽKAAA.3 ⌺ ŽK A..

Inductively the above construction produces a tower 6 6 ␣ ␣␣6 0 12 иии n иии 0 s A01ª A s PAAA 23ª ª AAnnq1 ª which we call the E-cellular tower of A, and commutative diagrams, ᭙ n G 1:

␥n 6

AAn

6 ␣n 6 5

␥ 6 nq 1 AAnq 1 .

The following result is a direct consequence of the above constructions. PURITY IN TRIANGULATED CATEGORIES 297

COROLLARY 5.2. For an object A g C, the following are equi¨alent: Ži. E-p.d A F n. Ž. ␥ ii The morphisms nq in: A qi ª A of the E-cellular tower of A constructed abo¨e are isomorphisms, ᭙ i G 1. Consequently E-gl.dim C F n iff the cellular tower of any object of C Ž. ␣ stabilizes after the n q 1 th-step; i.e. the morphisms mm: A ª A mq1 of the cellular tower are isomorphisms, ᭙ m G n q 1. If this is the case then ᭙A g C : A ( Am, ᭙ m G n q 1. Using the E-cellular tower we can prove the main result of this section.

THEOREM 5.3. Ž␣ . The following statements are equi¨alent:

Ži.The projecti¨ization functor S: C ª SE ŽC. is full and reflects isomorphisms. Žii. E-gl.dim C F 1.

Žiii. E generates C and Im S : ÄF g SE ŽC.: p.d F F 1.4

Ž␤ .Assume that idempotents split in PŽE.. ThenÄ F g SE ŽC.: p.d F F 14 : Im S. If E-gl.dim C F 1, then S induces an equiälence S: I ŽE.ª Inj SE ŽC.; further C has enough E-injectiës iff SE ŽC.has enough injec- tiës. Ž␥ .If C has enough E-injectiës, then the statements in Ž␣ .are also equiälent to the followingŽŽ the inclusion below is an equality if I E.has split idempotents.: Živ. ᭙A g C : E-i.d A F 1.

Žv. E generates C and Im S : ÄF g SE ŽC.: i.d F F 1.4 Proof. Ž␣ .Žii.« Ži. follows from Theorem 4.27.Ž i.« Žii. Suppose thatŽ i. is true. Applying to the cobase change diagram Žq. the projectivization functor S, we obtain the following commutative diagram of short exact sequences in SE ŽC.

0 0

6 SŽ g .6 SŽf . 6 6 10AA 0 SŽK AA.SŽP .SŽA. 0

1 6 6 ySŽh A. SŽ␣1. 5 6

6 SŽ␤ .6 SŽ␥ .6 6 2 22 0 SŽŽ⌺ K A..SŽA2 .SŽA. 0.

1 1 Since hAAis an E-phantom map, SŽh . s 0. Consider the cokernel ␻ : SŽA21.ª F of SŽ␣ .. By the above diagram, there exists a unique isomor- 2 phism ␨ : SŽŽ⌺ K A..ª F such that SŽ␤2 .( ␻ s ␨. Consider the morphism y1 2 2 ␻ (␨ : SŽA2 .ª SŽŽ⌺ K A... Since S is full, there exists ␾: A2 ª ⌺Ž K A. y10 such that SŽ␾.s ␻ (␨ . Next consider the morphism ␣1(␾: PA ª 298 APOSTOLOS BELIGIANNIS

2 y1 ⌺Ž K A.. Then SŽ␣11(␾.s SŽ␣ .( ␻ (␨ s 0. Hence the morphism ␣ 1(␾ 0 is E-phantom. Then ␣1(␾ s 0, since PA g PŽE.. From the diagram Žq., 1 0 ␣ 1 ␦1 1 0 we have that ⌺Ž g A. s ␤21(␦ , where PA ª A2 ª ⌺Ž PAA.ª ⌺ŽP . is a triangle in C. It follows that ␾ factors through ␦1; i.e., there exists ␳: 1 2 ⌺Ž PAA.ª ⌺ŽK ., such that ␦1212( ␳ s ␾. Then ␤ (␦ ( ␳ s ␤ (␾ « 1 y1 ⌺Ž g .␳ ␤ ␾. But SŽ␤ .␻ ␨ SŽ␤ .␻ ␨ 1 2 A ( s 22( ( s « 2( ( s SŽ⌺Ž K A.. « SŽ␤ .SŽ␾.1 22SŽ␤ ␾.1 . Since S reflects isomor- 2 ( s SŽ⌺Ž K AA.. « 2 ( s SŽ⌺Ž K .. 1 1 phisms, it follows that ␤2 (␾ s ⌺Ž g AA.( ␳ is invertible. Then ⌺Žg . is split 1 1 1 monic and consequently ⌺Ž fAA.: ⌺ŽP .ª ⌺ŽK A. is split epic. This implies 1 1 that K AAis E-projective as a direct summand of P . Hence E-p.d A F 1. PartŽ ii.m Žiii. follows fromŽ i. and Corollary 4.22, since if E generates, then E-p.d A s p.d SŽA., ᭙A g C.

Ž␤ .Let F g SE ŽC.with p.d F F 1. Since PŽE.has split idempotents, by Proposition 4.19, S induces an equivalence PŽE.f Proj SE ŽC.. SŽ ␣ . Hence F admits a projective resolution of the form 0 ª SŽP1.ª f ␣ SŽP01010.ª F ª 0 with P , P g PŽE.. Then the triangle P ª P ª A ª ⌺ŽP1.is in E and obviously F ( SŽA.. Hence Ä F g SE ŽC.: p.d F F 14 : Im S. If E-gl.dim C F 1, then by Ž␣ .we have that ÄF g SE ŽC.: p.d F F 14 s Im S. Hence Im S is closed under direct summands and the remaining assertions are consequences of Theorem 4.24. Part Ž␥ .is similar and is left to the reader.

5.3. The Category of Extensions and the Phantom Filtration We recall now a well-known construction fromwx 11 . If X is a class of objects of C, then the category X ) X of extensions of X is defined as follows:

X ) XsaddÄAg C : ᭚ triangle X12112ªAªX ª⌺ŽX .with X , X g X 4.

U n Inductively we define X [ X ) X ) X ) иии ) X Žn-factors. . It is shown in U n wx11 that the operation ) is associative; hence the above definition of X Uϱ U n U1 makes sense. Finally define X [ DnG1 X , where we set X s X. The U n category X is called the category of nth-extensions of X, for n s 1, 2, . . . , ϱ. Remark 5.4. Let A be an object of C, and consider the E-cellular tower of A:

␣ 6 ␣␣6 6 ␣6 012 иии n иии AAAA0123ª ª AAnnq1 ª .

n Then clearly An g PŽE.*,᭙ n G 1. The next corollary generalizes results of Kellywx 46 and Street wx 68 . PURITY IN TRIANGULATED CATEGORIES 299

COROLLARY 5.5. Ž␣ . If A g C, then ᭙ n G 0; consider the following statements:

Ži. E-p.d A F n. n 1 Žii.A g PŽE.*.Ž q . nq 1 Žiii. Ph E ŽA, ys.0.

Then Ži.« Žii.« Žiii. . In particular if E-p.d A -ϱ, then Ph E ŽA, y:. JacŽ A, y.and the ideal Ph E ŽA, A. of the ring End C ŽA. is nilpotent. Hence if an object X has a semiprime endomorphism ring, then either

Ph E ŽX, X .s 0 or else E-p.d X s ϱ. Ž␤ .If any object of C has finite E-projecti¨e dimension, then C is ϱ E-projecti¨ely generated and C s PŽE.*. Žnq1. nq1 Ž␥ .If E-gl.dim C F n, or if C s PŽE.*,then Ph E ŽC.s 0.

Proof. Ž␣ .The implicationŽ i.« Žii. follows from Corollary 5.2. Let fg h A be in PŽE.) PŽE., so there exists a triangle P101ª A ª P ª ⌺ŽP . with P01, P g PŽE.. If ␣: A ª B, ␤: B ª C are E-phantoms, then ␣ s g ( ␳ and ␳ ( ␤ s 0, for some ␳: P0 ª B. Then ␣ ( ␤ s g ( ␳ ( ␤ s 0. 2 Hence Ph E ŽA, ys.0. Now the implicationŽ ii.« Žiii. follows by induction. Parts Ž␤ .and Ž␥ .are direct consequences of Ž␣ ..

Using the proper class E, we define a new class of triangles E 2 as follows:

ⅷ fgh 22 A triangle A ª B ª C ª ⌺ŽA.is in E if h g Ph E ŽŽC, ⌺ A...

PROPOSITION 5.6. The class of triangle E 2 in C is proper. If C has 2 2 enough E-projectiës then C has enough E -projectiës and PŽE . s 22 PŽE.) PŽE.s PŽE.*.Moreoër if E generates C then also E generates C.

Proof. From Proposition 2.5, it follows that E 2 is a proper class of triangles. From the construction of the E-cellular tower we know that 1 2 ␻ A 2 2 ᭙A g C, there exists a triangle ⌺Ž K A. ª A2 ª A ª ⌺ Ž K A. such that 12 2 ␻ A g Ph E ŽC.and A2 g PŽE.) PŽE.. Hence the above triangle is in E 2 and it suffices to show that any object in PŽE.) PŽE.is E -projective. So fgh 2 let A ª B ª C ª ⌺ŽA.be a triangle in E and let ␣: X ª C be a morphism with X g PŽE.) PŽE.. Then the composition ␣ ( h is in 2 Ph E ŽŽX, ⌺ A... By Corollary 5.5, we have ␣ ( h s 0, so ␣ factors through 2 g. Hence X is E -projective and since PŽE.) PŽE.is closed under direct 22 2 summands, E has enough E -projectives and PŽE .s PŽE.) PŽE.. 2 Since E generates C iff Ph E ŽC.: JacŽC.and since Ph E 2 ŽC.s Ph E ŽC., the last claim follows trivially. 300 APOSTOLOS BELIGIANNIS

1 2 From now on we set E [ E. We note that Ph E 21ŽC.s Ph E ŽC.. The procedure described in the above proposition produces a filtration of proper classes of triangles

n ny1 2 1 ⌬ 0 : E : E : иии : E : E s E : ⌬, a filtration of phantom ideals

0 : Ph E n ŽC .: Ph E ny 1 ŽC .: иии : Ph E 2 ŽC .: Ph E 1 ŽC .

s Ph E ŽC .: C Žy, y.,

n where Ph E nŽC.s Ph E ŽC., and a filtration of subcategories

1 2 n 1 n 0 : PŽE .s PŽE .: PŽE .: иии : PŽE y .: PŽE .: C ,

Ž n.Ž.n ϱ n where P E s P E * . Finally setting E s FnG1 E we obtain a ϱŽ.nŽ. proper class of triangles such that Ph E C s FnG 1 Ph E C s n Ž.Žϱ.Ž.n Ž.ϱ FnG1 Ph E C and P E s DnG1 P E * s P E * . If C has enough n E-projectives then for 1 F n F ϱ, C has enough E -projectives. Moreover if E generates C then the same is true for E n. The above data induce a filtration of Steenrod categories

0 ¨ SE ϱ ŽC .¨ иии ¨ SE n ŽC .¨ SE ny 1 ŽC .¨ иии ¨ SE 2 ŽC .

¨ SE 1 ŽC .¨ AŽC . and for 1 F n F m F ϱ each inclusion SE mnŽC.¨ SE ŽC.has an exact right adjoint Sm, n: SE nmŽC.ª SE ŽC., with the kernel the Serre subcate- n m n m gory AŽPh E ŽC..rAŽPh E ŽC..f AŽPh E ŽC.rPh E ŽC... The following is a consequence of our previous results. COROLLARY 5.7. For any n with 1 F n F ϱ, the following are true. nn nn Ži. C is E -phantomless m Ph E ŽC.s 0 m C s PŽE.* m E - gl.dim C s 0. nn1 Žii. E -p.d A F 1 « ᭙B g C : Ph E ŽA, B.s E xtE nŽŽ⌺ A.., B , 2 n Ph E ŽA, B.s 0. n nиŽmq1. nиŽ mq1. Žiii. E -gl.dim C F m « C s PŽE.* and Ph E ŽC.s 0. We define the E-phantom dimension E-Ph.dim C of C by infÄn G nq 1 0 : Ph E ŽC.s 04 and the E-extension dimension E-Ext.dim C of C by n 1 infÄn G 0: PŽE.*Ž q . s C4. Trivially: E-Ph.dim C F E-Ext.dim C F E- gl.dim C. Using Corollary 5.7Ž i. , the octahedral axiom, and the E-cellular tower, it is not difficult to see that E-Ph.dim C s E-Ext.dim C. We thank PURITY IN TRIANGULATED CATEGORIES 301

J. D. Christensen for this nice observation. It is unclear under what conditions the last inequality is an equality.

5.4. Homotopy Colimits and Compact Objects In this subsection we study the behavior of the E-cellular tower under Žco-. homological functors. We assume throughout that C has countable coproducts. f 0 f1 f 2 Let A012ª A ª A ª иии be a tower of objects in C. The homotopy colimit A holim ª n of the tower is defined by the triangle

fg h Annª A ª holim A nª ⌺ A n ŽT. [[nG0 nG0 ª ž [nG0 /

Ž1,yfn.6 where f is induced by the morphisms AnnnA [ A q1 ¨ [nG 0 An. The homotopy colimit is uniquely determined up to a non-unique isomorphism. We include for completeness the proof of the following well-known result.

EMMA Ä4 L 5.8. Let Annn; fG 0 be a tower of objects in C. Ž1. If F: C ª Ab is a homological functor which preser¨es coproducts, then the canonical map FAŽ.FŽA. is in ertible lim ª n ª holim ª n ¨ . op Ž2. If F: C ª Ab is a cohomological functor con¨erting coproducts to products then the canonical map FŽA .FŽ A. induces a , holim ª n ª lim ¤ n Milnor short exact sequence Ž1. F ⌺ŽA .FŽA . :0ª lim ¤ n ª holim ª n ª lim ¤ FAŽn.ª 0. Proof. Ž1. The triangleŽ T. induces a short exact sequence 0 ª FAŽ.FAŽ.FŽA . b [nG 0 nnª [ G 0 n ª holim ª n ª 0 in A and the result follows.Ž 2. Applying F to the triangleŽ T. , we obtain the long exact иии FŽA .FAŽ.FAŽ. sequence ª holim ª nnª Ł G 0 nnª Ł G 0 n ª FŽ ⌺y1 ŽA .. иии b holim ª n ª in A , and the result follows from the con- Ži. i b struction of lim¤ , s 0, 1, of a tower in A ; seewx 71 . The following concepts are fundamental.

DEFINITION 5.9wx 56 . If C is a triangulated category, then an object A g C is called compact if the functor CŽA, y.: C ª Ab preserves all small coproducts. The compact part of C is by definition the full subcategory C b of all compact objects. The compact part C b of C is a full triangulated subcategory of C. The category C is compactly generated if C b b has all small coproducts, C is skeletally small, and CŽX, A.s 0, ᭙ X g C b implies that A s 0; i.e., the proper class EŽC . generates C. 302 APOSTOLOS BELIGIANNIS

By the construction of the E-cellular and the E-phantom tower of an n n object A, we have a diagram of towers ÄAnA4ª ÄA4ª Ä⌺ Ž K .4 ª n n Ä⌺ŽAnnAn.4, such that A ª A ª ⌺ Ž K .ª ⌺ŽA .is a triangle in C. It is ŽA. A A not known if the induced diagram E : holim ª n ª ª holim ª ⌺nŽ K n .⌺ŽA . A ª holim ª i is a triangle in C. This happens if C is a ‘‘homotopy’’ category of a suitable stable closed model categorywx 37 , for instance, a derived category or a stable module category. The next result follows from the construction of the cellular and phantom tower.

LEMMA 5.10.Ž 1. Let T be a compact E-projecti¨e object and let ⌺nŽ Kn . be a homotopy colimit of the phantom tower of A Then holim ª A . ŽT ⌺nŽ K n .. C , holim ª A s 0. t Ž2. If Ph E ŽA, ys.0, for some t G 0 or if C is generated by a class of compact projecti e objects then ⌺nŽ K n . X E- ¨ , holim ª A s 0. b PROPOSITION 5.11. Let T g C be a compact object. Ž. Let A and let A be a homotopy colimit of the 1 g C holim ª n cellular towerÄ A n 4of A Then we ha e an isomorphism E- n; G 0 . ¨ lim ª ŽT A .ŽT A . C , n ( C , holim ª n . Ž.If in addition T Ž.then ŽT A.ŽT A . 2 g P E , C , ( lim ª C , n ( ŽT A . C , holim ª n . Proof. PartŽ 1. follows from Lemma 5.8. PartŽ 2. follows fromŽ 1. , applying CŽT, y.to the cobase change diagrams, defining the E-cellular tower of A.

We have the following nice consequence.

COROLLARY 5.12. If C is generated by a class X of compact E-projecti¨e objects, then any object A g C is a homotopy colimit of its E-cellular tower: A A ( holim ª n. Proof. Since X generates C, the functors CŽT, y., T g X collectively reflect isomorphisms and the claim follows from Proposition 5.11.

5.5. The Phantom Topology ϱ Our purpose here is to describe the infinite E-phantom ideal Ph E ŽC.by means of the E-cellular tower. Fix two objects A, B g C.

DEFINITION 5.13. The E-phantom topology of CŽA, B.is the topology n with open sets ␾ q Ph E ŽA, B., n G 0. The topological group CŽA, B.is complete Žin the E-phantom topology. if it is Hausdorff and any Cauchy sequence has a limit. PURITY IN TRIANGULATED CATEGORIES 303

n n 0 Consider the E-phantom tower of A and let ␻ AA[ Žy1.h ( 1 2 2 n n nq 1 nq 1 ⌺ŽhAA.(⌺ Žh .( иии (⌺ Žh A.: A ª ⌺ Ž K A. as in Section 5.1. Then by n nq 1 Proposition 5.1, we have clearly Im CŽ␻ A, B. s Ph E ŽA, B.. Now applying the functor CŽy, B.to the E-cellular tower of A, we obtain a short exact sequence of inverse systems in Ab:

n n 0 ª ÄC ŽA, B.rPh E ŽA, B.4ª ÄC ŽAnA, B.4ª ÄIm C Ž␤ , B.4ª 0. Hence we have the exact sequence

n n 0 ª lim C ŽA, B.rPh E ŽA, B.ª lim C ŽAnA, B.ª lim Im C Ž␤ , B. ¤¤¤ Ž11.n Ž. ª lim C ŽA, B.rPh E ŽA, B.ª lim C ŽAn , B. ¤¤ Ž1. m ª lim Im C Ž␤A , B.ª 0.Ž 1. ¤ Assume now that C has countable coproducts and is generated by a class ⌺Ž.Ž. b A of objects X such that X s X : P E l C . Then since holim ª n ( A Ž. Ž1. ŽŽ⌺ A ..B , we have the following exact sequence 2 : 0 ª lim ¤ C n , ª ŽA B.ŽA B. C , ª lim ¤ C n, ª 0. From the short exact sequence of the inverse systems

ny1 n n n 0 ª ÄKer C Ž␻ AA, B.4ª ÄC Ž⌺ ŽK ., B.4ª ÄPh E ŽA, B.4ª 0

ny1 n using thatÄ Ker CŽ␻ AA, B.4s ÄIm CŽŽ⌺ ␤ .., B 4, we obtain the exact sequence

n n n 0 ª lim Im C Ž⌺Ž␤nA., B.ª lim C Ž⌺ ŽK ., B.ª lim Ph E ŽA, B. ¤¤¤ Ž11.nnnŽ. ª lim Im C Ž⌺Ž␤ ., B.ª lim C Ž⌺ ŽK A ., B. ¤¤ Ž1. n ª lim Ph E ŽA, B.ª 0.Ž 3. ¤ But ᭙B g C, the E-phantom tower of A induces a short exact sequence

Ž1. nq1 nnn 0 ª lim C Ž⌺ ŽK AA., B.ª C holim ⌺ ŽK ., B ¤ªž/ n n ª lim C Ž⌺ ŽK A ., B.ª 0. ¤ ⌺nŽ K n . Ž1. Ž⌺nq 1Ž K n..B Since holim ª A s 0, we have lim ¤ C A , s lim ¤ Ž⌺nŽ K n ..B ᭙B Ž.Ž␤ B. C A , s 0. Since this holds g C, by 3 , lim¤ Im C n, s Ž1. n ŽA B. n ŽA B. Ž1. ŽŽ⌺ ␤ ..B lim¤ Ph E , s 0, lim¤ Ph E , ( lim¤ Im C n , . Since Ž1. ŽA B.n ŽA B.Ž. lim ¤ C , rPh E , s 0, using 1 and collecting all the above information we deduce the following. 304 APOSTOLOS BELIGIANNIS

n ROPOSITION Ž.ŽA B.ŽA B. ŽA B. P 5.14. 1 lim ¤ C n, ( lim ¤ C , rPh E , . Ž. Ž1. n ŽA B. 2 lim¤ Ph E , s 0. Ž. ϱ ŽA B. n ŽA B. n ŽA B. Ž1. 3 Ph E , s FnG1 Ph E n, s lim¤ Ph E , ( lim ¤ CŽŽ⌺ An.., B . Ž.ŽA B. is complete in the phantom topology iff Ž1. 4 C , E- lim ¤ CŽŽ⌺ An.., B s 0. Ž1. G ÄG 4 It is well known that lim ¤ nns 0 for an inverse system of Artinian groups. Hence we have the following consequence. ϱ COROLLARY 5.15. If E-p.d A -ϱ or A g PŽE.* or CŽŽ⌺ An.., B is Artinian, ᭙ n G 0, then CŽA, B. is complete in the E-phantom topology, i.e., ϱ Ph E ŽA, B.s 0. Note that the above computations are useful for the construction and the study of convergence of the Adams spectral sequencewx 19, 68 for the proper class E.

6. LOCALIZATION AND THE RELATIVE DERIVED CATEGORY

We consider in this section a triangulated category C equipped with a proper class of triangles E and we assume always that C has enough E-projectives.

We have seen that the class of E-phantom maps Ph E ŽC.s Ä f : ᭙P g PŽE.; CŽP, f .s 04 is an ideal in C, and the condition Ph E ŽC.s 0 has nice consequences for the triangulated category C. So it is useful to find a way to ‘‘kill’’ the E-phantom maps. Define a class of morphisms by R [ Ä f ¬ SŽf .is invertible4 , i.e., ᭙A, B g C:

( ᭙ RA , B s ½f g C ŽA, B.¬ P g PŽE .; C ŽP, f .: C ŽP, A.ª C ŽP, B.5.

The easy proof of the following is left to the reader.

LEMMA 6.1. The class of morphisms R is a multiplicati¨e system of morphisms in C compatible with its triangulation. We call the elements of R E-quasi-isomorphisms. By the above lemma, the localized category Cw Ry1 x can be calculated by left and right fractions. We recall that Cw Ry1 x is a triangulated category and there exists an 1 exact functor of triangulated categories R: C ª Cw Ry x, which sends the elements of R to isomorphisms in Cw Ry1 x and has the following universal property. If F: C ª T is an exact functor to a triangulated category T PURITY IN TRIANGULATED CATEGORIES 305 which sends R to isomorphisms, then there exists a unique up to isomor- 1 phism exact functor G: Cw Ry x ª T with GR s F. From now on we y1 denote by DE ŽC.[ Cw R x and by R: C ª DE ŽC.the canonical functor. We call the triangulated category DE ŽC.the E-deri¨ed category of C with respect to E. In general DE ŽC.lives in a higher universe than C. If AE ŽC.s Ker R s Ä A g C : RŽA.s 04 is the full subcategory of E-acyclic objects, then we have a short exact sequence of triangulated categories:

j R 0 ª AE ŽC .ª C ª DE ŽC .ª 0.

LEMMA 6.2. The kernel-ideal ker R is contained in the ideal of E-phantom maps.

Proof. If RŽf .s 0, then there exists a morphism s g R with f ( s s 0. By the definition of R, we have CŽŽP E.., f s 0. Then by Lemma 4.2, f g Ph E ŽC.. LEMMA 6.3. ᭙P g PŽE.and ᭙B g C, the morphism

R P , B : C ŽP, B.ª DE ŽC .RŽP ., RŽB. is an isomorphism.

Proof. By Lemma 6.2, R P, B is injective. Let a˜: RŽP.ª RŽB.be a morphism in DE ŽC.. It is well known that a˜ can be represented as a sf diagram P ¤ X ª B with s g R. Then CŽŽP E.., s : C ŽŽP E.., X ª( CŽŽP E.., P . Let g g CŽP, X .be the unique morphism such that g ( s s 1P . By the definition of the morphisms in the category of fractions DE ŽC., we have RŽg ( f .s a˜, so R P, B is surjective. By Lemma 6.3 and induction we have the following direct consequence. n LEMMA 6.4. ᭙P g PŽE.*, n s 0, 1, 2, . . . , ϱ and ᭙B g C, the morphism

R P , B : C ŽP, B.ª DE ŽC .RŽP ., RŽB. ϱ is an isomorphism. In particular the induced functor R: PŽE.* ¨ DE ŽC.is fully faithful.

6.1. Relati¨e Deri¨ed Categories We seek conditions such that the functor R induces an equivalence between DE ŽC.and a full subcategory of C. We fix an object A g C, and consider the E-cellular tower of A: 6 6 ␣ ␣␣6 01 иии n иии ŽTAAA. 012ª ª AAnnq1 ª , n An g PŽE .* , ᭙ n G 0. 306 APOSTOLOS BELIGIANNIS

From now on we assume that the triangulated category C hasŽ countable. coproducts A . Consider a homotopy colimit holim ª n. ␨ ␪ ␩ Annª A ª holim A nª ⌺ A n Ž!. [[nG0 nG0 ª ž [nG0 / and the following triangle, where the morphism ␺ is naturally induced from ŽT .:

␺ ␾␹ E ŽA.: holim Annª A ª Aˆª ⌺ holim A .Ž !!. ªªž/ $ Let PŽE .*ϱ be the full additive subcategory generated by all objects which are isomorphic to a homotopy colimit of a tower ŽT .as above with n An g PŽE.*,n G 0: $ ϱ A A A A n ᭙ n PŽE .* [ add½g C ¬ ( holim nn, where g PŽE .*, G 0.5 ª Now we can prove the main result of this section. First we recall that a localizing subcategory of C is a full triangulated subcategory of C closed under coproducts. If U : C, then the localizing subcategory Loc²U : generated by U is the smallest localizing subcategory of C which contains U. Finally we set H U [ Ä A g C ¬ CŽA, U .s 0, ᭙U g U4 and AddŽU .to be the full subcategory of C consisting of all direct summands of arbitrary coproducts of objects of U.

THEOREM 6.5. Assume that there exists a set of compact E-projecti¨e objects X in C such that ⌺ŽX .s X and a morphism f g R iff the morphism CŽX, f. is an isomorphism. Then we ha¨e the following:

$ϱ H Ži.PŽE .* s AE ŽC.s Loc²ŽP E.: . $ϱ Žii. The functor R induces a triangle equi¨alence: R: PŽE .* ªf DE ŽC..

Žiii.The category DE ŽC.is compactly generated and any compactly generated triangulated category arises in this way.

Proof. Ži. Let P g PŽE.and L g AE ŽC.. If ␣: P ª L is a morphism, then RŽ␣ .s 0, since L is E-acyclic. Lemma 6.3 implies that ␣ s 0. H Hence PŽE.: AE ŽC.. By induction and using Lemma 6.4 it follows that Ž.n H Ž. A P E * : AE C . Now let holim ª n be a homotopy colimit of a tower n of objects An g PŽE.* . By Lemma 5.8, we have a short exact sequence Ž1. ŽŽ⌺ A ..L ŽA L.ŽA L. 0 ª lim ¤ C n , ª C holim ª n, ª lim ¤ C n, ª 0. L ŽA L. Since is an E-acyclic object, we have C holim ª n, s 0. Hence $ϱ HH PŽE .* : AE ŽC.. If A is in AE ŽC.and ÄAn; n G 04 is the E-cellular tower of A, then in the triangleŽ !!. above we have Aˆg AE ŽC.. Indeed PURITY IN TRIANGULATED CATEGORIES 307 applying CŽX, y.to the triangleŽ !!. and using Proposition 5.11, it follows Ž␺ .ŽA .ŽA. that C X, : C X, holim ª n ª C X, is an isomorphism. Hence by hypothesis, ␺ g R and this implies that RŽAˆˆ.s 0, i.e., A g AE ŽC.. Since A H A ŽC., we have ␾ 0. It follows that A is a direct summand of g E s $ holim A ; hence by definition A PŽE .*ϱ . We conclude that $ª n g PŽE .*ϱ H A ŽC., so it is a localizing subcategory of C. By construction $ s E PŽE .*ϱ is the localizing subcategory of C generated by PŽE.. Žii. It is well known and easy to see that R induces a full embedding H AE ŽC.¨ DE ŽC.s CrAE ŽC.. Hence R induces a full embedding $ϱ PŽE .* ¨ DE ŽC.. If A is an object in DE ŽC., then by the above argument the morphism ␺ : holim A A is in R. Since by construction $$ª n ª A ϱϱR Ž. holim ª n g PŽE .* , it follows that the functor : PŽE .* ª DE C is surjective on objects, so it is a triangle equivalence. $ϱ Žiii. If A g PŽE .* , then from the triangle EŽA.we deduce triv- ␺ A A ŽA. ially that the morphism : holim ª n ª is an isomorphism. If C X, ŽA .Ž␨ . s 0, then C X, holim ª n s 0. This implies that the morphism C X, in the triangleŽ !. is an isomorphism. Hence by hypothesis, ␨ is in R. It follows that RŽholim A .0, so A holim A A ŽC.. Hence A ª n s ( ª n g E $ s 0. Since X consists of compact objects and it is contained PŽE .*ϱ , it $ follows that PŽE .*ϱ is compactly generated. It remains to prove that any compactly generated category C arises in b b this way. Consider the proper class of triangles E s EŽC . in C. Since C is skeletally small it follows easilyŽ see Section 8. that C has enough b b E-projectives and PŽE.s AddŽC .. Since C generates C, the projectivization functor S reflects isomorphisms; hence R consists of isomorphisms. Choosing X IsoŽC b. to be the set of isoclasses of compact s $ y1 b ϱ objects above, we have C s CwxR s DE ŽC.s AddŽC .*. In some cases the compactness condition in Theorem 6.5 can be avoided:

Remark 6.6. Let H: C ª M be a coproduct preserving homological functor to an AB4 category M. Assume that H kills E-phantom maps and satisfies the property: HfŽ.is an isomorphism « f g R. Then partsŽ i. andŽ ii. in Theorem 6.5 are true. Indeed it suffices to show that HŽ␺ .: HŽA .HAŽ.HŽ holim ª n ª is an isomorphism. Observe first that holim ª A .HAŽ.HŽ⌺nŽ K n ..HŽ⌺nŽ K n .. n ( lim ª n and holim ª A ( lim ª A s 0. Hence applying H to the diagrams defining the E-cellular tower of A, we HŽ␺ .HŽA . ( HAŽ. have : holim ª n ª . Note that under the above assumptions, if f g R, then HfŽ.is an isomorphism. Indeed since H kills E-phantom maps, H s H*S where H*: SE ŽC.ª M is exact. Hence if f g R, i.e., SŽf .is invertible, then so is HfŽ.. It follows that E and H define the same phantom maps. 308 APOSTOLOS BELIGIANNIS

$ We call the objects of PŽE .*,ϱ E-cofibrant. Hence any object of C has an E-cofibrant resolution, i.e., admits an E-quasi-isomorphism to an E- cofibrant object. The following are direct consequences of Theorem 6.5 or Remark 6.6.

COROLLARY 6.7. Keep the hypothesis of Theorem 6.5 or Remark 6.6. $ϱ ! Then the inclusion i: PŽE .* ¨ C has a right adjoint i and the inclusion j: AE ŽC.¨ C has a left adjoint j*. The quotient functor R: C ª DE ŽC.has a fully faithful left adjoint. For any object A g C, there exists a functorial triangle

! ! i ŽA.ª A ª j*ŽA.ª ⌺Ži ŽA.., where the inölëd morphisms are the adjunction morphisms. Any triangle $ϱ P ª A ª L ª ⌺ŽP. with P g PŽE .* and L g AE ŽC.is isomorphic to the triangle aboë by a unique isomorphism of triangles which extends the identity of A.

COROLLARY 6.8. A full triangulated subcategory of DE ŽC.equals DE ŽC. if it contains X and is closed under coproducts.

COROLLARY 6.9. If C is compactly generated then any object A of C is the homotopy colimit holim A of a towerÄ A ; A ␣ n A 4, where $ ª nnnnª q1 bn An g AddŽC .*,᭙ n G 0.

6.2. Total Deriëd Functors Let C and B be triangulated categories with proper classes of triangles E and F, respectively. We assume that the pairs ŽC, E., ŽB, F .satisfy the conditions of Theorem 6.5 or Remark 6.6. Let F: C ª A be an additive functor to an additive category A. The total left E-deriëd functor of F is an additive functor L F: DE ŽC.ª A together with a morphism ␨ :LFR ª F such that for any additive functor G: DE ŽC.ª A and morphism ␣: GR ª F, there exists a unique natural morphism ␰ : G ª L F such that ␰ R(␨ s ␣. Similarly if F: C ª B is an additive functor, then the total left ŽE, F .-deriëd functor of F is an additive functor L F: DE ŽC.ª DF ŽB.which is the total left E-derived functor for the composite R B F: C ª DF ŽB.. THEOREM 6.10.Ž 1. Any additië functor F: C ª A to an additië category A has a total E-deriëd functor L F: DE ŽC.ª A. If F is homological then so is L F. Ž2.Any additië functor F: C ª B has a total left ŽE, F .-deriëd functor L F: DE ŽC.ª DF ŽB.. If F is exact then so is L F. PURITY IN TRIANGULATED CATEGORIES 309

Proof. In caseŽ 2. just set L F s R B Fi: DE ŽC.ª DF ŽB.. CaseŽ 1. is similar. Remark 6.11. Let U be a Serre subcategory of Ab. Define a class of Ž.Ž.Ž. morphisms R U in C as follows: R U s DA, B g C RA, B U , where a morphism f: A ª B belongs to RA, BŽU .if ᭙P g PŽE., the morphism CŽP, f .is an isomorphism in the Gabriel-quotient AbrU, i.e., ker CŽP, f ., coker CŽP, f .g U. As above the class RŽU .is a multiplicative system of U morphisms compatible with the triangulation of C. The class Ph E ŽC.of U-, E-phantom maps defined as the class of morphisms f: A ª B such that CŽP, f .s 0 in AbrU, i.e., Im CŽP, f .g U, ᭙P g PŽE., is then an ideal of C. One can develop the theory of the present section to this situation without important changes. The above theory then corresponds to the case U s 0. Note that this applies to the stable homotopy category of spectra, choosing as U suitable Serre subcategories of Ab associated with various sets of primes.

7. THE STABLE TRIANGULATED CATEGORY

Let C be a triangulated category and let E be a proper class of triangles in C. We assme that C has enough E-projectives. We denote by CrPŽE.the stable category of C modulo the full subcategory PŽE.of E-projective objects. If ␼ : C ª CrPŽE.is the canonical functor, then we set ␼ ŽA.s A and ␼ Žf .s f. Our aim is to show that CrPŽE.carries in a natural way a left triangulated structure, which in some cases it is useful to study. For the notion of left triangulated categories we refer towx 12 , or to wx 43 in which the dual notion is treated. For each object A C choose an E-projective presentation of A, i.e., a 0 g 0 0 1 g A 0 f A h A 1 0 triangle K AAª P ª A ª ⌺Ž K A. in E with P Ag PŽE.. By Schanuel’s lemma, we see that if we choose another E-projective presenta- 10 tion K ª P ª A ª ⌺ŽK .of A, then K AA[ P ( K [ P .Hencetheob- 1 1 ject K A is uniquely determined in CrPŽE.. We denote it by ⍀ E ŽA.s K A. 0 Let f: A ª B be a morphism in C. Then there are morphisms pf : 0 0 1 1 1 PABfABª P , k : K ª K and a morphism of triangles 0 0 0

g 6 f 6 h 6 10AAA 1 KPAAA⌺ŽK A.

10 1 6 6 6 kpff6 f ⌺Žk f.

000

gfh6 6 6 10BBB 1 KPBBB⌺ŽK B.. 0 0 0 1 If qfA: P ª P Bis another lifting of f which induces a morphism l f : 11 0 1 1 K ABª K in the above diagram, then h A(w⌺Žk f.y ⌺Žl f.x s 0, so there 310 APOSTOLOS BELIGIANNIS

0 1 1 1 exists a morphism ␣: ⌺Ž PAB.ª ⌺ŽK .such that ⌺Žk ff.y ⌺Žl . s 0 1 1 ⌺Ž g Aff.( ␣. Then the morphism ⌺Žk .y ⌺Žl . or equivalently the mor- 11 11 1 phism k ffy l is zero in CrPŽE., i.e., k ffs l . Hence setting ⍀ E Žf .s k f , we obtain an additive functor ⍀ E : CrPŽE.ª CrPŽE., which we call the ␣ ␤␥ E-loop functor of C with respect to E. If T: A ª B ª C ª ⌺ŽA.is a triangle in E, then T induces a morphism of triangles

0 0 0 6 6 g f h 6

10CCC 1

KPCCC⌺ŽK C.

6 6 6 c 6 d 5 ⌺Žc.

␣ 6 ␤␥6 6 ABC⌺ŽA..

1 It is easy to see as above that the morphism c: KC ª A is independent in CrPŽE.of the chosen E-projective presentation of C. We call the image of c in CrPŽE.the characteristic class of the triangle T and we denote it by c [ chŽT .. Hence the triangle T in E induces a diagram

␣ ␤ chŽT .6 T : ⍀ E ŽCA. ª B ª C in CrPŽE ..

We define a triangulation ⌬ E of the pair ŽŽCrP E.., ⍀ E as follows:

ⅷ A diagram ⍀ E ŽCЈ.ª AЈ ª BЈ ª CЈ in CrPŽE.belongs to ⌬ E iff it is isomorphic to a diagram of the form T, where T is a triangle in E.

THEOREM 7.1. The triple ŽŽCrP E.., ⍀ E , ⌬ E is a left triangulated category. Proof. The proof is similar to the proof of the main theorem ofwx 12 . We prove only how any morphism f: B ª C is embedded in a triangle in CrPŽE.. Consider the base-change diagram of the E-projective presentation of C along the morphism f:

6 ␤ 6 ␥ 6

11␣

KABCC⌺ŽK .

6 6 6 5 6 ␦ f 5

000

gfh6 6 6 10CCC 1 KPCCC⌺ŽK C..

Then consider the cobase-change diagram of the E-projective presentation 1 of C along the morphism ␣: KC ª A

0 0 0

g 6 f 6 h 6

10CCC 1

KPCCC⌺ŽK C.

6 6 6 ␣ 6 ␧ 5 ⌺Ž␣ .

␨␩6 ␪ 6 6 ADC⌺ŽA.. PURITY IN TRIANGULATED CATEGORIES 311

We know that the lower triangle in the above diagram is in E. Hence by the definition of ⌬ E we have a triangle ⍀ E ŽC.ª A ª D ª C in CrPŽE.. It is not difficult to see that the above two diagrams produce an isomor- 0 phism D ( B [ PC . Hence D ( B, such that the morphism D ª C is f isomorphic to f. Hence the diagram ⍀ E ŽC.ª A ª B ª C belongs to the triangulation ⌬ E of CrPŽE.. The stable category CrPŽE.is not necessarily triangulated. The next result describes when this happens. Its proof is left to the readerŽ compare wx12. . THEOREM 7.2. The stable category CrPŽE.is triangulated iff C has enough E-injecti¨es and PŽE.s I ŽE.. If CrPŽE.is triangulated, then E-gl.dim C s 0 or ϱ. Moreo¨er E-gl.dim C s 0 iff E s ⌬ 0.

8. PROJECTIVITY, INJECTIVITY, AND FLATNESS

Throughout this section we fix a triangulated category C with coproducts and a full skeletally small additive subcategory 0 /X : C, closed under isomorphisms, ⌺, ⌺y1, and direct summands. Our aim is to study the homological theory of C based on the proper class of triangles EŽX .:

ⅷ A triangle A ª B ª C ª ⌺ŽA.is in EŽX .iff ᭙ X g X, the induced sequence

0 ª C ŽX, A.ª C ŽX, B.ª C ŽX, C.ª 0 is exact in Ab.

LEMMA 8.1. C has enough EŽX .-projecti¨es and PŽŽE X ..s AddŽX ..

Proof. For any C g C, let ICC[ Ä X ª C ¬ X g IsoŽX .4. If X [ X , then X PŽŽE X ..since X PŽŽE X ..and PŽŽE X .. is [␭g IC ␭ C g : closed under coproducts. The set ICCinduces a morphism f: X ª C, which by construction is an EŽX .-proper epic. Then the triangle A ª XC f ª C ª ⌺ŽA.g EŽX .and C has enough EŽX .-projectives. Further AddŽX .: PŽŽE X .., since PŽŽE X .. is closed under coproducts and direct ␣ summands and contains X. If P g PŽŽE X .., let A ª XP ª P ª ⌺ŽA. be a triangle in EŽX .with XP a coproduct of objects of X. Since P is EŽX .-projective, ␣ splits and P is a direct summand of XP , so P g AddŽX .. Since C has enough EŽX .-projectives, we can apply the results of the previous sections to the pair ŽC, X .. An important example of the above situation occurs in the stable homotopy category of spectrawx 51 , choosing X to be the category of finite spectra. For other examples we refer to Section 12. 312 APOSTOLOS BELIGIANNIS

LEMMA 8.2.Ž 1.f: A ª B is an EŽX .-phantom map iff ᭙ X g X : CŽX, f .s 0.

Ž2.The ideal Ph E ŽX .ŽC.: JacŽC.iff ᭙ X g X : CŽX, A.s 0 « A s 0.

8.1. Projecti¨e and Flat Functors Since X is skeletally small, we can consider the category ModŽX .of right X-modules, i.e., the category of contravariant additive functors from X to the category Ab of abelian groups. Define a functor

SЈ: C ª ModŽX .by SЈŽA.s C Žy, A.< X and

SЈŽf .s C Žy, f .< X , where CŽy, A.< X denotes the restriction of the representable functor to X.

There is a well-defined tensor product functor ymX y: ModŽX .= op ModŽX . ª Ab, which satisfies all the usual propertieswx 53 . A functor F op in ModŽX .is called flat, if F myX : ModŽX . ª Ab is exact. The full subcategory of ModŽX .consisting of flat functors is denoted by FlatŽŽ Mod X ... Also we denote by ProjŽŽ Mod X .., resp. InjŽŽ Mod X .., the full subcategory of ModŽX .consisting of all projective, resp. injective, objects. Observe that ᭙ X g X, SЈŽX .is the representable functor X Žy, X ., so SЈŽX .g ProjŽŽ Mod X ... Remark 8.3. The functor SЈ: C ª ModŽX .is homological and a triangle T is in EŽX .iff SЈŽT .is a short exact sequence in ModŽX .. Moreover b SЈ preserves products, and SЈ preserves coproducts iff X : C . Obviously SЈ kills EŽX .-phantom maps, so there exists a unique exact functor H: SE ŽX .ŽC.ª ModŽX .with HS s SЈ, where SE ŽX .ŽC.is the Steenrod category and S is the projectivization functor.

PROPOSITION 8.4. The following conditions are equi¨alent:

Ži.The functor H: SE Ž X .ŽC.ª ModŽX . is an equi¨alence of categories. Žii.The functor SЈ induces an equi¨alence: PŽŽE X ..s AddŽX . f ProjŽŽ Mod X ... Ž. Ä4 iii For any object X g X and any family Xiig I : X, the natural Ž.Ž. morphism [ig IiC X, X ª C X, [ ig IiX is an isomorphism. PURITY IN TRIANGULATED CATEGORIES 313

If Ži.holds, then identifying SE ŽX .ŽC.s ModŽX . and SЈ s S, we ha¨e that S induces natural isomorphisms:

nn op E xtE ŽX .Žy, C.( E xtModŽ X . SŽy., SŽC.: C ª Ab, ᭙ C g C , ᭙ n G 0.

Further the ideal ker S s Ph E ŽX .ŽC., and if X generates C, then ᭙A g C : E ŽX .-p.d A s p.d SŽA.. In particular: E ŽX .-gl.dim C F gl.dim ModŽX .. Ž.Ž. Ä4 Proof. ii « iii Let Xiig I be any family of objects in X. Since ŽŽ..ЈŽ. [ig IiX is in P E X , by hypothesis we have [ig IiS X ( ЈŽ.Ž. S [ig IiX . It follows that for any X g X, [ ig IiC X, X s ЈŽ.Ž.ЈŽ.Ž.Ž. [ig IiS XX( S [ ig IiXXs C X, [ ig IiX . Žiii.« Žii. Since SЈ restricted to X is the Yoneda embedding, it follows that SЈ< X : X ¨ P is fully faithful where P is the category of finitely generated projective functors in ModŽX .. Since ProjŽŽ Mod X ..s AddŽP., the hypothesis implies easily that SЈ< X can be extended to a full embedding PŽŽE X ..s AddŽX .¨ ProjŽŽ Mod X ... Let F be a projective functor. Then Ä4 ЈŽ. there exists a family Xiig Ii: X such that F [ G s [ g IiS X s ЈŽ. S [ig IiX , for some functor G. This decomposition gives us an idempo- ЈŽ.ЈŽ.Ž. tent f: S [ig IiX ª S [ ig IiX with Im f s F. Then there exists an ЈŽ. idempotent e: [ig IiX ª [ ig IiX such that S e s f. Since C has coproducts, bywx 28 the idempotent e splits producing a direct sum decom- Ž.ЈŽ. position [ig IiX s K [ L with Im e s K. Then obviously S K ( F and consequently SЈ: PŽŽE X ..ª ProjŽŽ Mod X .. is an equivalence. Ži.m Žii. If conditionŽ i. holds, then H restricts to an equivalence between Proj SE Ž X .ŽC.and ProjŽŽ Mod X ... Since PŽŽE X ..s AddŽX . has split idempotents, by Proposition 4.19 we have an equivalence PŽŽE X .. f Proj SE Ž X .ŽC., induced by S. It follows that SЈ s HS induces an equivalence PŽŽE X ..f ProjŽŽ Mod X ... Conversely ifŽ ii. holds, then the functor f SЈ induces the desired equivalence H: SE Ž X .ŽC.s AŽŽŽP E X ... ª AŽŽŽProj Mod X ...s ModŽ X .. The last assertions follow from Section 4. We call X self-compact if X satisfies conditionŽ iii. of Proposition 8.4. From now on we write S s SЈ, so that if X is self-compact, SЈ is the projectivization functor and SE ŽX .ŽC.s ModŽX .. We would like to know when S has its image in the subcategory FlatŽŽ Mod X .. of flat functors. The motivation for this is that in case X is triangulated, the category CohŽX op, Ab., resp. HomŽX, Ab., of cohomological, resp. homological, functors over X and the category FlatŽŽ Mod X .., resp. FlatŽŽ Mod X op .., of 314 APOSTOLOS BELIGIANNIS flat right, resp. left, X-modules, coincidewx 13 . It is natural to compare C with the cohomological functors over X, via the homological functor S.

PROPOSITION 8.5. The following conditions are equi¨alent:

Ži. Im S : FlatŽŽ Mod X ... Žii. X has weak cokernels and the inclusion i: X ¨ C preser¨es them. Žiii. X has weak cokernels and any morphism in X is a weak kernel. Živ. X has weak cokernels and any injecti¨e right X-module is flat.

Proof. Žii.« Ži. Since X has weak cokernels, it follows fromwx 13 that a functor F g ModŽX .is flat iff F sends diagrams of the form Ž).: f g X ª Y ª Z, where g is a weak cokernel of f in X, to exact sequences in Ab. Hence we must show that if Ž).is a diagram as above, then SŽ A.Ž g . SŽ A.Ž f . SŽAZ.Ž. ª SŽAY.Ž. ª SŽAX.Ž. is exact, ᭙A g C. Obviously C Žg, A. this diagram is isomorphic to the complexŽ †. : CŽZ, A.ª CŽY, A. C Žf, A. fgЈ h ª CŽX, A.. Let X ª Y ª AЈ ª ⌺ŽX .be a triangle in C. Since g is a weak cokernel of f in X and the inclusion i: X ¨ C preserves weak cokernels, there are morphisms ␶ : Z ª AЈ and ␳: AЈ ª Z such that gЈ s g (␶ and g s gЈ( ␳. Then let ␻ g Ker CŽf, A.be such that f ( ␻ s 0. Since gЈ is a weak cokernel of f, there exists a morphism ␾: AЈ ª A with gЈ(␾ s ␻ . Then g (␶ (␾ s ␻ « ␻ g Im CŽg, A.. HenceŽ †. is exact in Ab, and consequently SŽA.is a flat functor in ModŽX .. fg Ži.« Žii. Let f: X ª Y be a morphism in X and let X ª Y ª A h ª ⌺ŽX .be a triangle in C. Since S is homological, we have an exact SŽf. SŽg. SŽh. sequence in ModŽX .: SŽX .ª SŽY .ª SŽA.ª SŽŽ⌺ X ... Let ␬: SŽY .ª F be the cokernel of SŽf ., and let ␭: F ¨ SŽA.be the canonical inclusion, such that SŽg.s ␬ ( ␭. Since SŽX ., SŽY .are representable, F is finitely presented. By hypothesis SŽA.is a flat functor. Hence the morphism ␭ factors through some representable functor X Žy, Z.s SŽZ.with Z g X. So there are morphisms m: F ª SŽZ.and ␯ : SŽZ.ª SŽA.such that ␭ s m(␯. Since Z g X and S< X is fully faithful, there exists a unique morphism ␥ : Z ª A such that ␯ s SŽ␥ .. Similarly for the morphism ␬ ( m: SŽY .ª SŽZ., there exists a unique morphism ␣: Y ª Z with ␬ ( m s S␣. Then SŽg.s ␬ ( ␭ s ␬ ( m(␯ s SŽ␣ .(SŽ␥ .s SŽ␣ (␥ .. Since S< X is fully faithful, g s ␣ (␥. We show that ␣: Y ª Z is a weak cokernel of f in X. First SŽf .(SŽg.s 0 « SŽf .(␬ ( ␭ s 0 « SŽf .(␬ s 0 « SŽf .(␬ ( m s 0 « SŽf .(SŽ␣ .s 0 « f ( ␣ s 0. Now let ␳: Y ª W be a morphism in X such that f ( ␳ s 0. Since g is a weak cokernel of f in C, there exists a morphism ␶ : A ª W with ␳ s g (␶. But then ␳ factors through ␣ as ␳ s ␣ (␥ (␶. Hence X has weak cokernels. To show that the inclusion i: X ¨ C preserves them, it suffices to show that if ␻ : PURITY IN TRIANGULATED CATEGORIES 315

Y ª B is a morphism in C with f ( ␻ s 0, then ␻ factors through ␣. If f ( ␻ s 0, there exists ␾: A ª B with g (␾ s ␻ . Then ␣ (␥ (␾ s ␻ . Žiii.m Živ. Follows fromwx 13 .Ž ii.m Žiii. The easy proof is left to the reader.

8.2. Injecti¨e Objects and Functors Assume, besides the default assumptions concerning the pair ŽC, X ., that X generates C and consists of compact objects. op Let F g FlatŽŽ Mod X ..; i.e., F: X ª Ab is a flat functor. Consider the functor

op ބ SŽym.X F : C ª Ab, Ž). where ބ s HomޚŽy, ޑrޚ.and ބwSŽym.X FAxŽ.s HomޚwSŽA.mX F, ޑrޚx. Since F is a flat, the above functor is cohomological and converts coproducts to products. Since C is compactly generated by X, by Brown representability theoremwx 55 there exists an object DŽF.g C, unique up to isomorphism, and a natural isomorphism

( ␻ : ބ SŽym.X F ª C y, DŽF .. Žq. Since S sends triangles in EŽX .to short exact sequences in ModŽX ., it follows that ބwSŽym.X Fxw, hence C y, DŽF.x, sends triangles in EŽX .to short exact sequences in Ab. We infer that DŽF.is an EŽX .-injective object in C. Choose F s X ŽX, y.in Ž)., where X g X. Then ބwSŽym.X X ŽX, y.x s ބCŽX, y.. In this case we denote the object DŽŽX X, y..by ބŽX .. Observe that if CŽŽC, ބ X ..s 0, ᭙ X g X, then ބCŽX, C. s 0 « CŽX, C.s 0, ᭙ X g X. Since X generates C, C s 0. Hence the set ÄބŽX .; X g IsoŽX .4 cogenerates C. For a class of objects Y : C, let ProdŽY .be the full subcategory of retracts of products of objects of Y. The following generalizes results of wx21, 50 . THEOREM 8.6. C has EŽX .-injectië enëlopes and I ŽŽE X .. s ProdŽŽބ X ... The functor S induces an equiälence S: I ŽŽE X .. ªf InjŽŽ Mod X ... Finally for any EŽX .-injectië object E, the ring ⌳ [ End C ŽE. is F-semiperfect; i.e., ⌳rJacŽ⌳.is Von Neumann regular and idempotents can be lifted modulo JacŽ⌳.. In particular an indecomposable EŽX .-injectië object in C has a local endomorphism ring.

Proof. For C g C, let JCC[ ÄC ª ބŽX .¬ X g IsoŽX .4. Set E s Ł ބŽX .. The set J induces a morphism g : C E . Let ŽT .: ␮g JC ␮ CCCª g C C ª EC ª A ª ⌺ŽC.be a triangle in C. Applying, ᭙ X g X, the functor 316 APOSTOLOS BELIGIANNIS

CŽŽy, ބ X .. to the triangle it follows by construction that 0 ª CŽŽA, ބ X ..ª CŽŽEC , ބ X ..ª CŽŽC, ބ X .. ª 0 is exact. But this sequence is isomorphic to the sequence 0 ª ބCŽX, A.ª ބCŽX, EC .ª ބCŽX, C.ª 0. Since ޑrޚ cogenerates Ab, the sequence 0 ª CŽX, C. ª CŽX, EC .ª CŽX, A.ª 0 is exact, i.e., ŽT .g EŽX .. Since EC is EŽX .-injective, we have that C has enough EŽX .-injectives and the characterization of EŽX .-injective objects follows. By Theorem 4.24, S: I ŽŽE X ..¨ InjŽŽ Mod X .. is fully faithful. Let F be an injective functor in ModŽX .. Applying Brown representability to the functor CŽŽS y.., F , we have an isomorphism ␻ : CŽŽS y.., F ª CŽy, EFF.and plainly E is in I ŽŽE X ... Then ␻ < X : CŽŽS y.., F < X s F ( SŽEF ., so S is surjective. By Theorem 4.24, C has EŽX .-injective envelopes. The last assertions are true since they are true in ModŽX .; seewx 39 . We close this subsection showing that any object of C admits a functorial ‘‘embedding’’ into an EŽX .-injective object. We assume from now on that X is a triangulated subcategory of C. By Proposition 8.5, for any object A g C, SŽA.is a flat right X-module. It is well known that ބSŽA. is an injective left X-module. Since X is a triangulated subcategory of C, by the dual of Proposition 8.5, we infer that ބSŽA.is a flat left X-module. Hence choosing F s ބSŽA.in Ž)., we obtain an EŽX .-injective object DŽŽބS A..in C, which we denote by DŽA., equipped with the natural ( isomorphism ␻ : ބwSŽym.X ބSŽA.x ª Cwy, DŽA.x. THEOREM 8.7. The assignment A ¬ DŽA. defines an additi¨e functor D: C ª C and ᭙A g C, the object DŽA. is EŽX .-injecti¨e. Further there exists a natural morphism ␦: Id C ª D, such that ␦A: A ª DŽA. is an EŽX .-proper monic.

Proof. It is well knownwx 65 that there is a canonical morphism F mX ބŽG.ª ބŽF, G.which is invertible if F is finitely presented. Hence we have a natural morphism ␾: SŽym.X ބSŽA.ª ބwSŽy., SŽA.x such that ␾ < X is invertible. We denote by ␺ : CŽy, A.ª Cwy, DŽA.x the composition ␮ S 6 y, A 2 C Žy, A.SŽy., SŽA.ª ބ SŽy., SŽA.

ބŽ␾ .6 ␻ ބ SŽym.X ބSŽA.ª C y, DŽA.,

where ␮ is the canonical monomorphism. Observe that ␺ < X : SŽA.ª SŽŽD A.. is a monomorphism. By Yoneda’s lemma there exists a unique morphism ␦AA: A ª DŽA.such that ␺ s CŽy, ␦ .. Then ␺ < X s CŽy, ␦AA.< X s SŽ␦ .and since ␺ < X is a monomorphism, the same is true for SŽ␦AA.; i.e., ␦ is an EŽX .-proper monic. We leave to the reader to Ž. Ä␦ 4 prove that A ¬ D A defines an additive functor D: C ª C and AAg C are the components of a natural morphism ␦: Id C ª D. PURITY IN TRIANGULATED CATEGORIES 317

2 Note that we have the formula SD ( ބ S, so D can be regarded as a ‘‘double dual’’ functor in C. The next result is a direct consequence of the above theorems.

COROLLARY 8.8. ᭙A, B g C, ᭙ X g X,PhE Ž X .ŽŽA, ބ X ..s 0 s Ph E Ž X .ŽŽA, D B... Moreo¨er f g Ph E Ž X .ŽC.iff DŽf .s 0 iff CŽŽf, ބ X .. s 0, ᭙ X g X.

8.3. Homological Functors Preser¨ing Coproducts Assume from now on that X is self-compact and satisfies the following condition

Ž††.X has weak kernels and the inclusion X ¨ C preserves them, which is dual to conditionŽ ii. of Proposition 8.5.

Fix a coproduct preserving homological functor H: C ª Ab. Set HЈ [ H < X : X ª Ab and let H* [ ymX HЈ: ModŽX .ª Ab be the unique colimit preserving functor extending HЈ through the Yoneda embedding X ¨ ModŽX .. Using Corollary 4.21, it is not difficult to see that H*S ( E Ž X . E Ž X . L00H. Let ␻ : L H ª H be the canonical morphism; in particular the restriction ␻ < PŽ E ŽX .. is invertible. Since H is homological and X satisfies conditionŽ ††. , bywx 13 it follows easily that HЈ is flat, so H* is E Ž X . exactŽ see alsowx 50. . We infer that H*S s L0 H is homological.

LEMMA 8.9. Assume that Ž␣ .: any object of C has finite EŽX .-projecti¨e b E ŽX . ( dimension or Ž␤ .: X : C and X generates C. Then ␻ : L0 H ª H.

Proof. Ž␣ .Assume first that EŽX .-p.d A F 1 and let P10ª P ª A ª ⌺ŽP1.be an EŽX .-projective resolution of A. Since ␻ < PŽ E ŽX .. is invert- E Ž X . ible, applying the homological functors L0 H, H to the above triangle E Ž X . and using 5-Lemma, it follows that ␻ A: L0 HAŽ.ª HAŽ.is invertible. Since any object of C has finite EŽX .-projective dimension, by induction we have that ␻ is an isomorphism. E Ž X . ( E Ž X . Ž␤ .Set U [ Ä A g C ¬ ␻ A: L00HAŽ.ª HAŽ.4. Since L H, H are coproduct preserving homological functors, U is a full triangulated subcategory of C, closed under coproducts, and contains the compact generating subcategory X of C. It follows that U s C and consequently ␻ is invertible.

THEOREM 8.10. Under the assumptions of Lemma 8.9, we ha¨e the following.

Ži.There exists an exact colimit preserïng functor H*: ModŽX .ª Ab, unique up to isomorphism, such that H*S s H. The functor H* admits a right adjoint which preserës injectiës and we can identify SŽym.X H < X s H. 318 APOSTOLOS BELIGIANNIS

Žii.HfŽ.s 0, ᭙ f g Ph E Ž X .ŽC.. b op Žiii. If FlatŽŽ Mod C .. is the category of coproduct preserïng homological functors C ª Ab, then the assignment F ¬ F*S induces an equiä- op b op lence FlatŽŽ Mod X .. ªf FlatŽŽ Mod C .., with the inërse giën by H ¬ H < X . Proof. Ži. Follows from the above analysis and Lemma 8.9.Ž ii. Follows fromŽ i. , since S kills EŽX .-phantom maps.Ž iii. The easy proof is left to the reader. Remark 8.11. In Theorem 8.10, Ab can be replaced by any abelian AB5 category.

8.4. The Roos Spectral Sequence

Let ÄFi4be a filtered system of functors in ModŽX .and let G be another E p, q Ž p. functor. Then bywx 65 we have a Roos’s spectral sequence: 2 s lim ¤ xtq F G xt n F G E wxi, « E wxlim ª i, . Assume now that X compactly generates C and satisfies conditionŽ ii. of Proposition 8.5. Let A, B g C. Since SŽA. is flat, there exists a functor I: I ª X from a small filtered category I SŽA.SŽX .X Ži. such that s lim ª ii, where s I . Trivially the Roos spectral sequence for the functors SŽA., SŽB.collapses, giving isomorphisms

Ž n. nn lim IiC ŽX , B.( E xt SŽA., SŽB.( E xtE Ž X .wx A, B , ᭙ n G 0. ¤

t OROLLARY Ž. B n iff Ž . ŽX B. for all t n C 8.12. E X -i.d F lim ¤ IiC , s 0, G q 1 and for all filtered direct systemsÄ Xi4 of objects of X.

Let A01ª A ª иии be a tower of objects of C. Since X is compact, the S SŽ functor preserves coproducts. Hence by Lemma 5.8 we have holim ª A .SŽA . ÄG 4 b Ž p. G ᭙ p i ( lim ª in. Since for a tower in A , lim ¤ n vanishes G 2, the next result follows from Roos’s spectral sequence by standard arguments. COROLLARY 8.13. ᭙B g C and ᭙ n G 0, there exists a short exact sequence

Ž1. ny1 n 0 ª lim E xtE Ž X .wx Ai , B ª E xtE Ž X . holim Ai , B ¤ª n ª lim E xtE Ž X .wx Ai , B ª 0. ¤

A A If the tower above is the E-cellular tower of , then since ( holim ª Ai, the above short exact sequence gives a method to compute the extension functors of A by means of the extension functors of its cells Ai. PURITY IN TRIANGULATED CATEGORIES 319

9. PHANTOMLESS TRIANGULATED CATEGORIES

Let C be a triangulated category with coproducts and let X : C be a skeletally small full additive subcategory, closed under ⌺, ⌺y1, direct summands, and isomorphisms. Our main result in this section characterizes the EŽX .-phantomless triangulated categories and generalizes results of Neemanwx 54 . First we need a simple lemma and some definitions.

b LEMMA 9.1. If C has products and X : C , then for any set of objects Ä4 Aiig Ii: C, the canonically defined triangle [ g IiA ª Ł ig IiA ª A ª ⌺Ž.Ž. [ig IiA is in E X . b Proof. Since S preserves products and also coproductsŽ since X : C ., иии Ž.Ž.Ž.иии the exact sequences ª S [ig IiA ª S Ł ig IiA ª S A ª and иии Ž.Ž.Ž. иии ª [ig IiS A ª Ł ig IiS A ª S A ª are isomorphic. Since in module categories the morphism from a coproduct to the product is a Ž.Ž.Ž. monomorphism, we see that 0 ª S [ig IiA ª S Ł ig IiA ª S A ª 0 is exact and the assertion follows.

We recallwx 13 that an additive category C is called weak abelian if C has weak kernels and weak cokernels, and any morphism is a weak kernel and a weak cokernel. Trivially triangulated categories are weak abelian. We recallwx 30 that a Grothendieck category G is called locally Noetherian, resp. locally Artinian, resp. locally finite, if G has a set of generators consisting of Noetherian, resp. Artinian, resp. finite length, objects. A functor category is called perfect wx39 if any flat functor is projective. We need the following result fromwx 13 .

PROPOSITION 9.2wx 13 . Let C be a skeletally small additi¨e category with split idempotents. Then the following are equi¨alent: Ži. ModŽC.is a Frobenius category. Žii.C is weak abelian and ModŽC.is locally Noetherian. Žiii.C is weak abelian and ModŽC.is perfect. Živ.C is weak abelian and ProjŽŽ Mod C.., resp. InjŽŽ Mod C.., is closed under products, resp. coproducts.

In this case the projectië, injectië, and flat functors coincide. Moreoër the module category ModŽC.is Frobenius and locally finite iff ModŽC op . is Frobenius and locally finite iff C is weak abelian and ModŽC., ModŽC op . are locally NoetherianŽ perfect..

Let F be an additive category. We recall that an object A g F is called finitely presented if the functor FŽA, y.commutes with filtered colimits. Followingwx 24 we say that F is locally finitely presented if F has filtered 320 APOSTOLOS BELIGIANNIS colimits, any object of F is a filtered colimit of finitely presented objects, and the full subcategory of finitely presented objects of F is skeletally small. Finally we recall that an additive category is a Krull᎐Schmidt category, if any of its objects is a finite coproduct of indecomposable objects and any indecomposable object has a local endomorphism ring. In b case C is compactly generated and X s C , then partsŽ 1. ,Ž 12. ,Ž 14. ,Ž 15. of the next result have been observed independently by Krausewx 50 . THEOREM 9.3. The following statements are equiälent: Ž1.The functor S: C ª ModŽX . induces an equiälence: C f FlatŽŽ Mod X ... b Ž2. X : C and EŽX .-gl.dim C s 0. b Ž3. X : C and Ph E Ž X .ŽC.s 0. b b Ž4. X : C and C s AddŽX ., resp. X : C generates C and C s ProdŽŽބ X ... b Ž5. X generates C, X : C , and the module category ModŽX .is Frobenius. b Ž6. X generates C, X : C is weak abelian, and AddŽX ., resp. I ŽŽE X .., is closed under products, resp. coproducts. b Ž7. X generates C, X : C is weak abelian, and C is EŽX .- Frobenius; i.e., C has enough EŽX .-injectiës and PŽŽE X ..s I ŽŽE X ... Ž8.The uniqueŽ exact. extension S*: AŽC.ª ModŽX .of S through the Freyd category AŽC.of C is an equiälence of categories. Ž9. For any abelian category M and for any homological functor H: C ª M, there exists a unique exact functor H*: ModŽX .ª M with H*S s H. b Ž10.X generates C, X : C is weak abelianŽ resp. and C has products., and the following conditions are true: Ž␣ . Ä4 Ž. For any family Xiig I : X the canonical mono- mor- phism

␮: C Žy, Xii.ª Ł C Žy, X .resp. ␯ : X iiª Ł X [igIigIiž[gIigI / splits. Ž␤ . Ä4 For any set I and directed family Xiig I : X and for any Y g X

Ž n. lim IiC ŽX , Y .s 0, ᭙ n G 1. ¤ PURITY IN TRIANGULATED CATEGORIES 321

b Ž11.X generates C, X : C is weak abelian, and ifÄ Xi; i g I4 and ÄYi; j g J4 are filtered direct systems of objects of X, then

Ž n. lim IJijlim C ŽX , Y .s 0, ᭙ n G 1. ¤ª

b Ž12.X : C is a Krull᎐Schmidt weak abelian category which generates f1 f 2 C and for any sequence X123ª X ª X ª иии of non-isomorphisms between indecomposable objects of X, there exists N g ގ, such that f12( f ( иии ( fN s 0. Ž. b generates and Ž. ⌬ i e a triangle is in 13 X : C C E X s lim ª 0; . ., EŽX .iff it is a filtered colimit of split triangles. Ž. b 14 X : C and if A g C then A s [ig IiX , where X ig X and End C ŽXi. is a local ring. Moreoër any two such decompositions are isomorphic. b Ž15.X s C and C is a locally finitely presented category. If one of the aboë conditions is true, then C is a pure semisimple locally b finitely presented category with products with X s C as its full subcategory of finitely presented objects. Further the projectiïzation functor S induces equi¨- alences

b b b b C s AddŽC .f ProjŽ ModŽC ..s FlatŽ ModŽC ..s InjŽ ModŽC ..

b and a triangle equi¨alence ⌬r⌬00f ModŽC ., where ⌬r⌬ is the stable triangulated category of all triangles modulo the split triangles wx13 and ModŽC b.is the stable triangulated category of the module category ModŽC b. modulo projecti¨es. Proof. In the first part of the proof we show that the first nine statements are equivalent by the scheme:Ž 1.« Ž2.« Ž3.« Ž4.« Ž5.« Ž1,.Ž 5.m Ž6. ,Ž 7. ,Ž 8. , andŽ 8.m Ž9. . In the second part we show that Ž5.m Ž10. ,Ž 11. ,Ž 12. , thatŽ 3.m Ž13. , thatŽ 4.m Ž14. and finally that Ž1.m Ž15. . IfŽ 1. is true, then since S is faithful, we have ker S s Ph E Ž X .ŽC.s 0. b Trivially X : C generates C. HenceŽ 1.« Ž2. . TriviallyŽ 2.« Ž3,.Ž 3.« Ž4. and by Theorem 4.25 we haveŽ 4.« Ž5. . IfŽ 5. is true then since b X : C , by Proposition 8.4, the functor S: PŽŽE X ..ª ProjŽŽ Mod X . s FlatŽŽ Mod X .is an equivalence. Since by Proposition 9.2, X is weak abelian, it follows by Proposition 8.5 that Im S consists of flat s projective modules. Now since X generates C, ᭙A g C, EŽX .-p.d A s p.d SŽA.s 0. This implies that E ŽX .-gl.dim C s 0; hence C s PŽŽE X ..f FlatŽŽ Mod X .., soŽ 1. is true.Ž 5.m Ž6. Since AddŽX .f ProjŽŽ Mod X .. and I ŽŽE X ..f InjŽŽ Mod X .., the category AddŽX ., resp. I ŽŽE X .., is closed 322 APOSTOLOS BELIGIANNIS under products, resp. coproducts, iff ProjŽŽ Mod X .., resp. InjŽŽ Mod X .., is closed under products, resp. coproducts. Then the assertion follows from Proposition 9.2.Ž 5.m Ž7. If X compactly generates C and X is weak abelian, then trivially ModŽX .is Frobenius iff C is EŽX .-Frobenius and the claim follows.Ž 5.« Ž8. SinceŽ 5.« Ž2. and ModŽX .is the Steenrod category of C,Ž 8. follows from Theorem 4.25. ThatŽ 8.« Ž5. is a consequence of the fact that AŽC.is Frobenius. FinallyŽ 8.m Ž9. follows from the fact that AŽC.is the universal homological category of C. So far we proved that the conditionsŽ 1. , . . . ,Ž 9. are equivalent. Observe b that the conditionsŽ 1. andŽ 4. imply that X s C . In particular X is a triangulated subcategory of C and C is compactly generated.

Ž.Ž.Ž.Ž. 5 m 10 If 5 is true then [ig IiC y, X is injective, since ModŽX .is Frobenius. Hence the canonical monomorphism ␮ splits. Let Ä4 Ž. Xiig I : X, with I a filtered set and let Y g X. Consider the flat functor F ŽX . s lim ª IiX y, . From the Roos spectral sequence of Section 8.4 for F ŽY . Ž n. ŽX Y . the functors , X y, ,itfollowsthatlim¤ IiC , ( n E xtw F, X Žy, Y .x, ᭙ n G 1. Since any flat is projective, we have n E xtŽŽ F, X y, Y ..s 0, ᭙ n G 1; hence conditionŽ 10.Ž␤ . holds. For the parenthetical case observe that since FlatŽŽ Mod X ..has products, byŽ 1. we have that C has products; hence the canonical morphism ␯ is defined. By

Lemma 9.1, ␯ is an EŽX .-proper monic, which splits since Ph E Ž X .ŽC.s 0 n byŽ 2. . Conversely ifŽ 10. is true, then by Ž␤ .we have E xtw F, X Žy, Y .x s 0, ᭙ n G 1, for any flat functor F and any object Y g X, again from Roos’s spectral sequence. Since X is weak abelian, X has weak cokernels so by wx40 , we have that X Žy, Y .is pure injective. Then by condition Ž␣ ., any free functor is pure injective. It follows that any projective is pure injective and then bywx 40 , the category ModŽX .is perfect; i.e., any flat functor is projective. By Proposition 9.2, ModŽX .is Frobenius. For the parenthetical b case observe that ␮ is isomorphic to SŽ␯ ., since X : C . Ž5.m Ž11. As in the proof ofŽ 5.m Ž10. , using Roos’s spectral sequence. Ž5.m Ž12. Suppose thatŽ 5. is true. Then ModŽX .is perfect and the assertion follows from the characterization of perfect functor categories in wx39 . IfŽ 12. is true, bywx 39 we have that ModŽX .is perfect and hence Frobenius since X is weak abelian. Ž4.m Ž14. IfŽ 4. is true then byŽ 5. , ModŽX .is perfect and C is identified with ProjŽŽ Mod X ... Then the assertion follows fromwx 39 . The converse is trivial. Ž3.m Ž13. If any triangle in EŽX .is a filtered colimit of split triangles, then obviously C is EŽX .-phantomless. Conversely if C is PURITY IN TRIANGULATED CATEGORIES 323

EŽX .-phantomless, then the assertion follows from the equivalence ofŽ 3. with the conditionŽ 1. . Ž1.m Ž15. IfŽ 1. is true, then C s FlatŽŽ Mod X .. which obviously is locally finitely presented. Moreover byŽ 4. , the full subcategory of finitely b presented objects of C coincides with X s C . Bywx 24 and conditionŽ 11. , we have that C is pure semisimple. Conversely ifŽ 15. is true, and if Y is b the category of finitely presented objects of C, then obviously Y : C and we know bywx 24 that the functor S: C ª FlatŽŽ Mod Y .. is an equivalence. By conditionŽ 1. with X replaced by Y it follows that C is EŽY .-phan- b b tomless. This implies as above that Y s C . Since by hypothesis C s X, it follows that X s Y andŽ 15. impliesŽ 1. . The last part follows from the above proof, noting that bywx 13 , we have always an equivalence ⌬r⌬0 f AŽC ., where the latter is the stable category of the Frobenius category AŽC.modulo projectives.

EXAMPLE 9.4. Any semisimple algebraic stable homotopy category in the sense of Hovey et al.wx 38 satisfies any one of the conditions of Theorem 9.3. For other examples we refer to Section 12.

We recall that an additive category D is called Ž¨on Neumann. regular if for any morphism f: X ª Y in D, there exists g: Y ª X such that f ( g ( f s f. It is not difficult to see that a triangulated category D is abelian iff D is semisimple abelian iff D is regular with split idempotents. It is well known that if D is regular and skeletally small, then all left or right modules over D are flat.

COROLLARY 9.5. If C is compactly generated, then the following are equiälent: Ži. All triangles in C are semi-split. b b Žii.C is EŽC .-phantomless and C is ön Neumann regular. Žiii.C is EŽC b.-phantomless and C b isŽ semisimple. abelian. Živ.The Steenrod category ModŽC b. is semisimple abelian. b Žv.The projectiïzation functor S: C ª ModŽC . is an equiälence.

9.1. Puppe Triangulations Let F be an abelian category with enough injectives. Bywx 43 , the stable category F modulo injectives is right triangulated with suspension functor 1 ⍀y : F ª F.

DEFINITION 9.6. A stable structure on F is a pair Ž⌺, ␦ ., where ⌺: 3 F ª F is an automorphism of F and ␦: ⌺ ª ⍀y is a natural isomor- 1 1 phism in F, such that ␦ ⍀y q ⍀y ␦ s 0. The last condition makes sense 324 APOSTOLOS BELIGIANNIS

1 1 since we can identify ⌺⍀y ( ⍀y ⌺. F is called ⌺-stable if F admits a stable structure Ž⌺, ␦ .. Observe that if F is ⌺-stable, then F is Frobenius. Now let P be an additive category equipped with an automorphism ⌺: P ª P. A class P of diagrams A ª B ª C ª ⌺ŽA.in P is called a Puppe-triangulation of the pair ŽP, ⌺., if P satisfies all the axioms of a triangulation, except possibly of the octahedral axiom. In this case P is called Puppe-triangulated; see wx57 where the terminology pre-triangulated is used. However, the last term has been used by many authors to mean different things and we shall avoid it. Our terminology is justified by the fact that according to Hellerwx 34 , D. Puppe was the first who introduced the axiomsŽ T12. ,Ž T. , andŽ T 3. of Section 2; see for instancewx 60 .

EXAMPLE 9.7. Let ŽT, ⌺, ⌬.be a Puppe-triangulated category. Then the Freyd category AŽT .is Frobenius and the suspension ⌺ of T induces an auto-equivalence ⌺#: AŽT .ª AŽT ., defined by ⌺#ŽF.s TŽ ,f. Coker TŽŽy, ⌺ f .., where TŽy, A. ª y TŽy, B.ª F ª 0 is a finite presentation of F g AŽT .. We leave to the reader to check that in fact AŽT .is ⌺#-stable. The following is due to Heller and was observed independently by Keller and Neeman.

THEOREM 9.8wx 34, 44 . Let F be a Frobenius category and ⌺: F ª F an automorphism of F. If P is the full subcategory of projectië᎐injectië objects of F, then there exists a bijectië correspondence between Puppe-triangulations P of the pair ŽP, ⌺.and ⌺-stable structures Ž⌺, ␦ .on F. In the next two paragraphs and for reasons of comparison, we rename triangulated categories, and we call them Verdier-triangulated categories. Hence a Verdier-triangulated category is a Puppe-triangulated category which satisfies the octahedral axiom. Theorem 9.8 suggests the following conjecture, which we believe is due to Keller and Neeman. Conjecture 9.9. There exists a ⌺-stable Frobenius category C, such that the Puppe-triangulated category ŽP, ⌺, P.of Theorem 9.8 is not Verdier- triangulated. The following provides support for the above conjecture to be true. Remark 9.10. By Example 9.7, a Puppe-triangulated category T is embedded in a ⌺-stable Frobenius category as a full subcategory of projective᎐injective objects. It follows that the above conjecture fails iff any Puppe-triangulated category is Verdier-triangulated. If the latter holds, then it would follow that for any Verdier-triangulated category T, the octahedral axiomŽ T4 . of Verdier is a formal consequence of the axioms PURITY IN TRIANGULATED CATEGORIES 325

ŽT12. ,Ž T. , andŽ T 3. . This is very unlikely to be true, but it is not known if there exists a counterexample.

9.2. Triangulations and Categories of Pro-objects and Ind-objects Let X be a skeletally small triangulated category. It is very natural to ask if the categories ProjŽŽ Mod X .., FlatŽŽ Mod X .. admit a triangulated structure in such a way that the inclusions X ¨ ProjŽŽ Mod X .., X ¨ FlatŽŽ Mod X .. are exact. We devote this subsection to a discussion of this question. By Theorem 9.3, if a triangulated category C is EŽX .-phantomless with b X as in Theorem 9.3, then X s C is triangulated, the Steenrod category ModŽX .is Frobenius, and its category of projectivesŽ equivalently the category of cohomological functors over X . has a triangulated structure which extends the triangulated structure of X. The following result provides a partial converse.

THEOREM 9.11. Let D be a skeletally small additi¨e category with split idempotents. Then for the statements Ži.the category FlatŽŽ Mod D.. is triangulated, Žii. ModŽD.is ⌺-stable, we ha¨e Ži.« Žii.and Žii.implies that FlatŽŽ Mod D.. is Puppe-triangulated. If Ži.holds, then D is triangulated, the inclusion i: D ¨ FlatŽŽ Mod D.. is exact, ModŽD.is Frobenius, and there are identifications:

AddŽD.s ProjŽ ModŽD..s FlatŽ ModŽD..s InjŽ ModŽD...

Proof. If C [ FlatŽŽ Mod D.. is triangulated with suspension ⌺, then C has coproducts and C b is a full triangulated subcategory of C. Bywx 24 , the category of finitely presented objects of FlatŽŽ Mod D.. can be identified b b with D. Trivially D : C generates C. Let ␣: F ª G be an EŽC .- b phantom map in C, i.e., ŽC , ␣ .s 0. Then ŽD, ␣ .s 0, which trivially by b Yoneda implies that ␣ s 0. By Theorem 9.3, it follows that D s C ; hence D is a triangulated subcategory of C and ModŽD.is Frobenius, which trivially is ⌺-stable. HenceŽ i.« Žii. . Conversely by Theorem 9.8,Ž ii. implies that ModŽD.is Frobenius and FlatŽŽ Mod D.. is Puppe-triangulated. For a skeletally small additive category D, let ProŽD., IndŽD.be the induced Pro-, Ind-categories; seewx 31 . In wx 13 , we gave necessary and sufficient conditions for ProŽD., IndŽD.to be abelian. It is useful to know when the Pro-, Ind-categories can be endowed with a triangulated struc- 326 APOSTOLOS BELIGIANNIS ture, since these categories are usually the codomain of derived functors defined on triangulated categories; seewx 25 .

COROLLARY 9.12.Ž a. For the statements Ži.the category IndŽD., resp. ProŽD., is triangulated, Žii. ModŽD., resp. ModŽDop ., is a ⌺-stable category, we haë Ži.« Žii.and Žii.implies that IndŽD., resp. ProŽD., is Puppe-triangulated. Žb. For the statements Ži.the categories ProŽD., IndŽD. are triangulated, op Žii. ModŽD., or equiälently ModŽD ., is a locally finite ⌺-stable category, we haë Ži.« Žii.and Žii.implies that IndŽD.and ProŽD. are Puppe-triangulated. If Ži.holds, then ModŽD., ModŽDop . are locally finite Frobenius categories, D is triangulated, and the inclusions D ¨ IndŽD., D ¨ ProŽD. are exact. Proof. It is well known that we can identify IndŽD.s FlatŽŽ Mod D.. op op op and ProŽD.s FlatŽŽ Mod D .. , and we have an equivalence: ProŽD. op f IndŽ D .. Then the assertions are consequences of Theorem 9.11 and Proposition 9.2. b Note that if Conjecture 9.9 is not true, then the map C ¬ ModŽC . gives a bijective correspondence between EŽC b.-phantomless compactly generated triangulated categories and ⌺-stable Frobenius module categories over skeletally small triangulated categories. The inverse map is given by ModŽD.¬ IndŽD..

10. BROWN REPRESENTATION THEOREMS

Throughout this section we assume that the triangulated category C has coproducts. We fix a skeletally small full additive subcategory X : C which is closed under isomorphisms, direct summands, and ⌺, ⌺y1. Con- sider the homological functor

SЈ: C ª ModŽX ., SЈŽA.s C Žy, A.< X and

SЈŽf .s C Žy, f .< X , where CŽy, A.< X denotes the restriction of the representable functor to X. We recall that an additive functor F: A ª B is called complete wx45 or a PURITY IN TRIANGULATED CATEGORIES 327 representation equi¨alence wx1 , if F is full, surjective on objects and reflects isomorphisms.

DEFINITION 10.1. The pair ŽC, X .satisfies the Brown representability theorem ŽBRT for short. if the canonical functor SЈ: C ª ModŽX .induces a representation equivalence between C and the flat functors FlatŽŽ Mod X .. over X. Our aim is to characterize when the pair ŽC, X .satisfies BRT. By Section 8, the functor SЈ satisfies Im SЈ : FlatŽŽ Mod X .. iff the following condition is true:

Ž†. X has weak cokernels and the inclusion functor X ¨ C preserves them. Further SЈ induces an equivalence between PŽŽE X ..and ProjŽŽ Mod X .. iff X is self-compact iff ModŽX .is the Steenrod category of C with respect to EŽX .and SЈ s S is the projectivization functor. Our main result in this section is the following:

THEOREM 10.2. The following statements are equi¨alent: Ži.The pair ŽC, X . satisfies BRT. Žii.Ž␣ .X is self-compact, generates C, and satisfies condition Ž†.. Ž␤ .FlatŽŽ Mod X .. s Ä F g ModŽX .¬ p.d F F 1.4 Žiii.Ž␣ .X is self-compact and satisfies condition Ž†.. Ž␤ .EŽX .-gl.dim C F 1. Ž␥ .Any F g FlatŽŽ Mod X .. has finite projecti¨e dimension. Proof. Ži.« Žii. By BRT, Im SЈ s FlatŽŽ Mod X ..; hence conditionŽ †. is true. Since SЈ: CrPh E Ž X .ŽC.f FlatŽŽ Mod X .., it follows directly that X is self-compact. Since idempotents split in PŽŽE X ..s AddŽX ., by Theo- rem 5.3, FlatŽŽ Mod X .. s Im SЈ s Ä F g ModŽX .¬ p.d F F 14 . Finally The- orem 5.3 ensures that X generates C and EŽX .-gl.dim C F 1. PartŽ ii.« Žiii. is trivial since if X generates C and conditionŽ †. is true, then EŽX .-gl.dim C F supÄ p.d F ¬ F g FlatŽŽ Mod X ..4. Žiii.« Ži. Since EŽX .-gl.dim C F 1, by Theorem 5.3 the functor SЈ s S: C ª FlatŽŽ Mod X .. is full and reflects isomorphisms. It remains to show that S is surjective on objects. Let F be a flat functor, let 0 ª SŽPn. a Ž.b Ž.иии Ž. ª S Pny1 ª S Pny20ª ª S P ª F ª 0 be a projective resolu- Ž. tion, and let b s d( e be the canonical factorization, where d: S Pny1 ª Ž.Ž. G and e: G ¨ S Pny2 . Then there exists f: Pnnª P y1 such that S f s a, since S: PŽŽE X ..f ProjŽŽ Mod X ... Since SŽf . is monic, the triangle f ⌺Ž.Ž.Ž. Pnnª P y1 ª Lny1 ª Pnnis in E X , and then S L y1 ( G. Since S Ž.Ž. is full, there exists a morphism g: Lny1 ª Pny2 with S g s e. Since S g 328 APOSTOLOS BELIGIANNIS

g ⌺Ž.Ž. is monic, the triangle Lny1 ª Pny2 ª Lny2 ª Lny1 is in E X , and Ž.Ž. then S Lny2 ( Im Pny2 ª Pny3 . Continuing in this way, it follows that F is in Im S, so S is surjective on objects.

If ŽC, X .satisfies BRT, then we have the classical version of the Brown representability theoremwx 51, 55 .

op COROLLARY 10.3. If the pair ŽC, X .satisfies BRT and if F: C ª Ab is a cohomological functor con¨erting coproducts to products, then F is representable.

Proof. Since the restriction functor F˜[ F < X g ModŽX .is flat, by BRT ( there exists A g C, such that ␣: SŽA.s CŽy, A.<

The author is indebted to the referee for the following nice remark.

Remark 10.4. If the pair ŽC, X .satisfies BRT, then C is compactly b generated. Indeed it suffices to show that X : C . Let ÄAi; i g I4be a set of objects in C and choose an EŽX .-projective resolution Pi1 ª Pi0 ª Ai ª ⌺ŽPi1.of Aii, ᭙ i g I. Then the sum [P 1 ª [Pi0 ª [Aiiª ⌺Ž[P 1. of these resolutions is a triangle which is an EŽX .-projective resolution of [Ai. If X g X, then applying CŽX, y.to the triangle and using 5-Lemma and the self-compactness of X, it follows easily that the canonical map [CŽX, Aii.ª CŽX, [A .is invertible. Hence X is compact. A direct consequence of the above representability result is the following.

COROLLARY 10.5. If the pair ŽC, X .satisfies BRT, then C has products.

The following is a consequence of Theorems 4.24 and 5.3.

COROLLARY 10.6. If BRT holds for the pair ŽC, X ., then C has EŽX .-in- jectië enëlopes and the functor S induces an equiälence S: I ŽŽE X .. ªf InjŽŽ Mod X ...

COROLLARY 10.7. Assume that X is self-compact, generates C, and satisfies condition Ž†..If Ž␣ .any flat right X-module is a direct union of countably presented pure submodules, Ž␤ .if any flat right X-module is countably presented, or Ž␥ .if the module category ModŽX .satisfies one of the following conditions: p.gl.dim ModŽX .F 1 or Kdim ModŽX .F 1, where p.gl.dim ModŽX .denotes pure global dimension and Kdim ModŽX .denotes Krull dimension wx30 of the Grothendieck category ModŽX ., then the pair ŽC, X .satisfies BRT. PURITY IN TRIANGULATED CATEGORIES 329

Proof. Bywx 39, 65 , we have supÄ p.d F; F g FlatŽŽ Mod X ..4 F p.gl.dim ModŽX .F Kdim ModŽX .. So the result follows from Theorem 10.2 in case Ž␥ .. In the first two cases it follows fromwx 40, 65 , since then for any flat F, p.d F F 1. COROLLARY 10.8. Let CrAddŽX ., CrI ŽŽE X .. be the stable categories of C modulo AddŽX ., I ŽŽE X ..and let FlatŽŽ Mod X .., FlatŽŽ Mod X .. be the stable categories of flat modules modulo projectiës, injectiës. If the pair ŽC, X .satisfies BRT, then S induces equiälences CrAddŽX .( FlatŽŽ Mod X ..and CrI ŽŽE X ..( FlatŽŽ Mod X ... CrAddŽX . has kernels and coproducts and CrI ŽŽE X .. has cokernels and products. Proof. By BRT the functor S induces a full and surjective on objects functor S: CrAddŽX .ª FlatŽŽ Mod X ... Let f: A ª B be a morphism in C such that SŽf .factors through a projective G s SŽP., P g AddŽX ., via the morphisms ␣ s SŽg.: SŽA.ª SŽP.and ␤ s SŽh.: SŽP.ª SŽB.. Then ␴ ␳␶ f y g ( h: A ª B is a phantom map. Let P10ª P ª A ª ⌺ŽP 1.be an EŽX .-projective resolution of A. Since f y g ( h is phantom, we have ␳ (Žf y g ( h.s 0; hence there exists ␻ : ⌺ŽP1.ª B such that ␶ ( ␻ s f y g ( h or f s ␶ ( ␻ q g ( h. Passing to the stable category we have f s 0 in CrAddŽX .. This means that S< C rAddŽX . is faithful, hence an equivalence. Finally the category FlatŽŽ Mod X .. is a left triangulated categorywx 12 , and since for any flat F we have p.d F F 1, it is trivial to see that FlatŽŽ Mod X .. has kernels. Since FlatŽŽ Mod X .. has coproducts it is easy to see that the same is true for FlatŽŽ Mod X ... The case of injectives is treated similarly.

There are pairs for which BRT failsŽ see Section 12. . Let C / 0 be the full subcategory of C consisting of objects A for which there exists a triangle P10ª P ª A ª ⌺ŽP 1.in EŽX .with P 10, P direct summands of countable / 0 coproducts of objects of X. Let Ph E Ž X .ŽC .be the restriction of Ph E ŽX .ŽC. to C / 0 and let Flat/ 0 ŽŽ Mod X .. be the full subcategory of countably presented flat functorswx 39, 65 . PROPOSITION 10.9. The functor S induces a representation equi¨alence S: / 0 / 0 / 0 / 0 / 0 C ª FlatŽŽ Mod X ... Hence C rPh E ŽX .ŽC . f FlatŽŽ Mod X ... Proof. By construction we have that S has image in Flat/ 0 ŽŽ Mod X .., and any object A g C / 0 has EŽX .-p.d A F 1. Sincewx 65 any countably presented flat functor F has p.d F F 1, we see that the functor S is surjective on objects. Recall fromwx 65 that the weight wŽX .of a skeletally small category X is defined by wŽX .[ cardÄ"X ŽX, Y .: X, Y g IsoŽX .4. PROPOSITION 10.10wx 65 . If X is a skeletally small additi¨e category with weight wŽX .F / t , for some t G 0, then any flat module X-module F has p.d F F t q 1. 330 APOSTOLOS BELIGIANNIS

The following consequence is a generalization of the main result ofwx 56 .

COROLLARY 10.11. If for the pair ŽC, X ., the subcategory X has weight wŽX .F / 0 and satisfies condition Ž†,.then the following are equi¨alent. Ži.The pair ŽC, X . satisfies BRT. Žii. X is a compact generating subcategory of C. For examples of pairs ŽC, X .satisfying BRT we refer to Sections 11 and 12.

11. PURITY

Throughout this section we fix a compactly generated triangulated category C. Our aim here is to apply the theory of Section 8, in case b X s C is the full subcategory of compact objects. The homological algebra in C based on the proper class of triangles EŽC b. is called the pure homological algebra of C. This kind of purity was also considered independently by Krausewx 50 . b TERMINOLOGY 11.1. A pure triangle in C is a triangle in EŽC .. The pure projectië, resp. pure injectië, objects PProjŽC., resp. PInjŽC., are the b b EŽC .-projective, resp. EŽC .-injective, objects. The pure projectië di- b mension p.p.d A of A g C is defined as EŽC .-p.d A. The pure global dimension p.gl.dim C is EŽC b.-gl.dim C and the pure extension functors U PE xt* are the functors E xtE ŽC b.. The ideal PhŽC.of pure phantom maps is the ideal Ph E ŽC b.ŽC., and so on. By Section 8 we know that C has enough pure-injectives and enough b b pure-projectives with PProjŽC.s AddŽC .and PInjŽC.s ProdŽŽބ C ... The pure Steenrod category of C is the module category ModŽC b. and the b projectivization functor is the restricted Yoneda functor S: C ª ModŽC ., which induces an equivalence between PProjŽC.and ProjŽŽ Mod C b.. and between PInjŽC.and InjŽŽ Mod C b... The motivating source of the above definitions and terminology is explained below. Its main theme is to try to describe objects of a larger category by filteredŽ homotopy. colimits of objects of a smaller and better behaved full subcategory.

11.1. Moti¨ation It is well known that the proper framework for the study of purity in a module category is that of a locally finitely presented Grothendieck categorywx 65 . If G is a locally finitely presented Grothendieck category, and f.pŽG.is the full subcategory of finitely presented objects, then the restricted Yoneda functor H: G ª ModŽŽ f.p G.. is fully faithful and identi- PURITY IN TRIANGULATED CATEGORIES 331 fies G with the full subcategory FlatŽŽŽ Mod f.p G... of flat functors and the pure-projective objects of G with the projective objects of ModŽŽ f.p G...A short exact sequence ŽE.in G is pure iff HŽE.is exact. The pure global dimension of the Grothendieck category G is computed as p.gl.dim G s supÄ p.d F ¬ F g FlatŽŽŽ Mod f.p G...4. In particular G is pure semisimple iff the module category ModŽŽ f.p G.. is perfect; i.e., any flat functor is projectivewx 65 . Hence the pure homological theory of G is strongly connected with the behavior of ModŽŽ f.p G.. via the functor H. Now if C is a triangulated category with coproducts such that C b is skeletally small, it is natural to consider C b as the full subcategory of b ‘‘finitely presented’’ objects, since ᭙ X g C , the functor CŽX, y.preserves coproducts. This analogy is justified by the fact that in the stable homotopy category the compact objects are the finite spectra and also in the derived category where the compact objects are the perfect complexes which behave bywx 69 , like finitely presented objects. See also Section 12, where the example of the stable module category provides further justification. If C has a set of compact generators, i.e., C is compactly generated, then C can be considered as the triangulated analogue of a locally finitely presented Grothendieck category. Then the Steenrod category ModŽC b. is b the analogue of ModŽŽ f.p G..and the functor S: C ª ModŽC . is the analogue of the functor H above. A triangle ŽT .is in EŽC b.iff SŽT .is short exact. Hence the triangles in EŽC b. can be considered as the ‘‘pure’’ triangles. The functor S is not fully faithful in general, due to the presence of phantom maps, and this is the major difference between the two theories. Indeed by Theorem 9.3, S is fully faithful iff C is phantomless iff S induces an equivalence between C and FlatŽŽ Mod C b... Also Theorem 9.3 characterizes the phantomless categories as the categories C such that ModŽC b. is perfect or equivalently all pure triangles are split. Hence the phantomless categories are the pure semisimple categories and the global dimension of C with respect to the pure triangles is the pure global dimension of C. b 1 We recall that a right C -module G is called FP-injecti¨e if E xtŽ F, G. b s 0 for any finitely presented module F. Since C is triangulated, bywx 13 orwx 50 , the category FlatŽŽ Mod C b.. of flat right C b-modules, the category CohŽŽC b.op, Ab. of cohomological functors over C b, and the category FPInjŽŽ Mod C b.. of FP-injective right C b-modules are identical.

PROPOSITION 11.2.

p.gl.dim C s supÄ p.p.d A; A g C ¬ p.p.d A -ϱ4 b s supÄ p.d F ; F g FlatŽ ModŽC ..¬ p.d F -ϱ4

b s supÄ p.d F ; F g FlatŽ ModŽC ..4 332 APOSTOLOS BELIGIANNIS

b s supÄ i.d F ; F g FPInjŽ ModŽC ..4

b s supÄ i.d F ; F g FlatŽ ModŽC ..4

b s supÄ p.d F ; F g FPInjŽ ModŽC ..4.

Proof. Since ᭙ A g C,p.p.dA s p.d SŽA.,andsinceImS : b b FlatŽŽ Mod C .., we have p.gl.dim C F supÄ p.d F; F g FlatŽŽ Mod C ..4. By b a result of Jensenwx 65 , supÄ p.d F; F g FlatŽŽ Mod C ..4s supÄ p.d F; F g FlatŽŽ Mod C b.. ¬ p.d F -ϱ4. We show that if F is a flat right C b-module of finite projective dimension, then there exists A g C such that p.p.d A s p.d F. If p.d F F 1, then we know that there exists A g C such that F s SŽA.. Suppose that ϱ )p.d F G 2. There exists a small filtered cate- I I b F SŽX . gory and a functor I: ª C such that s lim ª I i . Consider the triangle [I Xi ª [I Xi ª A ª ⌺Ž[I Xi.in C. Then we have a long exact y1 ␣ y1 ␤ ␥ sequence иии ª [I SŽ⌺ ŽXi.. ª SŽ⌺ ŽA.. ª [I SŽXi.ª [I ␦ SŽXi.ª SŽA.ª иии . The equivalence ⌺ induces an autoequivalence ⌺# b 1 of FlatŽŽ Mod C ..given by ⌺#ŽF. s F ⌺ with inverse ⌺#y defined by 1 1 1 ⌺#y ŽF.s F ⌺y . Then F s ImŽ␦ .and ImŽ␣ .s F ⌺y . Let H s ImŽ␤ . 1 and G s ImŽ␥ .. Obviously p.d ⌺#ŽF.s p.d F s p.d ⌺#y ŽF.and we have y1 y1 ␤ ␥ an exact sequence 0 ª ⌺# ŽF.ª SŽ⌺ ŽA.. ª [I SŽXi.ª [I SŽXi. ª F ª 0. From this exact sequence, since p.d F G 2 and [I SŽXi.is a 1 projective module, we have p.d H s p.d F y 2 s p.d ⌺#y ŽF.y 2. But then 1 1 from the short exact sequence 0 ª ⌺#y ŽF.ª SŽ⌺y ŽA.. ª H ª 0, and the fact that F, hence ⌺#y1 ŽF., has finite projective dimension, we deduce 1 1 that p.d F s p.d ⌺#y ŽF.s p.d SŽ⌺y ŽA..s p.d SŽA. s p.p.d A. Hence for any flat right C b-module F with p.d F -ϱ, there exists an object A g C, such that p.p.d A s p.d F. This implies trivially that supÄ p.d F; b F g FlatŽŽ Mod C .. ¬ p.d F -ϱ4 F p.gl.dim C. As a consequence we have that the first three equalities are true. Using that the flat and the FP-injective modules coincide, the proof of the remaining equalities is left to the reader. By Proposition 10.10, we have the following consequence.

b COROLLARY 11.3. If wŽC . F / t , t G 0, then p.gl.dim C F t q 1. If C is the stable homotopy category of spectra, then it is well known that C is compactly generated and the full subcategory C b of finite spectra has a countable skeleton. Hence by Corollary 11.3, p.gl.dim C F 1.

COROLLARY 11.4. If C is the stable homotopy category, then p.gl.dim C s 1. b Proof. If p.gl.dim C s 0, then by Theorem 9.3, ModŽC . is perfect. It is easy to seewx 65 that this implies that any compact object X has a perfect PURITY IN TRIANGULATED CATEGORIES 333

endomorphism ring. In C this is not true; e.g., choose X to be the sphere spectrum. Remark 11.5. The pure global dimension is invariant under triangle equivalence. Indeed any triangle equivalence F: C ªf D restricts to an b b b equivalence F : C ªf D . Hence p.gl.dim C s p.gl.dim D, by Proposi- tion 11.2. We denote the full subcategories of ModŽC b. consisting of the modules of finite projective, resp. injective, dimension by Proj-ϱŽŽ Mod C b.., resp. Inj-ϱŽŽ Mod C b... Since the flat and the FP-injective modules coincide, it is trivial to see that

-ϱ b b Proj ŽModŽC ..: FlatŽ ModŽC .. b -ϱ b s FPInjŽ ModŽC ..= Inj ŽModŽC ...

b Finally the finitistic projecti¨e dimension of ModŽC . is defined as b -ϱ b f.p.d ModŽC . s supÄ p.d F g ProjŽŽ Mod C ..4. The definition of the finitistic injective dimension f.i.d ModŽC b.of ModŽC b. is similar. Note that since C b is triangulated, bywx 13 the flat or FP-injective dimension of any left or right C b-module is 0 or ϱ.

11.2. Pure Hereditary and Brown Categories

Let I: I ª C be a functor from a small filtered category I. We use the notations: Aiijijs I Ži.for i g I and ␣ s I Ži ª j.: A ª A for an arrow i ª j in I.Aweak colimit of the functor I is an object A in C together with morphisms fii: A ª A, which are compatible with the system ÄA iij, ␣ 4 in the sense that for any arrow i ª j in I we have fiijjs ␣ ( f , and if g i: Ai ª B is another compatible family, then there exists aŽ not necessarily unique. morphism ␻ : A ª B such that fii( ␻ s g , for any object i g I. For example a homotopy colimit of a tower is a weak colimit. A weak A colimit of the functor I is denoted by w.lim ª I i. Bywx 51 a weak colimit ␪ of the functor I can be constructed as a weak cokernel I : [ig IiA ª w.lim A of the canonical morphism ␨ : 2 A A . ª I i I [␣ ijg Iiª [ ig Ii ␨ Hence it can be computed from the triangle 2 A I A [␣ ijg Iiª [ ig Ii ␪ I w.lim A ⌺Ž2 A .. ª ª I i ª [␣ ijg Ii EFINITION A D 11.6wx 51 . A weak colimit w.lim ª I i of a functor I: I ª C is called minimal if the canonical morphism

lim I SŽAi .ª S w.lim I Ai ªªž/

A is an isomorphism. We denote a minimal weak colimit by m.w.limª I i. 334 APOSTOLOS BELIGIANNIS

By Lemma 5.8 the homotopy colimit of a tower is a minimal weak colimit.

DEFINITION 11.7. A triangulated category C is called a Brown category, if C has coproducts, C b is skeletally small, and the pair ŽC, C b. satisfies b BRT; i.e., the projectivization functor S: C ª FlatŽŽ Mod C .. is a representation equivalence.

PartŽ 1.m Ž4. in the next result is due to Neemanwx 56Ž see alsowx 19, 21 . and partsŽ 1.m Ž9.« Ž11. were proved independently by Hovey et al.wx 38 ; see alsowx 51 . We believe that our proofs are simpler. Note that the characterizationsŽ 4. ,Ž 5. below are identical with the characterizations of flat modules over the Steenrod algebra; seewx 51 .

THEOREM 11.8. Let C be a compactly generated triangulated category. Then the following are equi¨alent. Ž1. C is a Brown category. b Ž2.The projecti¨ization functor S: C ª ModŽC . is full. Ž3. p.gl.dim C F 1. b b Ž4.Let F g ModŽC .. Then F g FlatŽŽ Mod C .. iff p.d F F 1. b b Ž5.Let F g ModŽC .. Then F g FlatŽŽ Mod C .. iff i.d F F 1. b Ž6. f.p.d ModŽC . F 1. b Ž7. f.i.d ModŽC . F 1. bb Ž8. If I: I ª C and J: J ª C are functors from small filtered categories, with I Ži.s Xij, JŽj.s Y , then

Žn. lim I lim J C ŽXij, Y .s 0, ᭙ n G 2. ¤ª

b Ž9. Any functor I: I ª C from a small filtered category I has a minimal weak colimit X in m.w.limª I i C. b Ž10. Any functor I: I ª C from a small filtered category I has a weak colimit X in with the following property For any coprod w.lim ª I i C . - uct preser¨ing homological functor F: C ª Ab, the canonical map ␾ below is in¨ertible: ( ␾ : lim I FXŽi .ª F w.lim I Xi . ªªž/

Ž11. Any object of C is a minimal weak colimit of a functor I: b I ª C from a small filtered category. If one of the aboë equiälent conditions is true, then any two minimal weak colimits are isomorphic and for any weak colimit X with X b w.lim ª I iig C , PURITY IN TRIANGULATED CATEGORIES 335 we haë

b w.lim I Xi ( m.w.lim I Xi [ P, where P g AddŽC .. ªª Proof. Using Theorems 5.3, 10.2 and Proposition 11.2 it follows trivially that the first seven conditions are equivalent. We include only a proof that Ž2.« Ž1. . If S is full, then by Theorem 5.3 we have that p.gl.dim C F 1. b Then Proposition 11.2 ensures that any flat functor F g ModŽC . has p.d F F 1. Finally by Theorem 10.2 it follows that C is a Brown category. Ž1.m Ž8. Let F, G be flat functors. Then F is a filtered colimit of b representables SŽXii.s C Žy, X .and G is a filtered colimit of repre- b sentables SŽYjj.s C Žy, Y ., where I, J are small filtered categories and bb I: I ª C , J: J ª C are functors with I Ži.s Xijand JŽj.s Y . By Roos’s spectral sequence of Section 8.4 for the functors F, G, we have Ž n. ŽX G. Žn. ŽX Y .xtnŽ F G.᭙ n Ž. lim ¤ I C i, ( lim¤ I lim ª J C ij, ( E , , G 0. If 1 holds, then the above isomorphism impliesŽ 8. , byŽ 3. . Conversely ifŽ 8. n holds, then E xtŽ F, G.s 0, ᭙ n G 2, for any two flat functors F, G. By wx40 , this implies that any flat G has pure injective dimension bounded by one. Since flat and FP-injective functors coincide and the pure injective dimension coincides with the injective dimension for FP-injective functors, Ž3. follows. Ž.Ž.F SŽX . I b 9 « 1 Let s lim ª I i be a flat functor, where I: ª C is a functor and I is a small filtered category. Consider a minimal weak X F SŽX . colimit m.w.limª I i of I in C. By hypothesis: s lim ª I i ( SŽX .S m.w.limª I i , so is essentially surjective. Since C is compactly generated, S reflects isomorphisms and it remains to prove that S is full. It b b suffices to show that if I: I ª C and J: J ª C are functors from small ␣ Ј SŽX .SŽY . filtered categories, then any map : m.w.limª I i ª m.w.limª J j is of the form SŽ␣ .. By construction we have projective presentations SŽX .SŽ ␨ I . SŽX .␰ I SŽX .SŽY .SŽ ␨ J . [ iiª [ ª m.w.limª I ijª 0 and [ ª SŽY .␰ J SŽY . [ j ª m.w.limª I j ª 0, and there exists a commutative diagram

SŽ␨␰I .6 I 6 6

SŽ[Xii.SŽ[X .SŽm.w.lim I Xi. 0

6 6 ᭚␤ Ј᭚6 ␥ Ј ␣ Ј

SŽ␨␰J .6 J 6 6 SŽ[Yjj.SŽ[Y .SŽm.w.lim J Yj. 0 ª

Since S induces an equivalence between AddŽC b.and ProjŽŽ Mod C b.., there exist morphisms ␤: [Xijª [Y , ␥ : [X ijª [Y such that SŽ␤ .s ␤Ј SŽ␥ . ␥ Ј ␤ ␨ ␨ ␥ X Y , s , and ( J s I ( . Since m.w.limª I i and m.w.limª J j 336 APOSTOLOS BELIGIANNIS

are weak cokernels of ␨ I and ␨ J , respectively, there exists a morphism ␣: X Y ␥ ␪ ␪ ␣ m.w.limª I i ª m.w.limª J j such that ( J s I ( . But then obviously SŽ␣ .s ␣ Ј. Hence S is full. Ž1.« Ž9. By the equivalence ofŽ 1. withŽ 3. , for any flat right b b C -module F, we have p.d F F 1. Let I: I ª C be a functor from a I F SŽX . small filtered category and consider the flat modules s lim ª I i G SŽX .SŽX .SŽX . and s w.lim ª I iii. Then identifying [ with [ , we have the following exact commutative diagram

SŽ␨ I .6 ␣ 6 6

[SŽXii.[SŽXF. 0

6 6 556 ᭚! ␻

SŽ␨ I .6 SŽ␪ I .6 SŽ[Xii.SŽ[XG.

A f A X By BRT there exists g C and a morphism : ª w.lim ª I i, such that SŽA.s F and SŽf .s ␻ . Similarly there exists aŽ unique. morphism ␮: [Xi ª A, such that SŽ␮.s ␣. Let SŽ␨ I .s ␤ (␥ be the canonical b factorization of SŽ␨ I .in ModŽC ., where ␤: SŽ[Xi.ª H and ␥ : H ª SŽ[Xi.. By BRT there exists Q g C andŽ unique. morphisms ␬: [Xi ª Q and ␭: Q ª [Xi such that SŽ␬ .s ␤ and SŽ␭.s ␥. Since SŽ␮.is epic the ␭ ␮␯ triangle Q ª [ Xi ª A ª ⌺ŽQ.is pure. Since p.d F F 1, the functor H s SŽQ.is projective; hence the morphism SŽ␬ .is split epic. Since S reflects isomorphisms, ␬ is split epic so there exists ␳: Q ª [Xi with ␳ (␬ s 1Qi inducing a direct sum decomposition [X ( Q [ P. Then we have the following morphisms of triangles:

␭␯6 ␮ 6 6

Q [XAi ⌺ŽQ.

6 6 6 ␳ 6 5 f ⌺Ž␳ .

␨␪I 6 I 6 ␩I 6

[Xii[X w.lim I Xii⌺Ž[X .

6 6 6 ␬ 6 5 ᭚g ⌺Ž␬ .

␭␯6 ␮ 6 6 Q [XAii⌺Ž[X ..

Since ␳ (␬ s 1Q , the morphism f ( g is invertible; in particular f is split Žf .⌺ŽP. X A monic. It is easy to see that Coker s ; hence w.lim ª I i ( [ ⌺ŽP.. Observe that ␨ I (␮ s ␬ ( ␭(␮ s 0. Let ␾: [Xi ª E be a morphism such that ␨ I (␾ s 0. Since by construction ␪I is a weak cokernel of ␨ ␾ ␪ ␺ ␺ X E I , we have s I ( for a morphism : w.lim ª I i ª . Hence ␾ s ␮( f (␺ and this implies that the morphism ␮: [Xi ª A is a weak b cokernel of ␨ I . Hence A is a weak colimit of the functor I: I ª C . Since SŽA.SŽX . A by the construction s lim ª I i , we conclude that is a minimal PURITY IN TRIANGULATED CATEGORIES 337 weak colimit of the functor I. If AЈ is another minimal weak colimit, then SŽA.( SŽAЈ.and by BRT we have A ( AЈ. The above argument shows b that if Aˆˆis any other weak colimit, then A ( A [ P where P g AddŽC .. Ž9.m Ž10. Note that by Theorem 8.10, a homological functor F: C ª Ab preserves coproducts iff F ( G*S, where G* is exact and preserves colimits. From this it follows directly thatŽ 9. impliesŽ 10. . Con- versely partŽ 9. follows fromŽ 10. , choosing F s CŽX, y.: C ª Ab, for b any X g C . Ž.Ž.A SŽA.SŽX . 1 m 11 If C is Brown and g C, then s lim ª I i , b where I: I ª C is a functor from a small filtered category I. ByŽ 9. , the AЈ X SŽA. functor I has a minimal weak colimit s m.w.limª I i. Then ( SŽAЈ.and since byŽ 1. , S is a representation equivalence, we have A ( AЈ. Ž. A A X Conversely if 11 holds, let g C. Then as above s m.w.limª I i. ␣ By construction of the weak colimit of I, there exists a triangle [Xj ª ␤ ␥ [ Xijª A ª ⌺Ž[X .. By the construction of the colimit of the functor b S I: I ª ModŽC ., it follows directly that ␥ is a pure phantom map, so the above triangle is pure. This implies that p.p.d A F 1. OROLLARY Suppose that If A X C 11.9. p.gl.dim C F 1. s m.w.limª I i b with Xi g C , then for any B g C there exists a short exact sequence Ž1. 0 ª lim I C Ž⌺ŽXi ., B.ª C ŽA, B.ª lim I C ŽXi , B.ª 0. ¤¤ Hence xt1ŽŽ⌺ A..B ŽA B. Ž1. ŽŽ⌺ X ..B and 2 Ž. PE , s Ph , s lim ¤ I C i , Ph C s 0. If A X B Y are representations of A B as s m.w.limª I i, s m.w.limª J j , b b minimal weak colimits of functors I: I ª C and J: J ª C , then ŽA B. Ž1. ŽŽ⌺ X ..Y and there exists a short exact se Ph , s lim¤ I lim ª J C ij, - quence Ž1. 0 ª lim I lim J C Ž⌺ŽXij., Y .ª C ŽA, B.ª limI lim J C ŽXij, Y .ª 0. ¤ª ¤ª Proof. By Theorem 4.27, there exists a short exact sequence 1 0 0 ª P E xt ŽŽ⌺ A.., B ª C ŽA, B .ª P E xtŽ A, B .ª 0. Since n n 1 E xt wSŽA., SŽB.x ( PE xtŽ A, B., ᭙ n G 0, we have E xt wSŽŽ⌺ A.., SŽB.x xt1 SŽŽ⌺ X ..SŽB. xt1ŽŽ⌺ A..B ( E wlim ª I i , x ( PE , . Using Roos’s spectral sequence as in the proof of Theorem 11.8 we see directly that Ž1. 1 E xt lim SŽ⌺ŽXii.., SŽB.( lim SŽ⌺ŽX ., SŽB. ªI ¤I Ž1. ( lim C Ž⌺ŽXi ., B.. ¤I xt0Ž A B.ŽX B. In the same way PE , ( lim ¤ I i, and the assertion follows. 338 APOSTOLOS BELIGIANNIS

COROLLARY 11.10. If C, D are Brown categories, then

b b C f D m CrPhŽC .f DrPhŽD.m PInjŽC .f PInjŽD..

b Proof. Trivial since CrPhŽC .f FlatŽŽ Mod C .., DrPhŽD. f FlatŽŽ Mod D b... Remark 11.11wx 21 . If p.gl.dim C F 1, then we have a q-linear extension of categories

S b 0 ª PhŽC .ª C ª FlatŽ ModŽC ..ª 0EŽ. in the sense ofwx 9 , which represents an element of the second Hochschild᎐Mitchell᎐Baues᎐Wirsching cohomology group

H 2 FlatŽ ModŽC b .., PhŽC ..

This element is zero if the above extension splits and this happens if there b exists a functor T: FlatŽŽ Mod C .. ª C such that ST s Id. If this is true, 1 then ᭙B g C, the short exact sequence 0 ª PE xt wx⌺y, B ª CŽy, B. 0 1 ª PE xt wxy, B ª 0 splits. This implies that PE xt wxy, B is a cohomological functor as a direct summand of CŽŽy, ⌺ B... Hence if P10ª P ª A ª ⌺ŽP110.is a pure triangle with P , P g PProjŽC., then we have a long 1 2 1 exact sequence иии ª P E xt w ⌺ ŽP1 ., B x ª P E xt w ⌺ŽA., B x ª 1 1 PE xt w⌺ŽP0 ., Bx ª иии and this shows that PE xt w⌺ŽA., Bx s 0. Hence B is pure injective. It follows that p.gl.dim C s 0. Hence if C is pure hereditary, then C is pure semisimple iff the extension ŽE.splits. Since the stable homotopy category of spectra C has p.gl.dim C s 1, we have H 2wFlatŽŽ Mod C b.., PhŽC.x /0. In this case all the above results are applied to this situation. In particular we recover the variousŽ topological. versions of the Brown representability theorem. Remark 11.12. If p.gl.dim C F 2, then the projectivization functor S: b SŽ f . C ª FlatŽŽ Mod C ..is surjective on objects. Indeed let 0 ª SŽP2 . ª SŽ g . ⑀ SŽP10.ª SŽP .ª F ª 0 be a projective resolution of the flat functor f ␤ F. If P21ª P ª B ª ⌺ŽP 2.is a triangle in C, then obviously Im SŽg. ( SŽB.and coim SŽg.s SŽ␤ .. Hence im SŽg.: SŽB.¨ SŽP0 .. Since p.p.d B F 1, by Theorem 4.27 it follows that im SŽg.s SŽ␣ .for some ␣ morphism ␣: B ª P00. If B ª P ª A ª ⌺ŽB.is a triangle in C, then obviously F ( SŽA.. In case C is the unbounded derived category DŽ⌳., where ⌳ is a right hereditary ring, then bywx 22 the converse is true. PURITY IN TRIANGULATED CATEGORIES 339

The interpretation of the pure global dimension in terms of properties of the projectivization functor S shows that we have, in general non- reversible, implications:

S is faithful « S is full « S is surjective on objects.

11.3. Homological Functors If p.gl.dim C s 0, then by Theorem 9.3, any homological functor H: C ª G to an abelian category G has a unique exact extension H*: b ModŽC . ª G, such that H*S s H. If the homological functor H preserves coproducts and G is Grothendieck, then without any restriction on p.gl.dim C we have the following consequence of Theorem 8.10, partŽ i. of which was observed also independently by Christensen and Stricklandwx 21 in the Brown case, and by Krausewx 50 in general. PartŽ ii. is due to Krause wx50 with a different proof. Note that Neeman proved in wx 57 that any product preserving homological functor C ª Ab is representable. How- ever, the proof of this general result is much more difficult.

COROLLARY 11.13. Let H: C ª G be a homological functor which preserës coproducts to a Grothendieck category G. Then: Ži.HŽ␾.s 0, for any pure-phantom map ␾ and there exists a unique b coproduct preserïng functor H*: ModŽC . ª G, such that H*S s H. Hence S is the uniërsal coproduct preserïng homological functor out of C to Grothendieck categories. The functor H* is exact and admits a right adjoint b which preserës injectiës, and we can identify H* s ymC b H< C and H s b SŽym.C b H < C . Žii.If G s ModŽ⌳. for a ring ⌳, then H preserës products iff b H ( CŽT, y., where T g C . In this case H* has a left adjoint which preserës projectiës. Proof. PartŽ i. follows from Theorem 8.10. If H: C ª ModŽ⌳. preserves products, then H* preserves limits, so it is representable: H* ( ŽM, y.. Since byŽ i. , H* is exact and preserves colimits, M is a finitely b generated projective functor, hence of the form M s SŽT ., for T g C . Then H ( H*S ( ŽŽS T ., SŽy(..CŽT, y..

b If H: C ª Ab is a homological functor, then Hˆ[ SŽym.C b H < C : C ª Ab is a homological functor preserving coproducts and there is a natural morphism ␾: Hˆª H. It follows by the above result that ␾ is invertible iff H preserves coproducts. Consider now the dual situation. Let op F: C ª Ab be a cohomological functor, and consider the functor F˜[ op ŽŽS y., F < C b .: C ª Ab. It is not difficult to see that F˜ sends coproducts to products and there exists a natural morphism ␺ : F ª F˜˜. However, F is 340 APOSTOLOS BELIGIANNIS not cohomological in general. Indeed F˜ is cohomological iff ␺ is invertible iff F s CŽy, E., where E is pure injective. If for A g C, DŽA.denotes b the category of morphisms Xaaª A with X g C , then HAˆŽ.s HXŽ.FA˜Ž.FXŽ. lim ª DŽ A. a and s lim ¤ DŽ A. a are the functors considered by Margoliswx 51 . If BRT holds, then the next corollary, which gives a trivial proof to a result of Adamswx 51 , shows that F˜ satisfies a weak exact condition. fg COROLLARY 11.14. If p.gl.dim C F 1 and P ª A ª B ª ⌺ŽP. is a triangle in C, where P is pure projecti¨e, then the sequence F˜˜Ž B.ª FAŽ.ª F˜Ž P. is exact. 0 Proof. By BRT F < C b s SŽC.. Hence F˜s wSŽy., SŽC.x s PE xt Žy, C.. Consider the complex Ž).: wSŽB., SŽC.xwª SŽA., SŽC.xwª SŽP., SŽC.x. Let ␣: SŽA.ª SŽC.be such that SŽf .( ␣ s 0. By BRT, ␣ s SŽ␤ .. Then f ( ␤ is pure phantom. Since P is pure projective, f ( ␤ s 0. Hence ␤ s g (␥, where ␥ : B ª C. Then ␣ factors through ŽŽS g., SŽC.. and this means that the complex Ž).is exact. Remark 11.15. Suppose C is monogenic wx38 ; i.e., C admits a compact object T such that C coincides with the localizing subcategory generated by T, for example a tilting complexwx 62 in the derived category of a ring or the sphere spectrumwx 51 in the stable homotopy category. The relative homological algebra based on the proper class EŽX ., where X s n addÄ⌺ ŽT .; n g ޚ4, equivalently on the proper class EŽH .where H s CŽT, y., can be considered as the ‘‘absolute homological algebra’’ of C and is studied inwx 14 ; see also Sections 12.4 and 12.5. The pure and the absolute theory are related by a Butler᎐Horrocks spectral sequencewx 18 ; seewx 14 . Remark 11.16. Assume that C has all small coproducts, and let X be an ␣-localizing subcategory of C in the sense ofwx 57 , where ␣ is an infinite cardinal. Then the Steenrod category of C with respect to EŽX .is op op the category E xŽX , Ab. of all functors X ª Ab converting coproducts of fewer than ␣ objects of X to products in Ab. Seewx 57 for a detailed analysis of this situation which can be regarded as a higher analogue of the present theory, which in turn corresponds to the case ␣ s / 0.

11.4. Pure Global Dimension under Smashing and Finite Localization Fix a compactly generated triangulated category C. We recall that a localizing subcategory L of C is called smashing if the inclusion functor i: L ¨ C has a right adjoint which preserves coproducts; equivalently the quotient functor ␲ : C ª CrL has aŽ fully faithful. right adjoint ␶ : CrL ª C which preserves coproducts. It is easy to seewx 50 that if L : C is smashing, then the Verdier quotient CrL is compactly generated and PURITY IN TRIANGULATED CATEGORIES 341

b bb the functor ␲ sends C to ŽCrL ., so ␲ induces an exact functor ␲ : b b C ª ŽCrL .. Since ␲ is exact and preserves coproducts, by Corollary U 11.13 orwx 50 , there exists a unique exact colimit preserving functor ␲ b : Ž b.ŽŽ.b. ␲ U ␲ Mod C ª Mod CrL such that b SC s SC r L , where SC , SC r L U are the projectivization functors. Observe that by the above relation, ␲ b preserves flat functors and moreover projective functors, since ␲ preserves compact objects. Similarly since ␶ is exact and preserves coproducts, there U b exists a unique exact colimit preserving functor ␶ b : ModŽŽCrL . . ª Ž b. ␶ ␶ U ␲␶ Mod C such that SC s b SC r L . Since s IdC r L , the above proper- UU ␲␶ b S ŽY . ties imply that bb( IdModŽŽ C r L .. . If lim ª C r L j is a flat functor Ž.b ␶ UU␶ ŽŽS Y .. over CrL , then since bbpreserves filtered colimits, lim ª C r L j ␶ U Ž.␶ Ž.␶ Ž. s lim b SC r L Yj ( lim SC Yj . Since SC Yj are flat functors and ª ª U a filtered colimit of flat functors is flat, it follows that ␶ b preserves flatness.

THEOREM 11.17.Ž 1. If L is a smashing subcategory of C, then p.gl.dim CrL F p.gl.dim C. Ž2. If L is a localizing subcategory of C which is generated by a set of compact objects from C, so L is compactly generated and smashing, then maxÄ p.gl.dim L , p.gl.dim CrL 4F p.gl.dim C.

b ⅷ U Proof. Ž1. Let F be a flat functor over ŽCrL .. Let P ª ␶ b ŽF.be a U b U projective resolution of the flat functor ␶ bbŽF.in ModŽC .. Since ␲ is U ⅷ UU exact and preserves projectives, ␲ bbbŽ P . ª ␲␶ ŽF.( F is a projective b resolution of F in ModŽŽCrL . . and the assertion follows. The second part is proved in a similar way. The following consequence is due to Hovey et al.wx 38 .

COROLLARY 11.18. Smashing or finite localizations of Brown, resp. phantomless categories are Brown, resp. phantomless.

11.5. Compactly Generated Triangulated Categories of Finite Type By Theorem 9.3 a compactly generated triangulated category C is pure semisimple iff its pure Steenrod category ModŽC b. is locally Noetherian or perfect. The pure homological theory of module categorieswx 39, 65 suggests the following definition.

DEFINITION 11.19. A compactly generated triangulated category C is said to be of finite type if its pure Steenrod category ModŽC b. is locally finite. Categories of finite type can be regarded as the ‘‘representation-finite’’ triangulated categories. For a justification of this statement we refer to the next section. 342 APOSTOLOS BELIGIANNIS

COROLLARY 11.20. If L is a smashing subcategory of C and C is of finite type, then so is CrL. If L is a localizing subcategory of C generated by a set of compact objects from C and C is of finite type, then so are L , CrL. b Proof. If L is smashing then bywx 50 , ModŽŽCrL . . is a localized quotient of ModŽC b.. Since C is of finite type, ModŽC b. is locally finite. b Hence bywx 59 , ModŽŽCrL . . is locally finite; i.e., CrL is of finite type. The second part is left to the reader. The pure homological theory of module categories suggests the following

QUESTION 11.21. Let C be a pure semisimple compactly generated triangulated category. Is it true that C is of finite type? A positive answer to Question 11.21 would follow from an affirmative answer to the following problem about Grothendieck categories posed by Rooswx 64 almost 30 years ago. Roos’s Problem 11.22. Let G be a locally Noetherian Grothendieck category in which any injective object is projective. Is it true that G is locally finite? Indeed if C is pure semisimple, then the module category ModŽC b. is a locally Noetherian Frobenius Grothendieck category. If Roos’s problem has an affirmative answer, then ModŽC b. is locally finite, so C is of finite type. In Section 12 we shall see that in general both Question 11.21 and Roos’s problem have a negative answer.

11.6. Krull᎐Gabriel Dimension and Dual Pure Global Dimension We close this section by presenting a characterization of categories of finite type, using two new dimensions which are invariant under triangle equivalence. Define the dual pure global dimension p.gl.dim C op of C as the supre- mum of the projective dimensions in ModŽŽC b.op . of the homological b functors C ª Ab:

op op b p.gl.dim C [ sup½ p.d F ¬ F g Flatž ModŽŽC ../5.

Note that p.gl.dim C op is not the pure global dimension of C op, since C op is never compactly generatedwx 57 . However, the above definition is reasonable, since if DŽ⌳.is the unbounded derived category of a ring ⌳, then op from the results of the next section it follows that p.gl.dim DŽ⌳. s p.gl.dim DŽ⌳op .. Similarly if ModŽD.is the stable category of a locally op finite Frobenius module category ModŽD., then p.gl.dim ModŽD.s p.gl.dim ModŽ Dop .. Finally if C is the stable homotopy category, then PURITY IN TRIANGULATED CATEGORIES 343

b b op using the Spanier᎐Whitehead duality functor D: C ªf ŽC . wx51 , it op follows that p.gl.dim C s p.gl.dim C s 1. More generally if T, S are b op b compactly generated and there exists an equivalence ŽT . f S , then op op p.gl.dim T s p.gl.dim S and p.gl.dim S s p.gl.dim T. Finally let KGdim C b be the Krull᎐Gabriel dimension wx32, 49 , of the b b abelian category AŽC .s modŽC .. Note that this dimension is the finitely presented version of the Krull dimension Kdim ModŽC b. of b b b ModŽC . defined by Gabrielwx 30 . By wx 49 , Kdim ModŽC . F KGdim C with equality if C is pure semisimple. It seems that the Krull᎐Gabriel dimension is more stable than the pure global dimensions and deserves b b op further study. For instance it is symmetric: KGdim C s KGdimŽC . ; seewx 32 . We denote this common value by KGdim C and by abuse of language we call it the Krull᎐Gabriel dimension of C. Note that bywx 32 it op follows easily that maxÄ p.gl.dim C , p.gl.dim C4 F KGdim C. The next result shows that Question 11.21 has an affirmative answer if op the implication p.gl.dim C s 0 « p.gl.dim C s 0 is true. op PROPOSITION 11.23.Ž 1. p.gl.dim C s 0 iff the Steenrod category ModŽC b. is locally Artinian. More generally assume that for any compact b b object X, any decreasing filtered family of subobjects of C Žy, X. in AŽC . op contains a cofinal subfamily of cardinality / t , t Gy1. Then p.gl.dim C F t q 1. Ž2. The following statements are equi¨alent: Ži. C is of finite type. op Žii. p.gl.dim C s 0 s p.gl.dim C. Žiii. KGdim C s 0. Proof. PartŽ 1. follows easily from results of Gruson and Jensen wx32 and Simson wx 65 . PartŽ 2. follows from partŽ 1. and Proposition 9.2, b op b op using the well-known duality modŽC . f modŽŽC . . given by bb Coker C Žy, f .¬ Ker C Žf, y.. QUESTION 11.24. What is the Krull᎐Gabriel dimension of a Brown category? In particular, what is KGdim C, if C is the stable homotopy category? Certainly KGdim C G 1 if C is Brown. It seems reasonable to conjecture that KGdim C s 1.

12. APPLICATIONS TO DERIVED AND STABLE CATEGORIES

In this section we apply our previous resultsŽ mainly about purity. to derived categories of rings and to stable module categories of quasi- Frobenius rings. 344 APOSTOLOS BELIGIANNIS

12.1. Deri¨ed Categories Let ⌳ be an associative ringŽ with 1 always. . We denote by DŽ⌳.the unbounded derived category of the abelian category ModŽ⌳.of all right ⌳-modules. We denote also by ProjŽ⌳.the full subcategory of all projec- tiuve modules and by P⌳ the full subcategory of finitely generated projec- b tive modules in ModŽ⌳.. Finally we denote by H ŽP⌳ .the bounded b homotopy category of P⌳, and by D ŽŽmod ⌳.. the bounded derived category of finitely presented modules. It is well knownwx 62 that DŽ⌳.is a b compactly generated triangulated category and H ŽP⌳ .is, up to equiva- b lence, the full subcategory of compact objects of DŽ⌳., i.e., DŽ⌳.f b H ŽP⌳ .. For the notion of pure semisimplicity in module categories, we refer towx 39, 65 . Our first result is a consequence of Corollary 11.3.

PROPOSITION 12.1. If <<⌳ F / t , for some t G 0, then p.gl.dim DŽ⌳.F t q 1. n EXAMPLE 12.2. Let DŽQcސ Žk.. be the derived category of quasi- coherent sheaves on the projective n-space over the field k. By Beilinson’s n descriptionwx 10 of the category DŽQcސ Žk.., it follows that if <

COROLLARY 12.4. Let ⌳ be a countable ring. If ⌳ is not right Artinian or right perfect, or r.gl.dim ⌳ s ϱ, then p.gl.dim DŽ⌳.s 1. In particular p.gl.dim DŽޚ.s 1. We recall that a ring ⌳ is called representation-finite if ⌳ is right Artinian and the set of isoclasses of indecomposable finitely presented right modules is finite. It is well known that the representation-finite commutative rings are exactly the Artinian principal ideal rings. Proposi- tion 12.3 implies easily the following.

COROLLARY 12.5. Let ⌳ be a commutati¨e ring. Then p.gl.dim DŽ⌳.s 0 iff ⌳ is a finite product of fields.

EXAMPLE 12.6. Let H be the category of commutative cocommutative connected graded Hopf algebras over a field k with charŽk.s p )0, i generated by elements of degrees 2 p , i G 0. Bywx 66 , H is a Grothendieck Žfunctor. category which is hereditary and pure semisimple. As in Proposi- tion 12.3, DŽH.is pure semisimple.

Remark 12.7. If p.gl.dim DŽ⌳.s 0, then it is not true that ⌳ is right hereditary. Indeed let ⌫ be a representation-finite piecewise hereditary algebra over an algebraically closed field, with gl.dim ⌫ G 2wx 33 . Then there exists a representation-finite hereditary algebra ⌳ and a derived equivalence 346 APOSTOLOS BELIGIANNIS

DŽ⌳.f DŽ⌫.. By Proposition 12.3, ⌫ is right pure semisimple but not hereditary. The author is indebted to the referee for the nice proof of the following result.

PROPOSITION 12.8. If ⌳ is a right hereditary ring, then

r.p.gl.dim ⌳ s p.gl.dim DŽ⌳..

Proof. By a result of Neemanwx 54 , any complex Aⅷ in DŽ⌳.is isomor- ⅷ ⌺yn Ž nŽ ⅷ .. phic to the coproduct of its cohomology groups: A s [ng ޚ HA. Hence it suffices to show that p.p.d A s p.p.d Aⅷwx0,᭙A g ModŽ⌳.. b Let Y: ModŽ⌳.ª ModŽH ŽP⌳ .. be the composition of the embedding ModŽ⌳.¨ DŽ⌳.given by A ¬ Aⅷwx0 , with the projectivization functor S: b 0 DŽ⌳.ª ModŽH ŽP⌳ ... The cohomology functor H : DŽ⌳.ª ModŽ⌳. preserves coproducts and kills pure phantom maps. Hence it can be extended uniquely to an exact colimit preserving functor H˜ 0 : b 0 ModŽŽH P⌳ ..ª ModŽ⌳., which is given by F ¬ FŽ⌳. and satisfies H˜ Y s IdModŽ⌳. . It is not difficult to see that Y commutes with filtered colimits. Using the well-known fact that a short exact sequence in a module category is pure iff it is a filtered colimit of split short exact sequences, it follows that Y preserves pure exactness. Since ⌳ is right hereditary, hence right coherent, it follows trivially that Y takes pure projective modules to projective functors. Hence the image under Y of a pure projective resolu- b tion of A is a projective resolution of YŽA.in ModŽH ŽP⌳ ... The latter can be lifted to a pure projective resolution of Aⅷwx0 in DŽ⌳.. Conversely H˜0 preserves pure exactness, since it is exact and colimit preserving. Since ⌳ is right coherent as a right hereditary ring, it follows that the cohomol- 00b 0 ogy functor H induces a functor H : H ŽP⌳ .ª modŽ⌳.. Since H˜ is 00 0 exact and preserves coproducts and H˜ S s H , it follows easily that H˜ preserves pure projectivity. Now any pure projective resolution of Aⅷwx0 in b DŽ⌳.induces via S aŽ pure. projective resolution of YŽA.in ModŽH ŽP⌳ .., which in turn induces via H˜0, a pure projective resolution of A in ModŽ⌳.. The above observations about pure projective resolutions in ModŽ⌳. and in DŽ⌳.imply trivially that p.p.d A s p.p.d Aⅷwx0. PROPOSITION 12.9Ž see alsowx 56. . Let ⌳ be a right coherent ring and assume that DŽ⌳.is a Brown category. Then for any flat right ⌳-module F, we ha¨e p.d F F 1. If moreo¨er ⌳ has finite weak global dimension, then r.p.gl.dim ⌳ F 1. Proof. Let A be a right ⌳-module, and consider the stalk complex ⅷ ⅷ ⅷ ⅷ f ⅷ g ⅷ A wx0 . By BRT there exists a pure triangleŽ †. : P10ª P ª A wx0 PURITY IN TRIANGULATED CATEGORIES 347

ⅷ h ⅷ ⅷⅷ b ª ⌺Ž P110., such that P , P are in AddŽH ŽP⌳ ... Applying the cohomol- 0 ogy functor H , we deduce a short exact sequenceŽ ††. : 0 ª K ª L ª A ª 0, where K, L are direct summands of coproducts of cohomology b modules of complexes in H ŽP⌳ .. Since ⌳ is right coherent, these coho- mologies are sums of finitely presented modules. This implies that K, L are pure projective modules. If A is flat, thenŽ ††. is pure; hence p.p.d A F 1. But then p.d A F 1, since p.d A s p.p.d A for A flat. If w.gl.dim ⌳ -ϱ, then it suffices to prove thatŽ ††. is pure. Let M be a finitely presented right ⌳-module and ␣: M ª A a morphism. Since ⌳ is right coherent, w.gl.dim ⌳ -ϱ, and M is finitely presented, it follows that M ⅷwx0 is compact. Since hⅷ is pure phantom, the composition M ⅷwx0 ⅷ ⅷ ␣ w0x ⅷ h ⅷ ⅷⅷ ⅷ ª A wx0 ª ⌺Ž P1 . is zero. So there exists a map ␤ : M wx0 ª P0 ⅷⅷ ⅷ 00ⅷ 0 ⅷ such that ␤ ( g s ␣ wx0 . Applying H , we have H Ž ␤ .( HgŽ . s ␣. Hence ␣ factors through L, soŽ ††. is pure. Remark 12.10. It is not difficult to see that if ⌳ is a right coherent ring with w.gl.dim ⌳ -ϱ, then r.p.gl.dim ⌳ F p.gl.dim DŽ⌳.. We refer to the work of Christensen et al.wx 22 for details and a thorough discussion of related topics. In particular inwx 22 it is proved that the above inequality can be strict. Since the pure global dimension coincides with the global dimension for a von Neumann regular ringŽ obviously coherent.wx 39 , we have the following consequence.

COROLLARY 12.11. Let ⌳ be a ¨on Neumann regular ring. Then p.gl.dim DŽ⌳.F 1 iff ⌳ is right hereditary. Moreo¨er p.gl.dim DŽ⌳.s 0 iff ⌳ is semisimple Artinian. By Theorem 3.2 ofwx 7 , we have the following.

COROLLARY 12.12. Let ⌳ s k² X: be the free algebra o¨er a set of ¨ariables X, with<< X F / 00. If <<⌳ s maxÄ/ , <

EXAMPLE 12.14. There exists a ring ⌳ with the following properties: Ži. p.gl.dim DŽ⌳./p.gl.dim DŽ⌳op .. Žii.DŽ⌳.is a Brown category, but DŽ⌳op . is not. Indeed Kaplansky inwx 41 constructed a von Neumann regular ring ⌳ of cardinality <<⌳ s 2/ 0 with r.gl.dim ⌳ s 1 and l.gl.dim ⌳ G 2. By Corollary op 12.11, p.gl.dim DŽ⌳.s 1 and by Remark 12.10, p.gl.dim DŽ⌳ . G 2. Hence by Theorem 11.8, DŽ⌳.is Brown and DŽ⌳op . is not. Assuming continuum / 0 op hypothesis 2 s /1, it follows from Proposition 12.1 that p.gl.dim DŽ⌳ . s 2.

12.2. The Deri¨ed Pure Semisimple Conjecture, Categories of Finite Type, and Roos’s Problem The next result shows that finite type is a symmetric condition for derived categories. PROPOSITION 12.15. The following are equi¨alent: Ži.DŽ⌳.is of finite type. Žii.DŽ⌳op . is of finite type. Žiii.DŽ⌳.and DŽ⌳op . are pure-semisimple. Živ. KGdim DŽ⌳.s 0. op b op Proof. The natural duality P⌳⌳f P op extends to a duality H ŽP ⌳.f b b op op b H ŽP⌳op ., so ŽŽD ⌳.. f DŽ⌳ . . Using the notation of Section 11, we op op have p.gl.dim DŽ⌳. s p.gl.dim DŽ⌳ . and the assertion follows from Proposition 11.23. PURITY IN TRIANGULATED CATEGORIES 349

COROLLARY 12.16.Ž 1.If DŽ⌳.is of finite type, then ⌳ is representation-finite of finite global dimension. Ž2.If ⌳ isŽ deriëd equiälent to. a representation-finite right hereditary ring, then DŽ⌳.is of finite type. Ž3. Let ⌳ be a PI-ring or a ring with a Morita duality, for instance, an Artin algebra or a quasi-Frobenius ring. If DŽ⌳.is pure semisimple, then ⌳ is of finite representation type and of finite global dimension. Ž4. If ⌳ is a representation-infinite Artin algebra of finite global dimension, then KGdim DŽ⌳./1. If ⌳ is deriëd equiälent to a hereditary algebra, then KGdim DŽ⌳./1, independently of the representation type. Proof. By a well-known result of Auslanderwx 39 , a ring is representation-finite iff it is left and right pure semisimple. ThenŽ 1. andŽ 2. follow from Propositions 12.3 and 12.15.Ž 3. follows from results of Simson, Herzogwx 35 claiming that right pure semisimple rings with polynomial identity or with Morita duality are representation-finite. Ž4. Since ⌳ has finite global dimension, the canonical full embed- b ding modŽ⌳.¨ DŽ⌳.has its image in H ŽP⌳ .. As in the proof of Proposition 12.3, it follows that ModŽŽ mod ⌳.. is a localized quotient of b ModŽ H ŽP⌳ ...ItfollowsfromresultsofGabriel30thatwx b Kdim ModŽŽ mod ⌳..F Kdim ModŽH ŽP⌳ ..F GKdim DŽ⌳.. Assume that GKdim DŽ⌳.s 1, so Kdim ModŽŽ mod ⌳..F 1. Since Kdim ModŽŽ mod ⌳.. /1 for any Artin algebraŽ seewx 36, 47. , it follows that Kdim ModŽŽ mod ⌳.. s 0. Then by a result of Auslanderwx 2 , it follows that ⌳ is representation- finite and this is not the case. Hence GKdim DŽ⌳./1. If ⌳ is derived equivalent to a hereditary algebra ⌫ and GKdim DŽ⌳.s 1, then GKdim DŽ⌫.s 1 and as above we deduce that ⌫ is representation finite. Hence byŽ 2. , DŽ⌳. is of finite type. Then by Proposition 12.15, GKdim DŽ⌳.s 0 and this is not true. Hence GKdim DŽ⌳./1. PartŽ 4. of the above corollary is a partial analogue of a module-theoretic result of Krausewx 47 and Herzog wx 36 . We do not know if for any Artin algebra ⌳ it holds that GKdim DŽ⌳./1. We recall that the still open pure semisimple conjecture asserts that a right pure semisimple ring ⌳ is left pure semisimple; equivalently ⌳ is representation-finite. We formulate an analogue of this conjecture in the derived category.

Deri¨ed Pure Semisimple Conjecture 12.17. Pure semisimplicity in DŽ⌳. op is left᎐right hand symmetric: p.gl.dim DŽ⌳.s 0 m p.gl.dim DŽ⌳ . s 0. Equivalently if DŽ⌳.is pure semisimple then DŽ⌳.is of finite type. Remark 12.18. The truth of the derived pure semisimple conjecture ŽDPSC. implies the truth of the pure semisimple conjectureŽ PSC. . Indeed 350 APOSTOLOS BELIGIANNIS if DPSC is true and PSC is not, then bywx 35 , there exists a right hereditary right pure semisimple ring which is not left pure semisimple. By Proposi- tion 12.3, DŽ⌳.is pure semisimple; hence also DŽ⌳op . is pure semisimple. By Proposition 12.3 again, ⌳ is left pure semisimple, which is not true. Hence PSC is true. Note that by Proposition 12.3, forŽ left and right. hereditary rings, DPSC is equivalent to PSC.

EXAMPLE 12.19. The following example provides a negatië answer to Roos’s Problem 11.22, to Question 11.21, and to the derived pure semisimple conjecture for ‘‘rings with several objects’’ in the sense of Mitchellwx 53 . Consider / t , t G 0 as a totally ordered set, let k be a field of cardinality / 0 , and let w/ t , ModŽk.x be the category of k-linear representations of / t. Let kŽ/ t .be the ringoid which yields an equivalence w/ t , ModŽk.x f op op ModŽŽk / tt. .. By a result of Brunewx 17 , ModŽŽk / . . is hereditary and pure semisimple and p.gl.dim ModŽŽk / t .. s t q 1 if t -/ 00and ϱ if t G / . As in Proposition 12.3, it follows that theŽ compactly generated. derived op category DŽŽŽMod k / t ... is pure semisimple but the derived category DŽŽŽMod k / t ... is not. Using Proposition 12.15 we infer that op DŽŽŽMod k / t ... is pure semisimple but not of finite type. op On the other hand since DŽŽŽMod k / t ... is pure semisimple, we have op b that the Grothendieck category ModŽŽD Mod ŽŽk / t ... . is Frobenius but op not locally finite since otherwise, DŽŽŽMod k / t ... would be of finite type. Note that by the Faith᎐Walker theoremwx 27 , the answer to Roos’s problem is affirmative for rings. For the representation-theoretic notions and arguments used in the following resultŽ which shows that for Artin algebras, DPSC is true. , we refer towx 33 . THEOREM 12.20. For an Artin algebra ⌳ the following are equiälent: Ži.DŽ⌳.is of finite type. Žii.DŽ⌳.is pure semisimple. Žiii. ⌳ is an iterated tilted algebra of Dynkin type. Živ. ⌳ is deriëd equiälent to a representation finite hereditary algebra. In this case, the set of isoclasses of indecomposable compact objects of DŽ⌳. up to shift is finite.

Proof. By Proposition 12.15,Ž i.« Žii. and bywx 33 ,Ž iii.« Živ. . By Corol- lary 12.16,Ž iv.« Ži. . So it remains to prove thatŽ ii. impliesŽ iii. . Žii.« Žiii. By Corollary 12.16, ⌳ is a representation finite algebra of finite global dimension. Let ⌳ˆ be the repetitive algebra of ⌳. Bywx 33 we bb have a triangle equivalence H ŽP⌳ .s D ŽŽmod ⌳..( modŽ⌳ˆ .. We shall show that ⌳ˆ is locally representation-finite. Let P⌳ˆ be the category of PURITY IN TRIANGULATED CATEGORIES 351 projective objects in modŽ⌳ˆˆ.. Since modŽ⌳.is a Frobenius category, by

Proposition 9.2, P⌳ˆ is a weak abelian category. Since modŽ⌳ˆ .is a length categoryŽ any object has finite length. , the module category ModŽP⌳ˆ .is locally Noetherian; hence it is perfect according to Proposition 9.2. Since b H ŽP⌳ .( modŽ⌳ˆˆ., we have that ModŽŽ mod ⌳.. is also perfect. By a result of Simsonwx 66 , since ModŽŽ mod ⌳ˆ ..and ModŽP⌳ˆ . are perfect, the module category ModŽŽ mod ⌳ˆ .. is also perfect or equivalently locally Noetherian. Let ⌫ : ⌳ˆ be a finite full convex subcategory. Then we have an inclusion modŽ⌫.: modŽ⌳ˆ .. This inclusion implies that ModŽŽ mod ⌫.. is a localizing subcategory of ModŽŽ mod ⌳ˆ ... Since the latter is a locally Noetherian category, so is ModŽŽ mod ⌫... But since ⌫ is an Artin algebra, by a well-known result of Auslanderwx 39 , we have that ⌫ is representation-finite. Hence ⌳ˆ is locally representation-finite. Bywx 33 , ⌳ is an iterated tilted algebra of Dynkin type. The last claim is trivial and is left to the reader.

The next corollaries follow fromwx 7, 8 and the above results.

COROLLARY 12.21. Let ⌳ be a finite dimensional hereditary k-algebra o¨er an algebraically closed field k of cardinality / t. Then we ha¨e the following:

Ži.If t s 0, then p.gl.dim DŽ⌳.s 0 or 1.Ž 0 occurs iff ⌳ is representation-finite.. Žii. If t )0, then: Ž␣ .If ⌳ is representation-finite, then p.gl.dim DŽ⌳.s 0. Ž␤ .If ⌳ is tame, then p.gl.dim DŽ⌳.s 2. Ž␥ .If ⌳ is wild, then p.gl.dim DŽ⌳.s t q 1.

COROLLARY 12.22. Let Q be a quiër with or without oriented cycles and let ⌳ s kwxQ be its quiër-algebra oër an algebraically closed field k of cardinality / t. Then:

Ži.If t s 0, then p.gl.dim DŽ⌳.s 0 or 1, the ¨alue 0 occurs iff Q is Dynkin. Žii. If t )0, then: Ž␣ .If Q is a Dynkin diagram, then p.gl.dim DŽ⌳.s 0. Ž␤ .If Q is an oriented cycle, then p.gl.dim DŽ⌳.s 1. Ž␥ . If Q is extended Dynkin not an oriented cycle, then p.gl.dim DŽ⌳.s 2. Ž␦ . In all the remaining cases, Q is a wild qui¨er and p.gl.dim DŽ⌳.s t q 1. 352 APOSTOLOS BELIGIANNIS

COROLLARY 12.23. Let ⌳ be a finite dimensional local or radical squared zero k-algebra o¨er an algebraically closed field of uncountable cardinality / t. If DŽ⌳.is Brown, then ⌳ is representation-finite.

12.3. Stable Module Caegories Let C be a skeletally small additive category such that the module category ModŽC.is Frobenius. It is not difficult to see that the stable b category ModŽC.is compactly generated and ModŽC.s modŽC.. If ModŽC op . is also Frobenius, then the Auslander᎐Bridger transpose duality op op b op functor Tr: modŽC.ªf modŽC ., seewx 3 , shows thatŽŽ Mod C.. op b f ModŽC . . PROPOSITION 12.24. p.gl.dim ModŽC.s p.gl.dim ModŽC.. In particular if wŽC.F / t , for some t G 0, then p.gl.dim ModŽC.F t q 1. Proof. Since any projective right C-module is pure projective, it follows trivially that M g ModŽC.is pure projective iff M is pure projective. Similarly it is not difficult to see that a short exact sequence 0 ª K ª L ª M ª 0 in ModŽC.is pure iff the induced triangle K ª L ª M ª ⌺ŽK . is pure in ModŽC.; further any pure triangle in ModŽC.arises in this way. It follows that a pure projective resolution of M induces a pure projective resolution of M and any pure projective resolution of M in the stable category arises in this way. This implies that p.p.d M s p.p.d M, ᭙M g ModŽC.. Hence we have p.gl.dim ModŽC.s p.gl.dim ModŽC.. THEOREM 12.25. If ModŽC.is Frobenius then the following are equiä- lent: Ži. ModŽC.is pure semisimpleŽ of finite type.. Žii. ModŽC.is pure semisimpleŽ and locally Artinian.. Žiii. ModŽŽ mod C..is locally Noetherian or perfectŽ and locally Ar- tinian.. If ModŽC.is of finite type, then ModŽC op .is Frobenius and ModŽC op . is of finite type. Finally if one of the aboë equiälent conditions is true, then:

ModŽC .( ProjŽ ModŽ modŽC ...s FlatŽ ModŽ modŽC ... s InjŽ ModŽ modŽC ....

Proof. By Theorem 9.3,Ž i.m Žiii. and by Proposition 12.24,Ž i.m Žii. . The easy proof of the remaining assertions and the parenthetical case is left to the reader. PURITY IN TRIANGULATED CATEGORIES 353

If ⌳ is a QF-ring, then ModŽ⌳.is Frobenius. Hence ModŽ⌳.is a b compactly generated triangulated category and ModŽ⌳.s modŽ⌳.. The following consequence of Theorem 12.25 generalizes a result ofwx 13, 48 . COROLLARY 12.26. Let ⌳ be a QF-ring. Then the following are equi¨a- lent: op Ži. ModŽ⌳., equi¨alently ModŽ⌳ ., is pure semisimpleŽ of finite type.. Žii.The set of isoclasses of indecomposable compact objects of ModŽ⌳. is finite. Žiii. ⌳ is representation finite. If ⌳ is an Artin algebra, we denote by TŽ⌳.s ⌳ h DŽ⌳.the trivial extension of ⌳ by the minimal injective cogeneratorwx 33 . The following consequence of Theorems 12.20 and 12.25 shows that the finite type property coincides for the three triangulated categories DŽ⌳., ModŽŽT ⌳.., ModŽ⌳ˆ ., associated to ⌳.

COROLLARY 12.27. For an Artin algebra ⌳ the following are equiälent. Ži.The deriëd category DŽ⌳. of ⌳ is of finite type. Žii.The stable module category ModŽŽT ⌳.. is of finite type. Žiii.The stable module category ModŽ⌳ˆˆ.of its repetitië algebra ⌳ is of finite type. Živ. ⌳ is an iterated tilted algebra of Dynkin type.

COROLLARY 12.28. Let ⌳ be a finite dimensional self-injectië local k-algebra, oër an algebraically closed field k. Then the following statements are equiälent: Ži. ModŽ⌳.is a Brown category, i.e., p.gl.dim ModŽ⌳.F 1. Žii. Either ⌳ is representation finite or else k is a countable field. Proof. ThatŽ ii. impliesŽ i. follows from Proposition 12.24. Suppose that ModŽ⌳.is Brown. If p.gl.dim ModŽ⌳.s 0, then ⌳ is representation-finite. If p.gl.dim ModŽ⌳.s 1, then by Proposition 12.24, r.p.gl.dim ⌳ s 1. Then bywx 8 , k is countable. Using induction on the p-Sylow subgroups, we have the following consequence first observed by Benson and Gnacadja. COROLLARY 12.29wx 15 . Let ⌳ s kG be the group algebra of a finite group Goër an algebraically closed field k with charŽk.s p. Then the following are equiälent: Ži. ModŽ⌳.is a Brown category, i.e., p.gl.dim ModŽ⌳.F 1. Žii. Either G has cyclic p-Sylow subgroups or else k is a countable field. 354 APOSTOLOS BELIGIANNIS

If ⌳ and ⌫ are derived equivalent rings, then p.gl.dim DŽ⌳.s p.gl.dim DŽ⌫.. If in addition ⌳, ⌫ are self-injective Artin algebras, then by wx63 , there exists a triangle equivalence ModŽ⌳.f ModŽ⌫.. Hence p.gl.dim ModŽ⌳.s p.gl.dim ModŽ⌫.. Finally if ⌳ is an Artin algebra of finite global dimension, then by Happel’s theoremwx 33 , there exists a triangle equivalence DŽ⌳.f ModŽ⌳ˆˆ., where ⌳ is the repetitive algebra. Since ModŽ⌳ˆ .is Frobenius, in case gl.dim ⌳ -ϱ, we have p.gl.dim DŽ⌳.s p.gl.dim ModŽ⌳ˆ .. It is interesting to have characterizations in representation-theoretic terms of Artin algebras, resp. self-injective Artin algebras, with Brown derived categories, resp. stable module categories. If the derivedŽ resp. stable module. category of a ringŽ resp. QF-ring. is a Brown category, then the results of Sections 10 and 11 are true for the triangulated category DŽ⌳.ŽŽresp. Mod ⌳... We leave the formulations of these results to the reader. Remark 12.30. The results of this section indicate that finite type is the appropriate notion of ‘‘representation-finiteness’’ for derived and stable categories. It is unclear what are the appropriate notions for tameness and wildness in these categories in analogy with the tame᎐wild dichotomy of finite dimensional algebras.

12.4. Localization and the Deri¨ed Category Let ⌳ be an associative ring. We denote by HŽ⌳.be the unbounded homotopy category of all right ⌳-modules and by HŽŽProj ⌳.. the homotopy category of all projective right ⌳-modules.

DEFINITION 12.31. A complex P ⅷ is called a Cartan᎐Eilenberg projecti¨e complexŽ CE-projective complex, for short. , if P ⅷ is homotopy equivalent to a complex having projective components and zero differential. The full subcategory of HŽ⌳.of all CE-projective complexes is denoted by PCE. Setting E [ EŽP CE ., we obtain a proper class of triangles in HŽ⌳., and HŽ⌳.has enough projectives and PŽE.s PCE Žsee for instance n wx23, 26. . Choosing C s HŽ⌳.and X s Ä⌺⌳Ž.¬ n g ޚ4 in Theorem 6.5 and keeping the terminology and notation of Section 6, we have the following result proved inwx 16, 42, 67, 71 , by using essentially the natural closed model structure of the category of complexes.

$ϱ THEOREM 12.32.Ž 1.PŽE .* is the full subcategory HP Ž⌳.of HŽ⌳. consisting of all complexes Aⅷ which are homotopy equi¨alent to complexes P ⅷ with the following property. There exists a tower of complexes

ⅷ ⅷ иии ⅷ ⅷ иии ⅷ 0 : P01: P : : Pnn: P q1 : : P PURITY IN TRIANGULATED CATEGORIES 355 such that:

Ž. ⅷⅷ ⅷⅷ † P s DnG 0 Pnnn, each inclusion P : Pq1 splits in each com- ⅷ ⅷ ponent, and each quotient complex Pnq 1rPn is a CE-projecti¨e complex. Ž2.DE ŽŽH ⌳..s DŽ⌳., the quotient functor HŽ⌳.ª DŽ⌳. admits a fully faithful left adjoint, and AE ŽŽH ⌳..is the subcategory HAcŽ⌳. of acyclic complexes.

Ž3.The canonical functor HP Ž⌳.ª DŽ⌳.is a triangle equi¨alence.

Ž4.The inclusion HAcŽ⌳.¨ HŽ⌳.has a left adjoint a, the inclusion ⅷ HP Ž⌳.¨ HŽ⌳.has a right adjoint p, and any complex A is contained in a triangle

pŽAⅷ .ª Aⅷ ª aŽAⅷ .ª ⌺ŽpŽAⅷ .., where

ⅷ ⅷ pŽA .g HP Ž⌳.and aŽA .g HAc Ž⌳..

Ž5.Any additi¨e functor F: ModŽ⌳.ª ModŽ⌫.has a total left deri¨ed functor

L F : DŽ⌳.ª DŽ⌫..

It is clear from Section 6 that the above theorem holds in much more general situations, for instance, in the stable homotopy category, and in the derived category of a Grothendieck category with projectives, where as noted inwx 16, 42, 67, 71 , the theory implies the existence of total left derived functors defined on unbounded complexes. The dual result concerning injectives and total right derived functors also holds, and can be applied for instance to the derived category of any Grothendieck category, e.g., to categories of sheaves, since the theory of the previous sections can be dualizedŽ compactness can be avoided using the cohomology functor as in Remark 6.6. . The main theorem of Section 6 also covers the construction of the various derived categories associated with projective classes inwx 20 .

12.5. The Stable Deriëd Category For simplicity we assume that ⌳ is an Artin algebra. We denote by D bŽ⌳.the bounded derived category D bŽŽmod ⌳... It is well known that b y, b q, b D Ž⌳.s H ŽP⌳⌳.s H ŽI ., where I ⌳is the full subcategory of y, b q, b modŽ⌳.consisting of injective modules and H ŽP⌳⌳.ŽH ŽI .. is the y q full subcategory of H ŽP⌳⌳.ŽH ŽI .., consisting of complexes with bounded cohomology. As in Section 12.4, the Cartan᎐Eilenberg projectives b PCE in D Ž⌳.form the full subcategory of E-projectives of a proper class 356 APOSTOLOS BELIGIANNIS of triangles E, and moreover using the above identifications, D bŽ⌳.has enough E-injectives, namely the full subcategory ICE of the Cartan᎐Eilen- berg injective complexes in D bŽ⌳.. The projectively stableŽ bounded. derived category D bŽ⌳.of ⌳ is b defined to be the stable category D Ž⌳.rPCE , and similarly the injectively stableŽ bounded. derived category D bŽ⌳.of ⌳ is defined to be the stable b b category D Ž⌳.rICE . By Section 7, we have that D Ž⌳.is left triangulated and D bŽ⌳.is right triangulated. bb PROPOSITION 12.33. ⌳ is self-injectië iff D Ž⌳.is triangulated iff D Ž⌳. bb is triangulated. If ⌳ is self-injectië, then D Ž⌳.s D Ž⌳.. Proof. Let 0 ª A ª B ª C ª 0 be an exact sequence in modŽ⌳.. It b b induces a triangle Awx0 ª B wx0 ª C wx0 ª A wx1 g E in D Ž⌳.. If D Ž⌳. is triangulated, then ⌳wx0 g PCE: I CE . Hence the sequence 0 ª ŽCwx0,⌳ wx0 .ª ŽBwx0,⌳ wx0 .ª ŽAwx0,⌳ wx0 . ª 0, equivalently the sequence 0 ª ŽC, ⌳.ª ŽB, ⌳.ª ŽA, ⌳.ª 0, is exact and ⌳ is self-injective. Con- versely if ⌳ is self-injective, then P⌳⌳s I . Then obviously PCEs I CE and by Theorem 7.2, D bŽ⌳.is triangulated. Our aim is to show that the stable derived category D bŽ⌳.is connected with the stable module category modŽ⌳., which is also left triangulated, in the same way as D bŽ⌳.is connected with modŽ⌳.. Consider the projective b resolution functor D: modŽ⌳.ª D Ž⌳., which assigns a deleted projective resolution to a ⌳-module. Trivially D sends projective modules to CE-pro- b jective complexes; hence it induces a functor ᑬ: modŽ⌳.ª D Ž⌳., mak- ing the following diagram commutative 6 6

P⌳ modŽ⌳.modŽ⌳.

6 6 6 D ᑬ

6 bb6 PCE D Ž⌳.D Ž⌳.

nb The cohomology functors H : D Ž⌳.ª modŽ⌳., n g ޚ, send the class of triangles E in D bŽ⌳.to short exact sequences in modŽ⌳.and they send nb PCE to P⌳; hence there are induced functors H : D Ž⌳.ª modŽ⌳.. We n n denote by H [ [ng ޚ H and H [ [ng ޚ H . Using the definition of the functor ᑬ and the construction of the triangulation of D bŽ⌳.as in Section 7, it is easy to deduce the following.

PROPOSITION 12.34. The functors ᑬ, H are exact, ᑬ is fully faithful, and b H ᑬ s IdmodŽ⌳. . Further E-gl.dim D Ž⌳.s gl.dim ⌳ and the functor ᑬ b induces an isomorphism K00ŽŽmod ⌳..( K ŽD Ž⌳.. between the Grothendieck groups of the left triangulated categories modŽ⌳.and D bŽ⌳.. PURITY IN TRIANGULATED CATEGORIES 357

ⅷⅷ ⅷ b LEMMA 12.35. A morphism f : A ª B in D Ž⌳.is an isomorphism in b n ⅷ n ⅷ n ⅷ D Ž⌳.iff ᭙ n g ޚ, HfŽ.: HAŽ.ª HŽ B. are isomorphisms in modŽ⌳..

ⅷ b ⅷⅷ ⅷ Proof. If f is an isomorphism in D Ž⌳., then there exists g : B ª A ⅷ ⅷ ⅷ ⅷ such that 1ABⅷ y f ( g ,1 ⅷ y g ( f factor through some CE-projective n ⅷ n ⅷ n ⅷ n ⅷ complex. Then 1H nnŽ AHⅷ . y HfŽ .( HgŽ . and 1 Ž B ⅷ . y HgŽ .( HfŽ . factor through some projective module. Hence H n Žf ⅷ .are isomorphisms b in modŽ⌳., ᭙ n g ޚ. Conversely since D Ž⌳.is a Krull᎐Schmidt category, we can assume that Aⅷ, B ⅷ do not have summands with projective homology. If Hfn Žⅷ .are isomorphisms in modŽ⌳., then so are HfnŽ ⅷ .. Hence f ⅷ is invertible in D bbŽ⌳., hence in D Ž⌳.. In the following result we denote by modŽ⌳.b the full subcategory of the ޚ-graded category modŽ⌳.ޚ consisting of all the bounded objects.

b b COROLLARY 12.36.Ž 1. The homology functor H: D Ž⌳.ª modŽ⌳.is a representation equi¨alence iff the functor ᑬ is an equi¨alence iff gl.dim ⌳ F 1. Ž2. If ⌳ is indecomposable, then D bŽ⌳.is abelian iff ⌳ is hereditary 1-Gorenstein.

Proof. PartŽ 1. follows as in Theorem 10.2, using Proposition 12.34 and the fact that ⌳ is hereditary iff any indecomposable complex in D bŽ⌳.is a stalk complex in some degreewx 33 . If ⌳ is indecomposable, then by wx 4 , ⌳ is hereditary 1-Gorenstein iff modŽ⌳.is abelian. PartŽ 2. follows from this, usingŽ 1. and Proposition 12.34.

The following generalizes results of Wheelerwx 72 , in which Lemma 12.35 and the next result are proved in the case of a self-injective algebra. We note that by Proposition 12.33, in case ⌳ is self-injective, the stable derived category is triangulated. This is also a result of Wheelerwx 73 , proved by different methods.

THEOREM 12.37. Let ⌳, ⌫ be Artin algebras, and letŽ F, G. be an adjoint pair of exact functors, F: modŽ⌳.~ modŽ⌫.: G. Then the following are equi¨alent:

Ži. The functors F, G preserë projectiës and the induced functors F: modŽ⌳.ª modŽ⌫.and G: modŽ⌫.ª modŽ⌳.are inërse triangle equiä- lences.

ⅷ b b ⅷ b Žii. The induced functors F : D Ž⌳.ª D Ž⌫.and G : D Ž⌫.ª b ⅷ b b D Ž⌳.preserë CE-projectiës and the induced functors F : D Ž⌳.ª D Ž⌫. ⅷ b b and G : D Ž⌫.ª D Ž⌳.are inërse triangle equiälences. 358 APOSTOLOS BELIGIANNIS

Proof. Fix counit ␧: FG ª IdmodŽ⌫. and unit ␦: Id modŽ⌳. ª GF. Let F ⅷ, G ⅷ be the induced functors on the derived categories. Then ␧, ␦ induce morphisms ␧ ⅷ, ␦ ⅷ which are counit and unit, respectively, of an adjoint pair ŽF ⅷⅷ, G .. If conditionŽ i. holds, then since F, G preserve projectives, the functors F ⅷ, G ⅷ preserve CE-projectives, so we have induced functors F ⅷ, G ⅷ on the stable derived categories. It is easy to see that ŽF ⅷⅷ, G . is also an adjoint pair, with counit ␧ ⅷⅷand unit ␦ . By Lemma 12.35, it suffices to show that H n Ž␧ ⅷ ., H n Ž␦ ⅷ .are isomorphisms. This is immediate, from the fact that the induced morphisms ␧, ␦ are isomorphisms in the stable module categories by hypothesis. The converse follows similarly. Since bywx 63 , derived equivalent self-injective algebras are stably equivalent by an equivalence satisfying conditionŽ i. of the above theorem, we have the following. COROLLARY 12.38. Let ⌳, ⌫ be deriëd equiälent self-injectië Artin algebras. Then the triangulated categories D bŽ⌳.and D bŽ⌫.are triangle equiälent. Remark 12.39. The results of this section generalize to the homotopy category of complexes of injective comodules, resp. stable category of comodules modulo injectives, over commutative, resp. finite dimensional, Hopf algebras over a field. Also most of the results concerning the stable derived category generalize to the unbounded case and are valid for any ring. Details are left to the reader.

ACKNOWLEDGMENTS

Most of the results of this paper were obtained and presented in lectures between January and April 1997 during a one year stay at the Instituto De Matematicas,´ U.N.A.M., Mexico, by a postdoctoral fellowship. The final version of this paper was written during a stay at the University of Bielefeld, thanks to a fellowship from the TMR-network ‘‘Algebraic Lie Representations.’’ The author thanks all the members of the Representation Theory Group at both universities, and especially Prof. J. A. de la Pena˜ and Prof. C. M. Ringel, for their warm hospitality. The author thanks H. Krause for the very enjoyable and interesting discussions and Prof. D. J. Benson and Dr. G. Gnacadja for their useful comments. Finally the author is grateful to Prof. J. D. Christensen, Prof. B. Keller, and especially the referee for their very valuable comments, suggestions, and corrections which resulted in a significant improvement of the paper.

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