We recall [8], [19], [27], that given a left or right triangulated categoryC, there exists a triangulated categoryS(C)and an exact functorS : C → S(C), which is universal for exact functors fromC to triangulated categories. In other words the pair(S,S(C))satisfies the following universal property: if F :C →Dis an exact functor to a triangulated categoryD, then there exists a unique exact functorF!:S(C)→D, such thatF!S=F. The triangulated categoryS(C)is thestabilizationofC and the functorSis thestabilization functor. ObviouslyC is triangulated iff the stabilization functorSis a triangle equivalence. The stabilization ofC is trivial iff the loop or suspension functor is locally nilpotent. We refer to [8] for the description and further information on the stabilization construction.
Definition6.1. LetC be a left or right triangulated category. We say that C admits a stable homotopy categoryif its stabilizationS(C)is compactly generated.
Definition 6.2. (i) A ring has a (right) projective stable homotopy categoryif the left triangulated category Mod()admits a stable homotopy category.
(ii)A ringhas a (right) injective stable homotopy categoryif the right triangulated category Mod()admits a stable homotopy category.
Example 6.3. If is a QF-ring, then has a left and right project-ive and injectproject-ive stable homotopy category. Indeed by Corollaries 5.9, 5.12, Mod()= Mod()is a compactly generated triangulated category, and co-incides with its stabilization.
Remark6.4. The projective stable homotopy category is invariant under derived equivalence. Indeed if , D are derived equivalent rings, then the stabilization of Mod()is triangle equivalent to the stabilization of Mod(D). Hencehas a projective stable homotopy category iffDhas a projective stable homotopy category.
Our aim in this section is to study rings which admit a projective or in-jective stable homotopy category. Before we proceed further we need some definitions and results from [8]. First we recall that a right-module Ais calledGorenstein-projectiveif there exists an exact sequence· · · →P−1→ P0 → P1 → · · ·of projective modules withIm(P−1 → P0)= Aand the sequence remains exact applying Hom(−,P). The full subcategory of all Gorenstein-projective modules is denoted byGP(Mod()), and the induced stable category modulo projectives is denoted byGP(Mod()). The full sub-categoryGI(Mod())of Gorenstein-injective modules is defined dually and the induced stable category modulo injectives is denoted byGI(Mod()). By the results of [8], the stable categoryGP(Mod())is a full triangulated subcat-egory of ModL()and dually the stable categoryGI(Mod())is a full triangu-lated subcategory of ModR(). We denote byGP(mod()), resp.GI(mod()), the full subcategory ofGP(Mod()), resp.GI(Mod()), consisting of finitely presented modules. Note thatGP(Mod()),GI(Mod())are Frobenius weak closed model categories. Obviously if is a QF-ring thenGP(Mod()) = Mod()= GI(Mod()). If any module has finite projective, resp. injective, dimension, thenGP(Mod()) = P, resp. GI(Mod()) = I. Ifis No-etherian (or coherent) ring, the Gorenstein-projective modulesGP(mod()), are exactly the modules in mod()with zero G-dimension in the sense of Auslander-Bridger [1].
Definition6.5. [8] A ringis calledright Gorensteinif any projective right module has finite injective dimension and any injective right module has finite projective dimension. Ifis right Gorenstein, thensup{p.d.I;I ∈ I} =sup{i.d.P;P ∈P}<∞. This number is called the(right) Gorenstein dimensionofand is denoted by G−dim.
We note that the Gorenstein property of a ring is a necessary and suffi-cient condition for the existence of Tate-Vogel (co)homology and complete projective or injective resolutions [8]. We need the following basic result from [8].
Lemma6.6.Letbe a right Gorenstein ring withG−dim=d. (1) The stabilization ofModR()is the stable categoryGI(Mod())with
stabilization functor0−d0d : ModR()→GI(Mod()).
(2) The stabilization ofModL()is the stable categoryGP(Mod())with stabilization functor/−d/d : ModL()→GP(Mod()).
(3) /d(Mod())=GP(Mod()),0d(Mod())=GI(Mod())and there exists a triangle equivalenceGP(Mod())≈GI(Mod()).
Our first main result in this section is the following.
Theorem 6.7. Any left coherent and right perfect or right Morita right Gorenstein ring, has a (right) projective and injective stable homotopy cat-egory.
Proof. By the above Lemma it suffices to show that the triangulated cat-egoryGP(Mod())or equivalentlyGI(Mod())is compactly generated. Sup-pose first that is coherent and right perfect. Obviously GP(Mod()) has coproducts. LetAbe a Gorenstein-projective module and suppose that∀G∈ GP(mod())we have(G, A)=0. Then any morphism from a finitely presen-ted Gorenstein-projective module toAfactors through a projective module.
Sinceis left coherent and right perfect by [8] we have thatGP(Mod())is covariantly finite in Mod()and it is not difficult to see that ifXis finitely presented then the leftGP(Mod())-approximation ofX is finitely presen-ted. Now let X be an arbitrary finitely presented module and f : X → A be a morphism. LetgX : X →GXbe its left Gorenstein-projective approx-imation. SinceA is Gorenstein-projective, the morphismf factors through gX. Hence there exists h : GX → A such that gX◦h = f. Since GX is in GP(mod()), the morphism hfactors through a projective. This implies that the morphism f factors through a projective module. This shows that (X, A) = 0,∀X ∈ mod(). By Theorem 5.7, we have that Ais projective, i.e.A = 0. We conclude that∀G ∈ GP(mod()): (G, A) = 0⇒A = 0.
SinceGP(mod())is skeletally small triangulated and consists of compact ob-jects, by Corollary 3.17 it follows that the triangulated categoryGP(Mod()) is compactly generated. Since by Lemma 6.6,GP(Mod())is triangle equival-ent toGI(Mod()), the latter is also compactly generated. Ifis right Morita, the proof is similar and is left to the reader.
Corollary6.8. Letbe an Artin algebra of finite selfinjective dimension, i.e.i.d. < ∞andi.d. opop < ∞. Thenhas a left and right projective and injective stable homotopy category.
Proof. Follows from Theorem 6.7, since a Noetherian ringis left Goren-stein iffis right Gorenstein iff i.d. <∞and i.d. opop <∞, see [8].
Corollary6.9. Letbe a left coherent and right perfect right Gorenstein ring and let{Hn;hn}n∈Zbe a cohomology theory inModR(). Then eachHn
sends coproducts to products⇔there exists a Gorenstein-projective moduleA such thatHn ∼=Mod()(−, /−n(A)). Hence with the notation of section3:
CohTh(ModR())≈GP(Mod()).
Proof. By Theorem 3.21,CohTh(ModR())is equivalent to the costabil-ization of ModL(). Sinceis right Gorenstein, by [8] this costabilization is triangle equivalent to the stabilizationGP(Mod()).
A similar result holds for cohomology theories defined over ModR(). If is left coherent and right perfect, resp. right Morita, then by Corollary 5.12, resp. Corollary 5.11, we know that cohomological functors sending coproducts to products defined over ModR(), resp. ModR(), are representable. The following result shows what happens in the Gorenstein case, for the remaining triangulations. First we need the following description of compacts objects.
Lemma6.10.Under the assumptions of Theorem6.7, we have:
GP(Mod())b=GP(mod()) and GI(Mod())b=GI(mod()).
In particular ifis Quasi-Frobenius, then:Mod()b=mod().
Proof. Since any finitely presented module becomes compact in the stable categories, we haveGP(mod())⊆ GP(Mod())b. By the above Theorem, GP(mod()) is a skeletally small generating epaisse subcategory of GP(Mod()). Then the assertion follows from Neeman-Ravenel Theorem (see Theorem 5.3 in [26]).
Corollary6.11.(α)Letbe a left coherent right perfect right Gorenstein ring.
(i)Let H : ModL()op → Abbe a cohomological functor which sends coproducts to products. Then there exists a Gorenstein-projective moduleG such thatH|GP(Mod()) =(−, G).
(ii)LetF : ModL() → Abbe a homological functor which preserves products and coproducts. Then there exists a finitely presented Gorenstein-projective moduleGsuch thatH|GP(Mod()) =(G,−).
(β)Letbe a right Morita right Gorenstein ring.
(i)Let H : ModL()op → Abbe a cohomological functor which sends coproducts to products. Then there exists a Gorenstein-injective module G such thatH|GI(Mod()) =(−, G).
(ii)LetF : ModL() → Abbe a homological functor which preserves products and coproducts. Then there exists a finitely presented Gorenstein-projective moduleGsuch thatH|GI(Mod()) =(G,−).
Proof. (α)(i)ObviouslyH restricts to a cohomological functor
GP(Mod())op→Ab, which sends coproducts to products. Then the asser-tion follows from Brown representability.(α)(ii) H restricts to a homological functorGP(Mod())→Ab, which preserves products and coproducts. Then the assertion follows from the above Lemma and [31]. Part(β)is treated sim-ilarly.
Examples of functors satisfying the condition of(α)(i) in the above Co-rollary, are the functors Exti(−, A), i ≥ 1, ∀A ∈ Mod(). Examples of functors satisfying the condition of (α)(ii) in the above Corollary, are the functors(F ,−),∀F ∈ mod(). Examples of functors satisfying the condi-tion of(β)(i)in the above Corollary, are the functors(−, A),∀A∈Mod(). Examples of functors satisfying the condition of(β)(ii)in the above Corollary, are the functorsExti(F,−), i≥1,∀F ∈mod().
We recall that a full subcategoryL of Mod()is calledresolving, resp.
coresolving, ifL is closed under extensions, kernels of epics, resp. cokernels of monics, and contains the projectives, resp. injectives.
Corollary6.12.Letbe a left coherent and right perfect right Gorenstein ring of dimensiond. LetL be a resolving, subcategory ofMod(), closed under coproducts and cokernels of left projective approximations. IfLconsists of Gorenstein-projective modules, thenL is contravariantly finite.
Proof. SinceLis resolving, the hypothesis implies that the stable category L is a localizing subcategory of the compactly generated triangulated category GP(Mod()). By Corollary 3.18, the inclusion functorL 4→ GP(Mod()) has a right adjoint R. Since by [8] the inclusion functor GP(Mod()) 4→ Mod() has a right adjoint/−d/d, it follows that the functor R/−d/d : Mod() → L is a right adjoint of the inclusion L 4→ Mod(), hence L is a coreflective subcategory of Mod(). This implies trivially thatL is contravariantly finite in Mod().
A similar result holds also for coresolving subcategories.
Remark6.13. If the ringis left coherent right perfect and right Goren-stein, then there exists a skeletally small generating compact left triangulated subcategory in Mod()), namely mod()), and a compactly generated trian-gulated category, namelyGP(Mod()), such that the stabilization of mod()) is identified with the full subcategory of compact objects ofGP(Mod()). This procedure is similar with the construction of the stable homotopy category of spectra in Algebraic Topology and explains our motivation. Here the role of Mod())is played by the homotopy category of spectra, the role of mod())is played by the homotopy category of finite spectra and the role ofGP(Mod())
is played by the stable homotopy category. However in contrast to the homo-topy theory of modules, the stable homohomo-topy category of spectra is not the stabilization of all spectra since the latter does not has arbitrary coproducts, so it is not compactly generated, see [32]. Similar remarks are applied for the injective stable category, ifis right Morita.
6.2. Phantomless and Brown Stable Homotopy Categories
LetC be a compactly generated triangulated category. A morphismf :A→ B in C is called phantom if C(X, f ) = 0, for any X ∈ Cb. The set of all phantom maps between A, B is denoted by Ph(A, B); setting Ph(C) =
A,B∈C Ph(A, B)we obtain an ideal inC. We denote by Phn(A, B)the set of all morphismsA→Bwhich may be written as a composition ofnphantom maps and by Phn(C)the induced ideal ofC. The categoryC is called phantom-lessif Ph(C)=0.
SinceCb is skeletally small, we can consider the Grothendieck category Mod(Cb). SinceCbis triangulated, the full subcategory Flat(Mod(Cb))of flat functors coincides with the category of cohomological functors overCb[6].
Define a functor
T:C →Mod(Cb) by T(A)=C(−, A)|Cb
whereC(−, A)|Cb denotes restriction. ThenTis a homological functor with image in Flat(Mod(Cb))and kernel the ideal Ph(C)of phantom maps. The categoryCis called aBrown categoryifTinduces a representation equivalence betweenC and Flat(Mod(Cb)). We recall that a functor is a representation equivalence if it is full, surjective on objects and reflects isomorphisms.
Definition6.14. LetC be a left or right triangulated category with cop-roducts.
(1)We say that C admits a phantomless stable homotopy categoryifC admits a stable homotopy categoryD which is phantomless.
(2)We say thatC admits a Brown stable homotopy categoryifC admits a stable homotopy categoryDwhich is a Brown category.
A ringhas a right injectivephantomless, resp. Brown, stable homotopy categoryif this holds for the right triangulated category Mod(). Similarly for the projective case. In this subsection we study ringswhich have injective or projective phantomless or Brown stable homotopy categories. We recall that a ringis called representation finite if is right Artinian and the set of isomorphism classes of indecomposable finitely presented right modules is finite. It is well known that representation finiteness is a symmetric condition and thatis representation finite iffis left and right pure-semisimple. The
proof of the following result is a consequence of the results of [6], [7]. For part (3), see also [11].
Proposition6.15. (1)Letbe a QF-ring.
(i) has a left or right projective or injective phantomless stable homotopy category⇔is representation finite.
(i) Ifis countable or|| ≤ ℵt for somet ≥0andis pure hereditary (i.e. if the right pure global dimensionr.pure.gl.dim≤1), thenhas a right projective or injective Brown stable homotopy category.
(2)Ifis a finite dimensional selfinjective localk-algebra over an algeb-raically closed field, thenhas a left and right projective or injective Brown stable homotopy category⇔is representation finite orkis countable.
(3)IfGis a finite group andkis a field with char(k) = p/|G|, thenkG has a left or right projective or injective Brown stable homotopy category⇔ kis countable orGhas cyclicp-Sylow subgroups.
We note that the stable homotopy category of spectra is a Brown category [32].
Proposition6.16. Letbe a ring as in Theorem 6.7. Assume that sup{r.pure.p.d. G;G∈GP(Mod())} =n <∞.
Then∀A, B ∈Mod():
Phn+1(/d(A), /d(B))=0=Phn+1(0d(A), 0d(B)).
In particularPhn+1(GP(Mod()))=0=Phn+1(GI(Mod())).
Proof. This follows from the results of [6], since the hypothesis implies that any object inGP(Mod())has a resolution of lengthnby proper triangles in the sense of relative homological algebra developed in [6].
Theorem6.17.Letbe a right Gorenstein ring. Ifis left coherent and right perfect or right Morita, then the following are equivalent.
(i) admits a right projective phantomless stable homotopy category.
(ii) Any Gorenstein-projective is pure projective (pure injective).
(iii) GP(Mod())is a pure-semisimple locally finitely presented Frobenius category.
(iv) GP(Mod())is a pure-semisimple locally finitely presented triangulated category.
(v) admits a right injective phantomless stable homotopy category.
(vi) Any Gorenstein-injective is pure injective (pure projective).
(vii) GI(Mod())is a pure-semisimple locally finitely presented Frobenius category.
(viii) GI(Mod())is a pure-semisimple locally finitely presented triangulated category.
Proof. Assume first thatis left coherent and right perfect ring. If con-dition(i)holds, thenGP(Mod())is phantomless. By [6], we have that any objectG∈GP(Mod())is a coproduct⊕i∈IGi, whereGi is inGP(mod()). But then in Mod()we haveG= ⊕i∈IGi⊕P, whereP is projective. Since is right perfect, this implies that G is a coproduct of finitely presented Gorenstein modules. HenceGis in particular pure-projective. Hence (i) ⇒ (ii). If(ii)holds, then obviouslyGP(Mod()) = Add(GP(mod())). Since GP(mod())is skeletally small and is contained in GP(Mod())b, by Pro-position 2.5 it follows thatGP(Mod())is a pure-semisimple locally finitely presented category. Hence (ii) ⇒ (iii). If (iii)holds, then any Gorenstein-projective module is a coproduct of finitely presented (Gorenstein-Gorenstein-projective) modules, and trivially the same is true inGP(Mod()). By [6],GP(Mod()) is a locally finitely presented (phantomless) pure-semisimple triangulated cat-egory and this shows that (iii) ⇒ (iv). Since by [6] any locally finitely presented triangulated category is phantomless, we have that(iv)⇒(i). Now we prove that any of the above equivalent conditions (i)−(iv) is equival-ent to the parequival-enthetical condition(ii). We use that an objectG in a locally finitely presented category with products is pure-injective iff for any index setI, the summation mapf : ⊕i∈IG → Gfactors through the pure mono µI : ⊕i∈IG →
i∈IG, see [13]. Suppose now that(i) is true and let G be a Gorenstein-projective module. LetI be any index set and consider the summation mapf : ⊕i∈IG →G. SinceGP(Mod())is a pure semisimple locally finitely presented category, the morphism f : ⊕i∈IG → G factors through the canonical morphismµ:⊕i∈IG→
i∈IG: there exists a morph-ismα :
i∈IG → Gsuch thatf = µ◦α. Then there exist a morphism β : ⊕i∈IG → P such that f −µ◦α = β ◦p, where p : P → G is the projective cover ofG. SinceP is projective andis right perfect, P is pure-injective. Since µ : ⊕i∈IG →
i∈IG is a pure mono, there exists a morphismt :
i∈IG→P such thatµ◦t = β. Thenf = µ◦(α+t ◦p). This shows that the summation map f factors through the pure mono µ. HenceG is pure-injective. Conversely if any Gorenstein-projective module Gis pure-injective, then using the same argument as above, it follows that all objects ofGP(Mod())are pure-injective in the sense of [31]. Hence by [31], GP(Mod())is phantomless.
SinceGP(Mod())is triangle equivalent toGP(Mod()), using the same
arguments as above we have that the remaining conditions are also equivalent to(i). Ifis right Morita, the proof is similar and is left to the reader.
From the (proof of the) above Theorem we deduce the following results.
Corollary 6.18. Let be a left coherent and right perfect or right Morita right Gorenstein ring. Thenadmits a phantomless projective or in-jective stable homotopy category iff any proin-jective or Gorenstein-injective module is a coproduct of finitely presented Gorenstein-projective or Gorenstein-injective modules with local endomorphism rings. In this case the coproduct decomposition is essentially unique.
Corollary6.19. Letbe a right pure-semisimple right Gorenstein ring.
Ifis left coherent or right Morita, thenhas a right projective and injective phantomless stable homotopy category. In particular this happens ifis a representation-finite Artin algebra of finite selfinjective dimension.
We close this section studying rings with Brown stable homotopy category.
Theorem6.20.Letbe a right Gorenstein ring of dimensiond. Suppose that|| ≤ ℵt, for somet ≥ −1andis left coherent and right perfect or right Morita.
Ifr.pure.p.dA≤1, for any Gorenstein-projective moduleAorr.pure.i.dB
≤1, for any Gorenstein-injective moduleB, thenhas a right projective and injective Brown stable homotopy category.
Proof. Since the Brown property is invariant under triangle equivalence, it suffices to prove the projective case. By the results of [7], it suffices to show that for any objectGin GP(Mod()), there exists a triangle G1 → G0 → G −→h /−1(G1)in GP(Mod()), wherehis a phantom map and G1, G0 ∈ Add(GP(mod())). Let(∗): 0→X1−→α X0−→β G→0 be a pure-projective resolution of the Gorenstein-projective moduleG. Then we have a triangle /(G) → X1
−α
→ X0
−β
→ Gin Mod(). Applying the stabilization functor S = /−d/d to this triangle, we obtain a triangleS(X1) −−→S(α) S(X0) −−→S(β) G −→h /−1S(X1)inGP(Mod())). Since the stabilization functor preserves coproducts and X0, X1 are pure projectives it follows that S(X0),S(X1) ∈ Add(GP(mod())). We remains to prove that h is phantom. Let H ∈ GP(mod())andg :H →Gbe a morphism. SinceH is finitely presented, there exists a morphismδ:H →X0such thatδ◦β=g. ThenS(δ)◦S(β)=g. This shows thathis phantom.
Corollary6.21.Letbe a left coherent and right perfect right Gorenstein ring of dimensiond. Thenadmits a Brown right projective stable homotopy
category iff for any filtered directed system of finitely presented right modules representations as minimal weak colimits as above, then there exists a short exact sequence
Proof. All the assertions are consequences of the results of [7].
A similar result is true for right Morita right Gorenstein rings.
Corollary6.22. Ifis a countable Artin algebra of finite self-injective dimension, thenhas a right projective and injective Brown stable homotopy category.
We refer to [6], [7] for further consequences of the phantomless or Brown property of projective or injective stable homotopy categories.
Acknowledgements. The author gratefully acknowledges support from the EC TMR Network “Orbits, Crystals and Representation Theory”, under the contract No. FMRX-CT97-0100. This work was done during a stay at the University of Bielefeld. The author expresses his gratitude to the members of the Representation Theory Group at Bielefeld, and especially to Prof. Dr. C. M.
Ringel, for the warm hospitality. Finally the author thanks Prof. T. Pirashvili, for bringing to his attention the results of the paper [34].
Note added in proof. The preprints [6] and [7] have appeared inRelative homological algebra and purity in triangulated categories, J. Algebra 227 (2000), 268–361.
REFERENCES
1. Auslander, M. and Bridger, M.,Stable Module Theory, Mem. Amer. Math. Soc. 94 (1969).
2. Auslander, M. and Smalø, S. O.,Preprojective modules over Artin algebras, J. Algebra 66 (1980), 61–122.
3. Auslander, M. and Reiten, I.,Cohen-Macaulay and Gorenstein Artin algebras, Prog. Math.
95 (1991), 221–245.
4. Beilinson, A., Bernstein, J. and Deligne, P.,Faisceaux Pervers, Astérisque 100 (1982).
5. Beligiannis, A.,On The Freyd categories of an additive category, Homology, Homotopy and Applications 2 (No. 11) (2000), 147–185.
6. Beligiannis, A.,Homological algebra in triangulated categories I: Relative homology and phantomless triangulated categories, Preprint, UNAM, Mexico, 1997.
7. Beligiannis, A.,Homological algebra in triangulated categories II: Representation theorems, localization and the stable category, Preprint, UNAM, Mexico, 1997.
8. Beligiannis, A.,The homological theory of contravariantly finite subcategories, Comm. Al-gebra 28 (10) (2000), 4547–4596.
9. Beligiannis, A. and Marmaridis, N.,Left triangulated categories arising from contravariantly finite subcategories, Comm. Algebra 22 (1994), 5021–5036.
10. Beligiannis, A. and Marmaridis, N.,Grothendieck groups arising from contravariantly finite subcategories, Comm. Algebra 24 (1996), 4415–4438.
11. Benson, D. J. and Gnacadja, G. PH.,Phantom maps and purity in modular representation theory I, Fund. Math. 161 (1999), 37–91.
12. Brown, E. H.,Abstract homotopy theory, Trans. Amer. Math. Soc. 119 (1965), 79–85.
13. Crawley-Boevey, W., Locally finitely presented additive categories, Comm. Algebra 22, (1994), 1644–1674.
14. Dwyer, W. G. and Spalinski, J.,Homotopy theories and model categories, I. M. James (Ed), Handbook of Algebraic Topology, North-Holland, 1995, 73–126.
15. Eckmann, B. and Kleisli, H.,Algebraic homotopy groups and Frobenius algebras, Illinois J.
Math. 6 (1962), 533–552.
16. Freyd, P.,Splitting homotopy idempotents, in: Proceedings of the Conference on Categorical Algebra, La Jolla, (1966), 173–176.
17. Gabriel, P. and Zisman, M.,Calculus of Fractions and Homotopy Theory, Springer Verlag, New York, (1967).
18. Heller, A.,The loop space functor in homological algebra, Trans. Amer. Math. Soc. 96 (1960), 382–394.
19. Heller, A.,Stable homotopy categories, Bull. Amer. Math. Soc. 74 (1968), 28–63.
20. Heller, A.,Completions in abstract homotopy theory, Trans. Amer. Math. Soc. 147 (1970), 573–602.
21. Hilton, P. J.,Homotopy Theory and Duality, Gordon and Breach, New York, 1965.
22. Hovey, M.,Model Categories, Math. Surveys Monographs 63, 1999.
23. Huber, P.,Homotopy theory in general categories, Math. Ann. 144 (1961), 361–385.
24. Jensen, C. U. and Simson, D.,Purity and generalized chain conditions, J. Pure Appl. Algebra 14 (1979), 297–305.
25. Jørgensen, P.,Brown representability for stable categories, Math. Scand. 85 (1999), 195–218.
26. Keller, B.,Deriving DG categories, Ann. Sci. École Norm. Sup. 27 (1994), 63–102.
27. Keller, B. and Vossieck, D.,Sous les cateégories dérivées, C. R. Acad. Sci. Paris 305 (1987), 225–228.
28. Keller, B. and Vossieck, D.,Aisles in derived categories, Bull. Soc. Math. Belg. 40 (1988), 239–253.
29. Kleisli, H.,Homotopy theory in abelian categories, Canad. J. Math. 14 (1962), 139–169.
30. Krause, H.,Functors on locally finitely presented categories, Colloq. Math. 75 (1998), 105–
132.
31. Krause, H.,Smashing subcategories and the telescope conjecture – An algebraic approach, Invent. Math. 139 (No. 1) (2000), 99–133.
32. Margolis, H. R.,Spectra and the Steenrod Algebra, North-Holland Math. Library 29, 1983.
33. Neeman, A.,The Grothendieck duality theorem via Bousfield’s techniques and Brown repres-entability, J. Amer. Math. Soc. 9 (1996), 205–236.
34. Pirashvili, T.,Closed model category structures on abelian categories(in Georgian), Bull.
34. Pirashvili, T.,Closed model category structures on abelian categories(in Georgian), Bull.