Homological And Homotopical Aspects Of Torsion Theories Apostolos .

INTRODUCTION3naturally associated with it two triangulated categories, and we show that throughthis construction there is a close relationship between (hereditary) torsion theories,a fact which is used for constructing new (co)homology theories in the last chapter.Tilting theory, a central topic in the representation theory of Artin algebras,is intimately related to torsion theories in several different ways. When T is atilting module with pd T 1, that is Ext1 (T, T ) 0 and there is an exact sequence0 Λ T0 T1 0 with T0 and T1 summands of finite direct sums of copiesof T , there is an associated torsion theory (T , F) where T Fac T (the factorsof finite direct sums of copies of T ). This torsion theory plays an important rolein tilting theory, and is closely related to a torsion theory for the endomorphismalgebra Γ End(T )op . The tilting in abelian categories referred to above gave away of constructing mod(Γ) from mod(Λ) via the torsion theory (T , F) and theone induced in Db (mod(Λ)). When T is more generally a tilting module withpd T , there is no natural associated torsion theory in mod(Λ) which plays asimilar role. But it is interesting that in this case we can show that T generatesa torsion theory in Db (mod(Λ)), whose heart is equivalent to mod(Γ). There is astill more general concept of tilting module, called Wakamatsu tilting module. Weconjecture that any Wakamatsu tilting module of finite projective dimension is infact a tilting module. We give interesting reformulations of this conjecture. Oneof them is that if (XT , YT ) is a hereditary torsion theory in D(Mod Λ) generatedby a Wakamatsu tilting module T of finite projective dimension, then XT is closedunder products.It is well-known that tilting theory can be regarded as an important generalization of classical Morita theory, which describes when two module categories areequivalent. During the last fifteen years investigations of many authors extendedMorita theory for module categories to derived categories, thus offering new invariants and levels of classification. These investigations culminated in a MoritaTheory for derived categories of rings and DG-algebras, which describes explicitly when derived categories of rings or DG-algebras are equivalent as triangulatedcategories. This important generalization can be regarded as a higher analogueof tilting theory and plays a fundamental role in representation theory, providinginteresting connections with algebraic geometry and topology. In this paper weinterpret Morita theory for derived categories in torsion theoretic terms, and wegive simple torsion theoretic proofs of (slight generalizations of) central results ofHappel [57], Rickard [91] and Keller [69], concerning the construction of derivedequivalences. In addition this interpretation via torsion theories gives interestingreformulations of several important open problems in various contexts, providingnew ways for their investigation.On the other hand there are the notions of contravariantly, covariantly andfunctorially finite subcategories of an additive category as introduced by Auslanderand Smalø in [13] and independently by Enochs in [46]. Special cases of these aresubcategories for which there exist a right or left adjoint of the inclusion. Contravariantly or covariantly finite subcategories play a fundamental role in the representation theory of Artin algebras. They provide a convenient setting for thestudy of several important finiteness conditions in various settings and there isagain a strong connection to tilting theory. For Artin algebras these categoriesoccur in pairs (X , Y), under some natural additional assumptions, namely X is

INTRODUCTION4contravariantly finite, closed under extensions and contains the projectives and Yis covariantly finite, closed under extensions and contains the injectives. Then Xand Y are in one-one correspondence, via the vanishing of Ext1 rather than ofHom. This correspondence was observed by Auslander and Reiten in [9] in thesetting of Artin algebras and by Salce in the setting of abelian groups in [97]. Bydefinition the subcategories X and Y involved in the above correspondence form acotorsion pair (X , Y). Cotorsion pairs have been studied recently by many peopleleading to the recent proof of the Flat Cover Conjecture by Enochs, Bashir andBican [17]. A central concept which is omnipresent in the paper and that it isclosely related to cotorsion pairs is that of a (relative) (Co)Cohen-Macaulay object.Their importance follows from the fact that for any cotorsion pair (X , Y) in anabelian category, the cotorsion class X , resp. the cotorsion-free class Y, consists ofrelative Cohen-Macaulay, resp. CoCohen-Macaulay, objects with respect to X Y.We would like to stress that prominent examples of cotorsion theories are comingfrom Gorenstein rings, and, more generally, from tilting or cotilting modules, whereCohen-Macaulay modules play an important role. Generalizing, we define and investigate in this paper Gorenstein and Cohen-Macaulay abelian categories and weprove structure results for them in the context of (co)torsion theories.For abelian and triangulated categories (in the latter case first formulated inthe language of t-structures), there is a close connection between X being a torsionclass of a torsion theory and X admitting a right adjoint (and between Y beinga torsion-free class of a torsion theory and Y admitting a left adjoint). Thereis a similar connection in the setting of pretriangulated categories, although theextension of the results does not work so smoothly in general. The main reasonis that the compatibility of the left and right triangles is not so well behaved asin the abelian or triangulated case. However we prove the corresponding resultunder some additional assumptions. Of particular interest for us is the case of astable category C/ω. Here we obtain at the same time the correspondence betweencontravariantly finite and covariantly finite subcategories via Ext1 discussed aboveand the torsion theory correspondence between the associated subcategories of C/ω,where ω X Y.As already mentioned such a stable category C/ω is the homotopy category ofa closed model structure in C in the sense of Quillen, which is defined via left/rightapproximations of objects of C by objects from ω. In this paper we show that there isa strong connection between the pairs of subcategories realizing the correspondencedescribed above, and closed model structures in the sense of Quillen in C. Recentlya similar connection from a different viewpoint was observed independently byHovey [65]. More precisely our results give a classification of all cotorsion pairs(X , Y) in an abelian category C with X Y contravariantly or covariantly finite, interms of suitable closed model structures. This leads to a classification of all torsiontheories in a stable category of an abelian category. As a consequence we obtainan interesting connection between (co)tilting modules and closed model structures.In general it is of central importance for the structure of the pretriangulatedcategory in question, that a reasonable subcategory is a torsion or torsion-free class.For instance when we have a torsion theory in a stable pretriangulated category ofan abelian category, we define new universal (co)homology theories on the abelian

INTRODUCTION5category which are complete with respect to the torsion or torsion free class inan appropriate sense. These new (co)homology theories are generalizations of theTate-Vogel (co)homology theories to the non-Gorenstein case. This procedure provides a useful interplay between torsion-theoretic properties of the stable categoryand homological properties of the abelian category. In this setting there is a strongconnection between the relative homological algebra, in the formulation proposedby Butler-Horrocks and later by Auslander-Solberg, induced by the torsion theoryand the behavior of the (co)homology theories. This connection has importantconsequences for the homological structure of the abelian category. For instancethese investigations allow us to generalize recent results of Avramov-Martsinkovsky,see [16], from finitely generated modules of finite Gorenstein dimension, to arbitrary modules, provided that a suitable torsion theory exists in the stable modulecategory. We show that such torsion theories exist in many cases, and in particularin the stable module category of, not necessarily finitely generated, modules over anArtin algebra. In this case we construct big Cohen-Macaulay modules and modulesof virtually finite projective dimension, and we show that these classes of modulesdefine a new relative cohomology theory for Artin algebras, called Cohen-Macaulaycohomology, which has intimate connections with many homological conjectures.We have provided background material, together with motivation, through recalling previous developments. Some of the main results of the paper have beenmentioned along the way. But to make it easier to focus on the essential points, wecollect a list of the main features below. Unify various concepts of torsion in different settings, and provide a general framework for studying torsion theories. Develop a general theory of torsion in pretriangualated categories, including interplay between the various settings. Provide methods for constructing torsion theories. Give applications to tilting theory. Give simpler (torsion theoretic) proofs of (slightly more general) resultsin the Morita theory of derived categories. Give interesting reformulations of homological conjectures for Artin algebras in terms of properties of torsion theories in the unbounded derivedcategory of all modules. Establish relationsip between cotorsion theories (X , Y) in an abelian category C and torsion theories in the stable category C/X Y. Give connections with closed model structures and classification of torsionand cotorsion theories in terms of these. Give structure results for (generalizations of) Gorenstein rings and, suitably defined, Gorenstein and Cohen-Macaulay abelian categories, in thecontext of (co)torsion theories. Give methods for constructing Gorenstein Categories out of certain Co–hen-Macaulay categories. Construct interesting (co)torsion theories in abelian categories equippedwith suitable Nakayama functors, where the Cohen-Macaulay objectsconstitute the (co)torsion class and the CoCohen-Macaulay objects constitute the (co)torsion-free class.

INTRODUCTION6 Construct new (co)homology theories generalizing Tate-Vogel (co)homo–logy using torsion theories.The article is organized as follows.In Chapter I, which is of preliminary nature, we recall some definitions and results, most of them well-known, concerning (hereditary/cohereditary) torsion theories in an abelian or triangulated category, which serve as motivation for the resultsof the rest of the paper. Here we observe that in the triangulated case the concept of a torsion theory essentially coincides with the concept of a t-structure inthe sense of Beilinson-Bernstein-Deligne [18]. We discuss briefly the connectionbetween torsion theories in abelian and derived categories investigated in [60] andindicate the relationship with tilting theory.In Chapter II we introduce the fairly general concept of torsion theory (X , Y)in a pretriangulated category. But first we recall basic results on left/right triangulated and pretriangulated categories, and provide a rather large source of examples.We give basic properties of torsion theories, and show that the notion specializes tothe concept of torsion theory in abelian and triangulated categories discussed in theprevious chapter. We also show that we can lift a hereditary or cohereditary torsiontheory in a pretriangulated category to such a pair in the left or right stabilization.In Chapter III we give a method for constructing torsion theories in triangulatedcategories containing all small coproducts. This is accomplished by starting with aset of compact objects. We investigate properties of the heart of this torsion theory,and give conditions for the heart to be a module category. Then we use these resultsto give applications to tilting theory, and to proving (a slight generalization of) aresult of Rickard on Morita theory of derived categories.In Chapter IV we deal with hereditary torsion theories generated by compactobjects in a triangulated category with arbitrary small coproducts. Following thefundamental work of Keller [69] on derived categories of DG-algebras, we describethe torsion class of the hereditary torsion theory generated by a set of compactobjects in terms of derived categories of appropriate DG-algebras. These resultsprovide simple torsion theoretic proofs of basic results of Happel [57], Rickard [91]and Keller [69]. We end with the relationship between homological conjectures inthe representation theory of Artin algebras and (hereditary) torsion theories in theunbounded derived category of the algebra.In Chapters V and VI we investigate torsion theories in pretriangulated categories of the form C/ω where C is abelian and ω is a functorially finite subcategoryof C. We give sufficient conditions for the existence of torsion theories, and give therelationship with cotorsion pairs in C. We also give applications to tilting theory. InChapter VI we discuss the problem of when a torsion or torsion free class in a stablecategory is triangulated. We show that this happens if and only if the functoriallyfinite subcategory ω is the category of projective or injective modules. In this casethe torsion subcategory is related to the subcategory of Cohen-Macaulay objectsand the torsion free subcategory is related to the subcategory of objects with finiteprojective dimension. We show that natural sources for such torsion theories are cotorsion triples induced by resolving and coresolving functorially finite subcategoriesof C.

INTRODUCTION7In Chapter VII we introduce and investigate in detail Gorenstein and CohenMacaulay abelian categories, which appear to be the proper generalizations of thecategory of (finitely generated modules) over a (commutative Noetherian) Gorenstein and Cohen-Macaulay ring. We give structure results for them in the contextof (co)torsion theories in connection with the finiteness of various interesting homological dimensions. In particular we show that, under mild conditions, the trivialextension, in the sense of [48], of a Cohen-Macaulay category is Gorenstein. Wealso study minimal Cohen-Macaulay approximations and we give applications toGorenstein and Cohen-Macaulay rings which admit a Morita self-duality.In Chapter VIII we investigate homotopy theoretic properties of torsion andcotorsion theories by studying the connections with closed model structures, thusgiving a homotopy theoretic interpretation of the results of the previous chapters.In particular we give classifications of cotorsion pairs and cotorsion triples in anabelian category and torsion pairs in a stable category in terms of closed modelstructures. As an application we give a closed model theoretic classification of(co)tilting modules over an Artin algebra.In the last two chapters we apply the results of the previous chapters to construct and investigate universal (co)homology extension functors on an abelian category C, when the stable category C/ω admits a (co)hereditary torsion theory. Weshow that these universal (co)homology extension functors are natural generalizations of the Tate-Vogel (co)homology functors studied in homological group theory,commutative algebra and representation theory. More importantly we show thatthe generalized Tate-Vogel (co)homology functors fit nicely in long exact sequencesinvolving the relative extension functors induced by the torsion or torsion-free class.Working in a Nakayama abelian category, which is a natural generalization ofthe module category of an Artin algebra, we show that there are well-behaved(co)hereditary torsion pairs which are intimately related to Cohen-Macaulay andCoCohen-Macaulay objects. We close the paper by studying the resulting universal(co)homology extension functors induced by the (Co)Cohen-Macaulay objects.Convention. From now on and following the current increasingly strongtrend, we use throughout the paper the terminology torsion pair instead of torsiontheory and TTF-triple instead of TTF-theory.Throughout the paper we compose morphisms in the diagrammatic order, i.e.the composition of morphisms f : A B and g : B C in a given category isdenoted by f g. Our additive categories admit finite direct sums.Acknowledgement. The authors would like to thank the referees for theiruseful comments.

CHAPTER ITorsion Pairs in Abelian and TriangulatedCategoriesIn this chapter we give definitions and useful properties of torsion pairs inabelian and triangulated categories. For triangulated categories we show that ourdefinition is closely related to the notion of t-structure in the sense of [18] and to thenotion of aisle in the sense of [74]. We also recall an interesting interplay betweentorsion pairs in abelian and triangulated categories related to tilting theory. Thisserves as background for more general results proved in later chapters.1. Torsion Pairs in Abelian CategoriesIn this section we recall some basic results concerning torsion pairs in an abeliancategory.We start by fixing some notation.If Z is a class of objects in an additive category C, we denote by Z : {X C C(X, Z) 0}the left orthogonal subcategory of Z, and byZ : {Y C C(Z, Y ) 0}the right orthogonal subcategory of Z.Assume now that C is an abelian category.Definition 1.1. A torsion pair in an abelian category C is a pair (X , Y)of strict (i.e. closed under isomorphisms) full subcategories of C satisfying thefollowing conditions:(i) C(X, Y ) 0, X X , Y Y.(ii) For any object C C there exists a short exact sequence:gCfC0 XC C Y C 0(1)Cin C such that XC X and Y Y.If (X , Y) is a torsion pair, then X is called a torsion class and Y is called atorsion-free class.It is well–known and easy to see that for a torsion pair (X , Y) in C we havethat X is closed under factors, extensions and coproducts and Y is closed underextensions, subobjects and products, and moreover: X Y and Y X . Conversely if C is a locally small complete and cocomplete abelian category, then anyfull subcategory of

Introduction The concept of torsion is fundamental in algebra, geometry and topology. The main reason is that torsion-theoretic methods allow us to isolate and therefore to study better, important phenomena having a local structure. The proper frame-work for the study of torsion is the context of torsion theories in a homological or homotopical .