Henning Krause

the stable derived category of a noetherian scheme

for a noetherian scheme, we introduce its unbounded stable derived category. this leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. some applications are included, for instance an analogue of maximal cohen–macaulay approximations, a construction of tate cohomology, and an extension of the classical grothendieck duality. in addition, the relevance of the stable derived category in modular representation theory is indicated.

Triangulated Categories: Localization theory for triangulated categories

Introduction These notes provide an introduction to the theory of localization for triangulated categories. Localization is a machinery to formally invert morphisms in a category. We explain this formalism in some detail and we show how it is applied to triangulated categories. There are basically two ways to approach the localization theory for triangulated categories and both are closely related to each other. To explain this, let us fix a triangulated category T . The first approach is Verdier localization . For this one chooses a full triangulated subcategory S of T and constructs a universal exact functor T → T / S which annihilates the objects belonging to S . In fact, the quotient category T / S is obtained by formally inverting all morphisms σ in T such that the cone of σ belongs to S . On the other hand, there is Bousfield localization . In this case one considers an exact functor L : T → T together with a natural morphism η X : X → LX for all X in T such that L (η X ) = η( LX ) is invertible. There are two full triangulated subcategories arising from such a localization functor L . We have the subcategory Ker L formed by all L -acyclic objects, and we have the essential image Im L which coincides with the subcategory formed by all L -local objects. Note that L , Ker L , and Im L determine each other.

The spectrum of a module category

Introduction The functor category Definable subcategories Left approximations duality Ideals in the category of finitely presented modules Endofinite modules Krull-Gabriel dimension The infinite radical Functors between module categories Tame algebras Rings of definable scalars Reflective definable subcategories Sheaves Tame hereditary algebras Coherent rings Appendix A. Locally coherent Grothendieck categories Appendix B. Dimensions Appendix C. Finitely presented functors and ideals Bibliography.

Handbook of tilting theory

1. Introduction 2. Basic results of classic tilting theory L. Angeleri Hugel, D. Happel and H. Krause 3. Classification of representation-finite algebras and their modules T. Brustle 4. A spectral sequence analysis of classical tilting functors S. Brenner and M. C. R. Butler 5. Derived categories and tilting B. Keller 6. Fourier-Mukai transforms L. Hille and M. Van den Bergh 7. Tilting theory and homologically finite subcategories with applications to quasihereditary algebras I. Reiten 8. Tilting modules for algebraic groups and finite dimensional algebras S. Donkin 9. Combinatorial aspects of the set of tilting modules L. Unger 10. Cotilting dualities R. Colpi and K. R. Fuller 11. Infinite dimensional tilting modules and cotorsion pairs J. Trlifaj 12. Infinite dimensional tilting modules over finite dimensional algebras O. Solberg 13. Representations of finite groups and tilting J. Chuang and J. Rickard 14. Morita theory in stable homotopy theory B. Shipley.

Maps between tree and band modules

The indecomposable modules over special biserial algebras are known to be of a specially simple form, so-called “tree” and “band” modules (cf. [6,2]). Crawley-Boevey has shown how to describe the homomorphisms between tree modules (cf. [3]) and our objective in this paper is to extend this result to band modules. Tree and band modules occur as indecomposable representations of arbitrary algebras. Nevertheless we follow Crawley-Boevey and consider zero-relation algebras as the appropriate context. Given two tree or band modules we first study certain quiver homomorphisms between their underlying trees or bands and it turns out that a map between the representations is completely described by k-linear maps which are associated to these quiver homomorphisms. It is interesting to note that in general the classification of maps between representations of a special biserial algebra A is a wild problem. In fact the maps are equivalent to the representations of T,(A). For instance take for A the path algebra of an A,, n 2 6 then T,(A) is wild although the maps between indecomposable representations of A are easily to describe. Throughout, k will be a fixed algebraically closed field. Maps are written on the left. This paper results from a stay at the University of Liverpool supported by the Deutscher Akademischer Austauschdienst. I am very grateful to Sheila Brenner for her suggestions.

A Brown representability theorem via coherent functors

Abstract In this paper we discuss the Brown Representability Theorem for triangulated categories having arbitrary coproducts. This theorem is an extremely useful tool and various versions appear in the literature. All of them require a set of objects which generate the category in some appropriate sense. Depending on the proof, there are essentially two types: the first type is based on the analogue of iterated attaching of cells which is used in the topological case; the second type is based on solution sets and applies a variant of Freyds Adjoint Functor Theorem. Motivated by recent work of Neeman (A. Neeman, Triangulated Categories, Annals of Mathematics Studies, 148, Princeton University Press, Princeton, NJ, 2001) and Franke (On the Brown representability theorem for triangulated categories, Topology, to appear), we prove a new theorem of the first type (Theorem A) and add, as an application, a Brown Representability Theorem for covariant functors (Theorem B). The final Theorem C establishes a filtration of a triangulated category which clarifies the relation between results of the first and the second type.

Generic modules over Artin algebras

Generic modules have been introduced by Crawley-Boevey in order to provide a better understanding of nite dimensional algebras of tame representation type. In fact he has shown that the generic modules correspond to the one-parameter families of indecomposable nite dimensional modules over a tame algebra 5]. The Second Brauer-Thrall Conjecture provides another reason to study generic modules because the existence of a generic module over an artin R-algebra (R= rad R an innnite eld) implies that has strongly unbounded representation type, i.e. there are innnitely many pairwise non-isomorphic-modules of length n for innnitely many n 2 N 6]. The aim of this paper is to develop further the analysis of existence and properties of generic modules. Our approach depends to a large extent on the embedding of a module category into a bigger functor category. These general concepts are explained in the rst two sections. We continue in Section 3 with a new characterization of the pure-injective modules which occur as the source of a minimal left almost split morphism. This is of interest in our context because generic modules are pure-injective. Next we consider indecomposable endoonite modules. Recall that a module is endoonite if it is of nite length when regarded in the natural way as a module over its endomorphism ring. Changing slightly the original deenition, we say that a module is generic if it is indecomposable endoonite but not nitely presented. Section 4 is devoted to several characterizations of generic modules in order to justify the choice of the non-nitely presented modules as the generic objects. We prove them for dualizing rings, i.e. a class of rings which includes noetherian algebras and artinian PI-rings. Existence results for generic modules over dualizing rings follow in Section 5. Several results in this paper depend on the fact that a functor f: Mod(?) ! Mod(() which commutes with direct limits and products, preserves certain niteness conditions. For example, if a ?-module M is endoonite then f(M) is endoonite. If in addition End ? (M) is a PI-ring, then End (N) is a PI-ring for every indecomposable direct summand N of f(M). This material is collected in Section 6 and 7. In Section 8 we introduce an eeective method to construct generic modules over artin algebras from so-called generalized tubes. The special case of a tube in the Auslander-Reiten quiver is discussed in the following section. We obtain 1

The spectrum of a locally coherent category

A topology on the spectrum of a locally coherent Grothendieck category is introduced. The closed subsets are related to certain localizing subcategories which are characterized in terms of Serre subcategories of the full subcategory of finitely presented objects.

Stratifying triangulated categories

A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences which follow when T is stratified by R. Among them are a classification of the localizing subcategories of T in terms of subsets of the set of prime ideals in R; a classification of the thick subcategories of the subcategory of compact objects in T; and results concerning the support of the R-module of homomorphisms Hom_T^*(C,D) leading to an analogue of the tensor product theorem for support varieties of modular representation of groups.

Applications of Cotorsion Pairs

Fo ra na rtin algebra Λ, cotorsion pairs are studied for the category mod Λ of finitely presented Λ-modules and for the category Mod Λ of all Λ-modules. It is shown that every cotorsion pair for mod Λ induces ac otorsion pair for Mod Λ. This has some interesting applications, even for the category of finitely presented modules. Another theme of the paper is the interplay between cotorsion and torsion pairs. This leads to a conjecture which is an analogue of the telescope conjecture in stable homotopy theory.