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For any essentially small triangulated category, the centre of its lattice of thick subcategories is introduced; it is a spatial frame and yields a notion of central support. A relative version of this centre recovers the support theory for tensor triangulated categories and provides a universal notion of cohomological support. Along the way, we establish Mayer–Vietoris sequences for pairs of commuting subcategories. [ABSTRACT FROM AUTHOR]

For a tensor‐triangulated category and any regular cardinal α$\alpha$, we study the frame of α$\alpha$‐localizing tensor ideals and its associated space of points. For a well‐generated category and its frame of localizing tensor ideals, we provide conditions such that the associated space is obtained by refining the topology of the corresponding space for the triangulated subcategory of α$\alpha$‐compact objects. This is illustrated by several known examples for α=ℵ0$\alpha =\aleph _0$, and new spaces arise for α>ℵ0$\alpha >\aleph _0$. [ABSTRACT FROM AUTHOR]

For any ring we propose the construction of a cover which increases the finitistic dimension on one side and decreases the finitistic dimension to zero on the opposite side. This complements recent work of Cummings. [ABSTRACT FROM AUTHOR]

The notion of support provides an analogue of Stone duality, relating lattices to topological spaces. This note aims to explain in lattice theoretic terms what has been developed in the context of triangulated categories. In particular, the parallel between support via closed and open sets is addressed in terms of Hochster duality. As an application we indicate some consequences for tensor exact categories. [ABSTRACT FROM AUTHOR]

Given a tilting object of the derived category of an abelian category of finite global dimension, we give (under suitable finiteness conditions) a bound for the global dimension of its endomorphism ring. [ABSTRACT FROM AUTHOR]

We prove that every flat tensor-idempotent in the module category Mod-Tc of a tensor-triangulated category comes from a unique smashing ideal in. We deduce that the lattice of smashing ideals forms a frame. [ABSTRACT FROM AUTHOR]

Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor Ext¹(C,−) into modules over the endomorphism ring of C admits a partially defined right adjoint when C is a finitely presented object. This result seems to be new even for module categories. For appropriate schemes over a field, the connection with Serre duality is discussed. [ABSTRACT FROM AUTHOR]

We present a categorification of the non-crossing partitions given by crystallographic Coxeter groups. This involves a category of certain bilinear lattices, which are essentially determined by a symmetrisable generalised Cartan matrix together with a particular choice of a Coxeter element. Examples arise from Grothendieck groups of hereditary artin algebras. [ABSTRACT FROM AUTHOR]

We describe a procedure for constructing morphisms in additive categories, combining Auslander's concept of a morphism determined by an object with the existence of flat covers. Also, we show how flat covers are turned into projective covers and we interpret these constructions in terms of adjoint functors. [ABSTRACT FROM AUTHOR]

We compute the Krull–Gabriel dimension of the category of perfect complexes for finite dimensional algebras which are derived discrete. [ABSTRACT FROM AUTHOR]

We discuss the notion of strongly unbounded type for abelian length categories; this is closely related to the Second Brauer–Thrall Conjecture for Artin algebras. A new ingredient is the space of characters in the sense of Crawley-Boevey. [ABSTRACT FROM AUTHOR]

Inclusion-preserving maps from modules over an Artin algebra to complete partially ordered sets are studied. This yields a filtration of the Ziegler spectrum that is indexed by all Gabriel–Roiter measures. Another application is a compactness result for the set of subcategories of finitely presented modules that are closed under submodules. [ABSTRACT FROM AUTHOR]

This is a report on recent work of Chałupnik and Touzé. We explain the Koszul duality for the category of strict polynomial functors and make explicit the underlying monoidal structure which seems to be of independent interest. Then we connect this to Ringel duality for Schur algebras and describe Serre duality for strict polynomial functors. [ABSTRACT FROM PUBLISHER]

Abstract: Expansions of abelian categories are introduced. These are certain functors between abelian categories and provide a tool for induction/reduction arguments. Expansions arise naturally in the study of coherent sheaves on weighted projective lines; this is illustrated by various applications. [Copyright &y& Elsevier]

Abstract: For the module category of a hereditary ring, the Ext-orthogonal pairs of subcategories are studied. For each Ext-orthogonal pair that is generated by a single module, a 5-term exact sequence is constructed. The pairs of finite type are characterized and two consequences for the class of hereditary rings are established: homological epimorphisms and universal localizations coincide, and the telescope conjecture for the derived category holds true. However, we present examples showing that neither of these two statements is true in general for rings of global dimension 2. [ABSTRACT FROM AUTHOR]

We define support varieties in an axiomatic setting using the prime spectrum of a lattice of ideals. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. We assign to each ideal its support and can classify ideals in terms of their support. Applications arise from studying abelian or triangulated tensor categories. Specific examples from algebraic geometry and modular representation theory are discussed, illustrating the power of this approach which is inspired by recent work of Balmer. [ABSTRACT FROM AUTHOR]

Abstract: An Auslander–Reiten formula for complexes of modules is presented. This formula contains as a special case the classical Auslander–Reiten formula. The Auslander–Reiten translate of a complex is described explicitly, and various applications are discussed. [Copyright &y& Elsevier]

Abstract: We characterize additive functors which are epimorphisms up to direct factors. [Copyright &y& Elsevier]

The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S, a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic K-theory and demonstrates the relevance of the telescope conjecture for derived categories. Another application leads to a derived analogue of an almost module category in the sense of Gabber-Ramero. It is shown that the derived category of an almost ring is of this form. [ABSTRACT FROM AUTHOR]

A classical theorem of Zimmermann describes the relation between almost split sequences in the category of finitely presented modules and those in the category of all modules over some fixed ring. An analogue of Auslander–Reiten triangles in triangulated categories is proved in this paper. This is used to explain the relation between different existence results for Auslander–Reiten triangles, which are based either on Brown's representability theorem, or on the existence of Serre functors. 2000 Mathematics Subject Classification 18E30, 16G70 (primary), 14F05, 55U35 (secondary). [ABSTRACT FROM PUBLISHER]

Tilting and cotilting modules are classified for the completed path algebra of a quiver of type A˜n with linear orientation. This classification problem arises naturally in the classification of cotilting modules over certain associative algebras (Pacific J. Math. 211 (2003) 41–60). The combinatorics of the collection of all tilting and cotilting modules is described in terms of Stasheff associahedra. [Copyright &y& Elsevier]

A uniqeness result for direct sum decompositions into uniform objects is proved. The result is shown to be equivalent to the classical Krull–Remak–Schmidt Theorem for decompositions into indecomposable objects with local endomorphism rings. [Copyright &y& Elsevier]

We give a short proof of Auslander’s formula relating the covariant and the contravariant defect of a short exact sequence in a module category. An immediate consequence is the classical Auslander–Reiten formula relating Ext and Hom. [Copyright &y& Elsevier]

For a right artinian ring A we show that for every n ≥ 0 there exists a pure-injective A-module P[subn] such that the A-modules of projective dimension at most n are precisely the direct factors of A-modules having a finite filtration in products of copies of P[subn]. This is a consequence of a general description of certain contravariantly finite resolving subcategories of Mod A. It leads in addition to a one-to-one correspondence between equivalence classes of (not necessarily finitely generated) cotilting modules and resolving subcategories of Mod A which are closed under products and admit finite resolutions and special right approximations. As an application it is shown that every finitely presented partial cotilting module over an artin algebra admits a complement. [ABSTRACT FROM AUTHOR]

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 Freyd''s 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. [Copyright &y& Elsevier]

The notion of tameness for the derived category of a finite dimensional algebra is introduced and standard properties are established. This is based on classical tameness definitions of Drozd and Crawley-Boevey for the category of finite dimensional representations. [ABSTRACT FROM AUTHOR]

Let Λ and Γ be finite dimensional algebras. It is shown that any stable equivalence f:mod¯Λ→mod¯Γ between the categories of finitely generated modules induces a bijection M ↦ Mf between the sets of isomorphism classes of generic modules over Λ and Γ such that the endolength of Mf is bounded by the endolength of M up to a scalar which depends only on f. Using Crawley-Boevey's characterization of tame representation type in terms of generic modules, one obtains as a consequence a new proof for the fact that a stable equivalence preserves tameness. This proof also shows that polynomial growth is preserved. 2000 Mathematics Subject Classification 16G60, 16D90. [ABSTRACT FROM AUTHOR]

We develop a support theory for elementary supergroup schemes, over a field of positive characteristic p ≥ 3, starting with a definition of a π-point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and π-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra k[t,τ]/(tp−τ2), where t has even degree and τ has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme. [ABSTRACT FROM AUTHOR]

A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the p-local and p-torsion subcategories of the stable category, for each homogeneous prime ideal p in the cohomology ring of the group scheme. [ABSTRACT FROM AUTHOR]

Local cohomology functors are constructed for the category of cohomological functors on an essentially small triangulated category ⊺ equipped with an action of a commutative noetherian ring. This is used to establish a local-global principle and to develop a notion of stratification, for ⊺ and the cohomological functors on it, analogous to such concepts for compactly generated triangulated categories. [ABSTRACT FROM AUTHOR]

The Hom closed colocalizing subcategories of the stable module category of a finite group are classified. Along the way, the colocalizing subcategories of the homotopy category of injectives over an exterior algebra, and the derived category of a formal commutative differential graded algebra, are classified. To this end, and with an eye towards future applications, a notion of local homology and cosupport for triangulated categories is developed, building on earlier work of the authors on local cohomology and support. [ABSTRACT FROM AUTHOR]

Abstract: The localising subcategories of the derived category of the cochains on the classifying space of a finite group are classified. They are in one to one correspondence with the subsets of the set of homogeneous prime ideals of the cohomology ring . [Copyright &y& Elsevier]

A notion of stratification is introduced for any compactly generated triangulated category ⊤ endowed with an action of a graded‐commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences that follow if ⊤ is stratified by R. Among them are a classification of the localizing subcategories of ⊤ 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 ⊤; and results on the support of the graded R‐module of morphisms Hom*⊤(C, D) leading to analogs of the tensor product theorem for support varieties of modular representation of groups. [ABSTRACT FROM AUTHOR]

A notion of stratification is introduced for any compactly generated triangulated category  endowed with an action of a graded-commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences that follow if  is stratified by R. Among them are a classification of the localizing subcategories of  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 ; and results on the support of the graded R-module of morphisms Hom*(C, D) leading to analogs of the tensor product theorem for support varieties of modular representation of groups. [ABSTRACT FROM PUBLISHER]