The model-theoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their model-theoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se. Our default is that the term “module ” will mean (unital) right module over a ring (associative with 1) R. The category of such modules is denoted Mod-R, the full subcategory of finitely presented modules will be denoted mod-R, the

: 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. A Grothendieck category A is said to be locally coherent provided that A has a generating set of finitely presented objects and the full subcategory fp(A) of finitely presented objects in A is abelian. The spectrum sp(A) of A is a representative set of indecomposable injective objects in A. We show that this set carries a natural topology and it is the purpose of this paper to establish a natural and bijective correspondence between the following structures which arise for each locally coherent category A: -- Serre subcategories of fp(A), -- hereditary torsion theories of finite type for A, -- closed subsets of sp(A). This analysis is motivated by some construction which reduces the theory of purity for a locally finitely ...

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this paper is to show that certain properties of-modules become more transparent if one views them as exact functors. In particular, one can use the machinery of localization theory for locally coherent Grothendieck categories because Ex(C

. Let mod kG be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a Krull-Schmidt theorem for thick subcategories of modkG. It is shown that every thick tensor-ideal C of mod kG (i.e. a thick subcategory which is a tensor ideal) has a (usually infinite) unique decomposition C = ` i2I C i into indecomposable thick tensor-ideals. This decomposition follows from a decomposition of the corresponding idempotent kG-module EC into indecomposable modules. If C = CW is the thick tensor-ideal corresponding to a closed homogeneous subvariety W of the maximal ideal spectrum of the cohomology ring H (G; k), then the decomposition of C reflects the decomposition W = S n i=1 W i of W into connected components. Introduction In modular representation theory of finite groups, one frequently passes to the stable module category which is a triangulated category. Following ideas from stable homotopy theory, Benson, Carlson, and Rickard s...

R is Iwanaga-Gorenstein if and only if the class of Gorenstein projective right modules is induced by a finitely generated cotilting module (Corollary 3.8). This work was started while the first author was a Ramon y Cajal Fellow at the Universitat Aut`onoma de Barcelona. The research of the second author was funded by the DGI (Spain) and FEDER through Project BFM2002-01390, and by the Comissionat per Universitats i Recerca of the Generalitat de Catalunya. Third author supported by grant SAB2001-0092 of Secretaria de Estado de Education y Universidades MECD at CRM IEC Barcelona, and by MSM 113200007. In the final section, we provide for an explicit construction of tilting and cotilting modules over any commutative 1-Gorenstein ring R (Example 4.1). By [32], it is consistent with ZFC that these are the only tilting and cotilting modules up to equivalence in case R is a small Dedekind domain. 1 Preliminaries Let R be a ring. Denote by ModR the category of all (right R-) modules, an

Abstract. Over an associative ring we consider a class X of left modules which is closed under set-indexed coproducts and direct summands. We investigate when the triangulated homotopy category K(X) is compactly generated, and give a number of examples.

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this paper we show that mod determines the representation type of . Recall that the algebra is either tame, i.e. all finite dimensional indecomposable-modules belong to oneparameter families, or is wild, i.e. there are two-parameter families of finite dimensional indecomposable-modules [8]. Of course, one feels that this dichotomy should not depend on the deletion of finitely many objects in the category mod , and this is precisely one of the main results of this paper. More precisely, given another algebra \Gamma and an equivalence mod ! mod \Gamma, then \Gamma is tame if is tame. Moreover, we show that the equivalence sends the one-parameter families in mod to one-parameter families in mod \Gamma. The fact that mod determines the representation type of also follows, for some classes of symmetric algebras, from recent work of Assem, de la Pe~na and Erdmann [2, 9]; however their methods are completely different. Equivalences between stable module categories have been studied by many authors. They naturally occur for instance in representation theory of finite groups. Another source of examples, which includes every algebra of Loewy length 2, is the class of algebras stably equivalent to a hereditary algebra. Usually the analysis concentrates on homological properties of the category mod which are preserved by an equivalence mod ! mod \Gamma. In this paper we follow a different approach. We investigate pure-injective modules which are not necessarily finitely presented. Among them the endofinite modules are of particular interest. Recall that a module is endofinite if it is of finite length when regarded in the natural way as a module over its endomorphism ring. In order to study the non-finitely presented -modules we introduce a new category

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