Results 1  10
of
79
Model Theory and Modules
, 1988
"... The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic 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 ..."
Abstract

Cited by 96 (23 self)
 Add to MetaCart
The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic 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 ModR, the full subcategory of finitely presented modules will be denoted modR, the
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. A Grothendieck category A is said ..."
Abstract

Cited by 31 (12 self)
 Add to MetaCart
: 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 ...
Stable equivalence preserves representation type
 COMMENTARII MATHEMATICI HELVETICI
, 1997
"... ..."
Decomposing Thick Subcategories Of The Stable Module Category
 Math. Ann
, 1999
"... . Let mod kG be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a KrullSchmidt theorem for thick subcategories of modkG. It is shown that every thick tensorideal C of mod kG (i.e. a thick subcategory which is a tensor ideal) has a (usu ..."
Abstract

Cited by 15 (6 self)
 Add to MetaCart
(Show Context)
. Let mod kG be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a KrullSchmidt theorem for thick subcategories of modkG. It is shown that every thick tensorideal 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 tensorideals. This decomposition follows from a decomposition of the corresponding idempotent kGmodule EC into indecomposable modules. If C = CW is the thick tensorideal 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...