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14
Model Theory and Modules
, 2006
"... 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 ..."
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Cited by 64 (20 self)
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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
Smashing Subcategories And The Telescope Conjecture  An Algebraic Approach
 Invent. Math
, 1998
"... . We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a cl ..."
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Cited by 25 (6 self)
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. We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pureinjective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory. Introduction Smashing subcategories naturally arise in the stable homotopy category S from localization functors l : S ! S which induce for every spectrum X a natural isomorphism l(X) ' X l(S) between the localization of X and the smash product of X with the localization of the sphere spectrum S. In fact, a localization functor has this property if and only if it preserv...
Realizability Of Modules Over Tate Cohomology
, 2001
"... Let k be a eld and let G be a nite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology G 2 HH 3; 1 ^ H (G; k) with the following property. Given a graded ^ H (G; k)module X, the image of G in Ext 3; 1 ^ H (G;k) (X; X) vanishes if and only if X is is ..."
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Cited by 18 (1 self)
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Let k be a eld and let G be a nite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology G 2 HH 3; 1 ^ H (G; k) with the following property. Given a graded ^ H (G; k)module X, the image of G in Ext 3; 1 ^ H (G;k) (X; X) vanishes if and only if X is isomorphic to a direct summand of ^ H (G; M) for some kGmodule M . The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a dierential graded algebra A, there is also a canonical element of Hochschild cohomology HH 3; 1 H (A) which is a predecessor for these obstructions.
Failure Of Brown Representability In Derived Categories
"... Let T be a triangulated category with coproducts, T c T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors fT c g op ! Ab are the restrictions of representable functors on T, and all natural tr ..."
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Cited by 14 (0 self)
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Let T be a triangulated category with coproducts, T c T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors fT c g op ! Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [36], it was proved that Adams' theorem remains true as long as T c is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T? In a recent paper, Beligiannis [5] made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T = D(R) of rings, and homological functors fT c g op ! Ab which are not restrictions of representables. Contents
Pure Injectives And The Spectrum Of The Cohomology Ring Of A Finite Group
 J. reine angew. Math
, 1999
"... This paper grew out of an attempt to understand this phenomenon. The purpose of this paper is to investigate a certain functor T from injective modules I over the cohomology ring H ..."
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Cited by 11 (3 self)
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This paper grew out of an attempt to understand this phenomenon. The purpose of this paper is to investigate a certain functor T from injective modules I over the cohomology ring H
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 ..."
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Cited by 8 (3 self)
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. 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...
Modules with injective cohomology, and local duality for a finite group
 New York J. Math
, 2001
"... Abstract. Let G be a finite group and k a field of characteristic p dividing G. Then Greenlees has developed a spectral sequence whose E2 term is the local cohomology of H ∗ (G, k) with respect to the maximal ideal, and which converges to H∗(G, k). Greenlees and Lyubeznik have used Grothendieck’s ..."
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Cited by 5 (2 self)
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Abstract. Let G be a finite group and k a field of characteristic p dividing G. Then Greenlees has developed a spectral sequence whose E2 term is the local cohomology of H ∗ (G, k) with respect to the maximal ideal, and which converges to H∗(G, k). Greenlees and Lyubeznik have used Grothendieck’s dual localization to provide a localized form of this spectral sequence with respect to a homogeneous prime ideal p in H ∗ (G, k), and converging to the injective hull Ip of H ∗ (G, k)/p. The purpose of this paper is give a representation theoreticinterpretation of these local cohomology spectral sequences. We construct a double complex based on Rickard’s idempotent kGmodules, and agreeing with the Greenlees spectral sequence from the E2 page onwards. We do the same for the GreenleesLyubeznik spectral sequence, except that we can only prove that the E2 pages are isomorphic, not that the spectral sequences are. Ours converges to the Tate cohomology of the certain modules κp introduced in a paper of Benson, Carlson and Rickard. This leads us to conjecture that
Brown Representability And Flat Covers
, 1999
"... this paper is devoted to proving the main result. To this end we need to recall our assumptions on the triangulated category T : ..."
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Cited by 3 (0 self)
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this paper is devoted to proving the main result. To this end we need to recall our assumptions on the triangulated category T :
Ghosts in modular representation theory
 Advances in Mathematics
"... Abstract. A ghost over a finite group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis—the statement that ghosts between finitedimensional Grepresentations factor through a projective—we define the compact ..."
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Cited by 3 (2 self)
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Abstract. A ghost over a finite group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis—the statement that ghosts between finitedimensional Grepresentations factor through a projective—we define the compact ghost number of kG to be the smallest integer l such that the composition of any l ghosts between finitedimensional Grepresentations factors through a projective. In this paper we study ghosts and the compact ghost numbers of pgroups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation k, holds for all pgroups. We do this by proving that a map between finitedimensional Grepresentations is a ghost if and only if it is a dual ghost. We then compute the compact ghost numbers of all cyclic pgroups and all abelian 2groups with C2 as a summand. We obtain bounds on the compact ghost numbers for abelian pgroups and for all 2groups which have a cyclic subgroup of index 2. Using these bounds we determine the finite abelian groups which have compact ghost number at most 2. Our methods
Generic Idempotent Modules For A Finite Group
, 1999
"... . Let G be a finite group and k an algebraically closed field of characteristic p. Let FU be the Rickard idempotent kGmodule corresponding to the set U of subvarieties of the cohomology variety VG which are not irreducible components. We show that FU is a finite sum of generic modules corresponding ..."
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Cited by 2 (0 self)
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. Let G be a finite group and k an algebraically closed field of characteristic p. Let FU be the Rickard idempotent kGmodule corresponding to the set U of subvarieties of the cohomology variety VG which are not irreducible components. We show that FU is a finite sum of generic modules corresponding to the irreducible components of VG . In this context, a generic module is an indecomposable module of infinite length over kG but finite length as a module over its endomorphism ring. 1. Introduction The purpose of this paper is to draw attention to a particular infinite dimensional module for a finite group algebra, which plays the role of a generic module in a certain precise sense which will be described. The module in question is a particular case of Rickard's construction [14] of idempotent modules in the stable category. These idempotent modules have played a pivotal role in some recent developments in modular representation theory, see for example [1, 3, 4, 5]. Theorem 1.1. Let U ...