Results 1  10
of
16
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 ..."
Abstract

Cited by 64 (20 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
InfiniteDimensional Modules in the Representation Theory of FiniteDimensional Algebras
, 1998
"... this article. Throughout, we restrict to studying finitedimensional associative algebras (with 1) over an algebraically closed field K, and write D for duality with the field. Except where stated, all modules are left modules. We write Amod for the category of finitedimensional Amodules, and AM ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
this article. Throughout, we restrict to studying finitedimensional associative algebras (with 1) over an algebraically closed field K, and write D for duality with the field. Except where stated, all modules are left modules. We write Amod for the category of finitedimensional Amodules, and AMod for the category of all
Generic Modules Over Artin Algebras
 Proc. London Math. Soc
, 1995
"... 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 first two sections. We continue in Section 3 with a n ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
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 first two sections. We continue in Section 3 with a new characterization of the pureinjective modules which occur as the source of a minimal left almost split morphism. This is of interest in our context because generic modules are pureinjective. Next we consider indecomposable endofinite modules. 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. Changing slightly the original definition, we say that a module is generic if it is indecomposable endofinite but not finitely presented. Section 4 is devoted to several characterizations of generic modules in order to justify the choice of the nonfinitely presented modules as the generic objects. We prove them for dualizing rings, i.e. a class of rings which includes noetherian algebras and artinian PIrings. 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(\Gamma) ! Mod() which commutes with direct limits and products, preserves certain finiteness conditions. For example, if a \Gammamodule M is endofinite then f(M) is endofinite. If in addition End \Gamma (M) is a PIring, then End (N) is a PIring for every indecomposable direct summand N of f(M ). This material is collected in Section 6 and 7. In Section 8 we introduce an effective method to construct generic modules over artin algebras from socalled generalized tubes. The special case of a tube in the AuslanderReiten quiver is discussed in t...
Stable equivalence preserves representation type
 COMMENTARII MATHEMATICI HELVETICI
, 1997
"... ..."
Some Algebraically Compact Modules. I
 I, Proceedings Padua Conference
, 1995
"... . Given a finite dimensional monomial algebra, one knows that some finite dimensional indecomposable modules may be described by words (finite sequences of letters) using as letters the arrows of the quiver and their formal inverses. To every word w, one can attach a socalled string module M(w). He ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
. Given a finite dimensional monomial algebra, one knows that some finite dimensional indecomposable modules may be described by words (finite sequences of letters) using as letters the arrows of the quiver and their formal inverses. To every word w, one can attach a socalled string module M(w). Here, we are going to construct certain infinite dimensional modules: We will consider Nwords and Zwords (thus infinite sequences of letters) satisfying suitable periodicity conditions. To every such Nword or Zword x, we describe an algebraically compact module C(x): This module C(x) is obtained from the corresponding string module M(x) as a kind of completion. Keywords. Algebraically compact modules, pure injective modules. Linearly compact modules. Quivers. Words, Nwords, Zwords. String modules, Pr ufer modules, padic modules. Special biserial algebras. Kronecker modules. Shift endomorphism. 1. Finite Words and Finite Dimensional Modules Let k be a field. Let A be a finite dimensio...
Stable Module Categories and Their Representation Type
, 1996
"... this paper we show that mod determines the representation type of . Recall that the algebra is either tame, i.e. all finite dimensional indecomposablemodules belong to oneparameter families, or is wild, i.e. there are twoparameter families of finite dimensional indecomposablemodules [8]. Of cours ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
this paper we show that mod determines the representation type of . Recall that the algebra is either tame, i.e. all finite dimensional indecomposablemodules belong to oneparameter families, or is wild, i.e. there are twoparameter families of finite dimensional indecomposablemodules [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 oneparameter families in mod to oneparameter 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 pureinjective 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 nonfinitely presented modules we introduce a new category
Endofiniteness In Stable Homotopy Theory
, 1998
"... . We study endofinite objects in a compactly generated triangulated category in terms of ideals in the category of compact objects. Our results apply in particular to the stable homotopy category. This leads, for example, to a new interpretation of stable splittings for classifying spaces of finite ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
. We study endofinite objects in a compactly generated triangulated category in terms of ideals in the category of compact objects. Our results apply in particular to the stable homotopy category. This leads, for example, to a new interpretation of stable splittings for classifying spaces of finite groups. Introduction A CWcomplex is said to be endofinite if all stable homotopy groups ß s n (X) = fS n ; Xg are finite length modules over the ring fX; Xg of stable selfmaps of X. In this paper we study CWcomplexes having this finiteness property. One motivation for this work is to understand stable splittings of certain spaces, for instance the classifying space BG of a finite group G. We shall give an algebraic interpretation of such splittings in terms of certain ideals in the category of finite spectra. A natural framework for the study of endofinite objects are compactly generated triangulated categories. Thus we consider a triangulated category C, for example the stable homo...
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
. 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 ...
HOW TO CONSTRUCT A ‘CONCRETE’ SUPERDECOMPOSABLE PUREINJECTIVE MODULE OVER A STRING ALGEBRA
"... Abstract. We construct an element in a direct product of finite dimensional modules over a string algebra such that the pureinjective envelope of this element is a superdecomposable module. A nonzero module M is said to be superdecomposable if M has no indecomposable direct summands. For an examp ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. We construct an element in a direct product of finite dimensional modules over a string algebra such that the pureinjective envelope of this element is a superdecomposable module. A nonzero module M is said to be superdecomposable if M has no indecomposable direct summands. For an example, let R be the endomorphism ring of a countable dimensional vector space V, and let I be the ideal of R consisting of all endomorphisms with finite dimensional images. If R ′ = R/I and e is a projection on a subspace of V whose image and coimage are infinite dimensional, then R ′ ∼ = eR ′ ⊕ (1 − e)R ′ as a right module over itself and R ′ ∼ = eR ′ ∼ = (1 − e ′)R ′. Furthermore, every nontrivial decomposition of R ′ as a right module is of this form, therefore R ′ is superdecomposable. For more examples, let R = k〈X, Y 〉 be a free algebra over a field k, and let E = E(RR) be the injective envelope of R considered as a right module over itself. It is easily verified (see [11, Prop. 8.36]) that RR has no nonzero uniform submodules. Therefore the same is true for E, hence E is a superdecomposable injective module. More generally, this is a common feature of finite dimensional wild algebras, that they usually (conjecturally always) have a superdecomposable pureinjective module (see [11, Ch. 8] for a list of existing results). Here a module M over a finite dimensional algebra A is pureinjective if M is a direct summand of a direct product of finite dimensional Amodules. If A is a tame finite dimensional algebra over a field, it has been believed for a while (see [24, p. 38]) that every pureinjective Amodule has an indecomposable direct summand. But recently Puninski [19] showed that every nondomestic string algebra over a countable field has a superdecomposable pureinjective module (note that every string algebra is tame). The main