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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 ..."
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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
The Ziegler and Zariski spectra of some domestic string algebras
 Theory
, 1999
"... It was a conjecture of the second author that the CantorBendixson rank of the Ziegler spectrum of a nitedimensional algebra is either less than or equal to 2 or is undened. Here we refute this conjecture by describing the Ziegler spectra of some domestic string algebras where arbitrary nite va ..."
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It was a conjecture of the second author that the CantorBendixson rank of the Ziegler spectrum of a nitedimensional algebra is either less than or equal to 2 or is undened. Here we refute this conjecture by describing the Ziegler spectra of some domestic string algebras where arbitrary nite values greater than 2 are obtained. We give a complete description of the Ziegler and GabrielZariski spectra of the simplest of these algebras. The conjecture has been independently refuted by Schroer [21] who, extending his work [20] on these algebras, computed their KrullGabriel dimension. 1 Indecomposable pureinjectives over domestic string algebras 12 Let R be a domestic string algebra over an arbitrary eld k. Modules will generally be left Rmodules: the category of these we denote by RMod. 1 This work was done while the rst author was supported by EPSRC grant number GR/K19686. Both authors thank the EPSRC for this nancial support. 2 Primary: 16G20; Secondary: 03C60,...
Exactly Definable Categories
"... this paper is to show that certain properties ofmodules 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 ..."
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this paper is to show that certain properties ofmodules 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
Cotorsion theories and splitters
, 2000
"... Let R be a subring of the rationals. We want to investigate self splitting Rmodules G that is Ext R(G, G) = 0 holds and follow Schultz [22] to call such modules splitters. Free modules and torsionfree cotorsion modules are classical examples for splitters. Are there others? Answering an open prob ..."
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Let R be a subring of the rationals. We want to investigate self splitting Rmodules G that is Ext R(G, G) = 0 holds and follow Schultz [22] to call such modules splitters. Free modules and torsionfree cotorsion modules are classical examples for splitters. Are there others? Answering an open problem by Schultz [22] we will show that there are more splitters, in fact we are able to prescribe their endomorphism Ralgebras with a free Rmodule structure. As a byproduct we are able to answer a problem of Salce [21] showing that all rational cotorsion theories have enough injectives and enough projectives. 647 revision:19991026 modified:19991026 1
Module and Comodule Categories  A Survey
 Proc. of the Mathematics Conference (Birzeit University 1998), World Scientific
, 2000
"... The theory of modules over associative algebras and the theory of comodules for coassociative coalgebras were developed fairly independently during the last decades. In this survey we display an intimate connection between these areas by the notion of categories subgenerated by an object. After a re ..."
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The theory of modules over associative algebras and the theory of comodules for coassociative coalgebras were developed fairly independently during the last decades. In this survey we display an intimate connection between these areas by the notion of categories subgenerated by an object. After a review of the relevant techniques in categories of left modules, applications to the bimodule structure of algebras and comodule categories are sketched.
The Ziegler spectrum of a tame hereditary algebra
 Coll. Math
"... . Let A be a finite dimensional hereditary algebra of tame representation type. Let ComA be a complete set of indecomposable algebraically compact Amodules (one from each isomorphism class). We are going to give an explicit description of the closed subsets of ComA in the sense of Ziegler. 1. Intro ..."
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. Let A be a finite dimensional hereditary algebra of tame representation type. Let ComA be a complete set of indecomposable algebraically compact Amodules (one from each isomorphism class). We are going to give an explicit description of the closed subsets of ComA in the sense of Ziegler. 1. Introduction Let k be a field, let A be a connected finite dimensional hereditary kalgebra of tame representation type. For a description of the finite dimensional hereditary kalgebras we refer to [DR2]. Let ModA denote the category of all Amodules and modA the full subcategory of Amodules of finite length. The global structure of the category of all Amodules is wellknown, see [DR1] and [R1]. Let ComA be a set of indecomposable algebraically compact Amodules, one from each isomorphism class. Let ind A be the subset of elements of ComA of finite length (since any finite length module is algebraically compact, ind A is just a complete set of indecomposable Amodules of finite length). If ...
ON CENTRAL EXTENSIONS AND DEFINABLY COMPACT GROUPS IN OMINIMAL STRUCTURES
"... Abstract. We prove several structural results on definably compact groups G in ominimal expansions of real closed fields such as (i) G is definably an almost direct product of a semisimple group and a commutative group, (ii) (G, ·) is elementarily equivalent to (G/G 00, ·). We also prove results on ..."
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Abstract. We prove several structural results on definably compact groups G in ominimal expansions of real closed fields such as (i) G is definably an almost direct product of a semisimple group and a commutative group, (ii) (G, ·) is elementarily equivalent to (G/G 00, ·). We also prove results on the internality of finite covers of G in an ominimal environment, as well as deducing the full compact domination conjecture for definably compact groups from the semisimple and commutative cases which were already settled. These results depend on key theorems about the interpretability of central and finite extensions of definable groups, in the ominimal context. These methods and others also yield interpretability results for universal covers of arbitrary definable real Lie groups. 1. Introduction and
Abstract elementary classes
 Annals of Pure Appl. Logic
"... We show that the concept of an Abstract Elementary Class (AEC) provides a unifying notion for several properties of classes of modules and discuss the stability class of these AEC. An abstract elementary class consists of a class of models K and a strengthening of the notion of submodel ≺ K such tha ..."
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We show that the concept of an Abstract Elementary Class (AEC) provides a unifying notion for several properties of classes of modules and discuss the stability class of these AEC. An abstract elementary class consists of a class of models K and a strengthening of the notion of submodel ≺ K such that (K, ≺ K) satisfies the properties described below. Here we deal with various classes ( ⊥ N, ≺N); the precise definition of the class of modules ⊥ N is given below. A key innovation is that A≺NB means A ⊆ B and B/A ∈ ⊥ N. We define in the text the main notions used here; important background definitions and proofs from the theory of modules can be found in [EM02] and [GT06]; concepts of AEC are due to Shelah (e.g. [She87]) but collected in [Bal]. The surprising fact is that some of the basic model theoretic properties of the class ( ⊥ N, ≺N) translate directly to algebraic properties of the class ⊥ N and the module N over the ring R that have previously been studied in quite a different context (approximation theory of modules, infinite dimensional tilting theory etc.). Our main results, stated with increasingly strong conditions on the ring R, are:
Topological and Geometric aspects of the Ziegler Spectrum
, 1998
"... The aim here is to emphasise the topological and geometric structure that the Ziegler spectrum carries and to illustrate how this structure may be used in the analysis of particular examples. There is not space here for me to give a survey of what is known about the Ziegler spectrum so there are ..."
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The aim here is to emphasise the topological and geometric structure that the Ziegler spectrum carries and to illustrate how this structure may be used in the analysis of particular examples. There is not space here for me to give a survey of what is known about the Ziegler spectrum so there are a number of topics that I will just mention in order to give some indication of what lies beyond what is discussed here. 1. The Ziegler spectrum 2. Various dimensions 3. These dimensions for artin algebras 4. These dimensions in general 5. Duality 6. The complexity of morphisms in modR 7. The GabrielZariski topology 8. The sheaf of locally definable scalars 1 The Ziegler spectrum 1.1 A reminder on purity and pureinjectives Suppose that M is a submodule of N . Consider a finite system \Sigma n i=1 x i r ij = a j (j = 1; :::m) of Rlinear equations over M : that is, the r ij are in R, the 1 a j are in M and the x i are indeterminates. Suppose that there is a solution b 1 ; ...
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 ..."
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. 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...