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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 ..."
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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
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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Cited by 25 (0 self)
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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
Canonical Matrices for Linear Matrix Problems
 Bielefeld University
, 1999
"... We consider a large class of matrix problems, which includes the problem of classifying arbitrary systems of linear mappings. For every matrix problem from this class, we construct Belitskiĭ’s algorithm for reducing a matrix to a canonical form, which is the generalization of the Jordan normal form, ..."
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We consider a large class of matrix problems, which includes the problem of classifying arbitrary systems of linear mappings. For every matrix problem from this class, we construct Belitskiĭ’s algorithm for reducing a matrix to a canonical form, which is the generalization of the Jordan normal form, and study the set Cmn of indecomposable canonical m × n matrices. Considering Cmn as a subset in the affine space of mbyn matrices, we prove that either Cmn consists of a finite number of points and straight lines for every m × n, or Cmn contains a 2dimensional plane for a certain m × n. AMS classification: 15A21; 16G60. Keywords: Canonical forms; Canonical matrices; Reduction; Classification; Tame and wild matrix problems. All matrices are considered over an algebraically closed field k; k m×n denotes the set of mbyn matrices over k. The article consists of three sections. In Section 1 we present Belitskiĭ’s algorithm [2] (see also [3]) in a form, which is convenient for linear algebra. In particular, the algorithm permits to reduce pairs of nbyn matrices to a canonical form by transformations of simultaneous similarity: (A, B) ↦ → (S −1 AS, S −1 BS); another solution of this classical problem was given by Friedland [15]. This section uses rudimentary linear algebra (except for the proof of Theorem 1.1) and may be interested for the general reader. This is the author’s version of a work that was published in Linear Algebra Appl. 317 (2000) 53–102. 1 In Section 2 we determine a broad class of matrix problems, which includes the problems of classifying representations of quivers, partially ordered sets and finite dimensional algebras. In Section 3 we get the following geometric characterization of the set of canonical matrices in the spirit of [17]: if a matrix problem does not ‘contain ’ the canonical form problem for pairs of matrices under simultaneous similarity, then its set of indecomposable canonical m × n matrices in the affine space k m×n consists of a finite number of points and straight lines (contrary to [17], these lines are unpunched). A detailed introduction is given at the beginning of every section. Each introduction may be read independently. 1 Belitskiĭ’s algorithm
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 ..."
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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
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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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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
Generic representation theory of quivers with relations, in preparation
"... Let Λ be a basic finite dimensional algebra over an algebraically closed field K. Tameness of the representation type of Λ – the only situation in which one can, at least in principle, meaningfully classify all finite dimensional representations of Λ – is a borderline phenomenon. However, even in t ..."
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Cited by 10 (9 self)
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Let Λ be a basic finite dimensional algebra over an algebraically closed field K. Tameness of the representation type of Λ – the only situation in which one can, at least in principle, meaningfully classify all finite dimensional representations of Λ – is a borderline phenomenon. However, even in the wild scenario, it is often possible to obtain a good
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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Cited by 7 (5 self)
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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 ; ...
Representation Type And Stable Equivalence Of Morita Type For Finite Dimensional Algebras
"... this paper. The representation type of a finite dimensional algebra is traditionally defined using the concept of a continuous oneparameter family in the category mod of finite dimensionalmodules. For instance, is of tame representation type if for every n 2 N only finitely many such families are ..."
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this paper. The representation type of a finite dimensional algebra is traditionally defined using the concept of a continuous oneparameter family in the category mod of finite dimensionalmodules. For instance, is of tame representation type if for every n 2 N only finitely many such families are needed to parametrize all indecomposable modules of dimension n. An alternative approach uses socalled generic modules. This was suggested by CrawleyBoevey and he established for tame algebras a correspondence between continuous oneparameter families and generic modules [3]. In [6], we used generic modules to show that a stable equivalence mod ! mod \Gamma induces a bijection between continuous oneparameter families in mod and mod \Gamma. However, without any extra assumption on the stable equivalence it remains an open question how the dimensions of the modules in mod and mod \Gamma are related. In this paper we settle the problem for stable equivalences which are induced by an appropriate functor mod ! mod \Gamma. Stable equivalences of Morita type are of this form, and therefore we can prove that two finite dimesional algebras have the same representation type if they are stably equivalent of Morita type. I would like to thank Thorsten Holm for drawing my attention to the representation type problem for derived equivalent algebras. We now begin with some notation and recall briefly the definitions which are needed in this paper. Let be an associative ring with identity. Denote by Mod the category of (right) modules and let mod be the full subcategory of finitely presentedmodules. A sequence of morphisms 0 ! L ! M ! N ! 0 in Mod is pureexact if the induced sequence 0 ! Hom(X;L) ! Hom(X;M) !