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46
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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Cited by 96 (23 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
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 31 (1 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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Cited by 22 (10 self)
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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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Cited by 20 (5 self)
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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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Cited by 13 (5 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
A homological bridge between finite and infinite dimensional representations of algebras, to appear in Algebras and Representation Theory
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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