The model-theoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their model-theoretic 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 Mod-R, the full subcategory of finitely presented modules will be denoted mod-R, the

A quiver is just a directed graph. 1 Formally, a quiver is a pair Q=(Q0,Q1) where Q0 is a finite set of vertices and Q1 is a finite set of arrows between them. If a∈Q1 is an arrow, then ta and ha

this article. Throughout, we restrict to studying finite-dimensional 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 A-mod for the category of finite-dimensional A-modules, and A-Mod for the category of all

this paper. Consider the exact functor

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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 m-by-n matrices, we prove that either Cmn consists of a finite number of points and straight lines for every m × n, or Cmn contains a 2-dimensional 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 m-by-n 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 n-by-n 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

By Drozd's Tame and Wild Theorem, ®nite-dimensional algebras over an algebraically closed ®eld k may be divided into two disjoint classes: one class consists of tame algebras whose indecomposable modules can be parametrized using only one continuous parameter, the other class consists of wild algebras

For a class of generalized Weyl algebras which includes the Weyl algebras A n the tame criteria is given for the problem of describing indecomposable weight and generalized weight modules with supports from a fixed orbit. In tame cases all the indecomposable modules are described. Introduction Let an algebra A =\Omega n 1 A i be the tensor product over an algebraically closed field K of generalized Weyl algebras A i = D i (oe i ; a i ) of degree 1 (see subsec. 1.1 and 1.4) with basic polynomial rings D i = K[H i ] in one variable, where oe i is any automorphism of D i and a i is any non-zero element of D i : Then A is a generalized Weyl algebra of degree n: If a i = H i and oe i (H i ) = H i \Gamma 1 for i = 1; : : : ; n the algebra A is isomorphic to the Weyl algebra A n of degree n (subsec. 1.4). Remark that considered generalized Weyl algebras of degree 1 include the quantum Weyl algebra A 1 (q); the quantum plane, the algebra of functions A(S 2 q ) on the quantum 2-dimens...

Let u(L; Ø) be the reduced enveloping algebra associated to a finite dimensional restricted Lie algebra (L; [p]) and a linear form Ø 2 L . It is shown that a connected component \Theta of the stable Auslander-Reiten quiver of u(L; Ø) is of type Z[A1 ], whenever its support variety VL (\Theta) has dimension 3. Various applications concerning AR-components of Lie algebras of algebraic groups and the structure of hearts of principal indecomposable u(L; Ø)-modules are given. 1. Introduction and Preliminaries In recent work [10] K. Erdmann has shown that the nonperiodic components of the stable Auslander-Reiten quivers belonging to wild blocks of a modular group algebra are isomorphic to Z[A1 ]. Accordingly, the Auslander-Reiten theory of group algebras is now very well understood. By contrast, relatively little is known about the AR-quivers of the family (u(L; Ø)) Ø2L of reduced enveloping algebras associated to a restricted Lie algebra (L; [p]). One main problem in this context is...

this paper we show that mod determines the representation type of . Recall that the algebra is either tame, i.e. all finite dimensional indecomposable-modules belong to oneparameter families, or is wild, i.e. there are two-parameter families of finite dimensional indecomposable-modules [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 one-parameter families in mod to one-parameter 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 pure-injective 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 non-finitely presented -modules we introduce a new category