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41
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 ..."
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Cited by 64 (20 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
Schröer: Radical embeddings and representation dimension
 Adv. Math
"... Abstract. Given a representationfinite algebra B and a subalgebra A of B such that the Jacobson radicals of A and B coincide, we prove that the representation dimension of A is at most three. By a result of Igusa and Todorov, this implies that the finitistic dimension of A is finite. DEDICATED TO T ..."
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Cited by 16 (7 self)
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Abstract. Given a representationfinite algebra B and a subalgebra A of B such that the Jacobson radicals of A and B coincide, we prove that the representation dimension of A is at most three. By a result of Igusa and Todorov, this implies that the finitistic dimension of A is finite. DEDICATED TO THE MEMORY OF SHEILA BRENNER 1. Introduction and
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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Cited by 12 (9 self)
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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,...
Degenerations For Modules Over RepresentationFinite Biserial Algebras
 Department of Mathematics, Beijing Normal University, Beijing 100875
"... this paper form a part of my doctoral dissertation written under supervision of professor A. Skowro'nski. ..."
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Cited by 8 (1 self)
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this paper form a part of my doctoral dissertation written under supervision of professor A. Skowro'nski.
Stable endomorphism algebras of modules over special biserial algebras
 Math. Z
"... Abstract. We prove that the stable endomorphism algebra of a module without selfextensions over a special biserial algebra is a gentle algebra. In particular, it is again special biserial. As a consequence, any algebra which is derived equivalent to a gentle algebra is gentle. Dedicated to Idun Rei ..."
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Cited by 8 (0 self)
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Abstract. We prove that the stable endomorphism algebra of a module without selfextensions over a special biserial algebra is a gentle algebra. In particular, it is again special biserial. As a consequence, any algebra which is derived equivalent to a gentle algebra is gentle. Dedicated to Idun Reiten on the occasion of her 60th birthday 1.
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 6 (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 ; ...
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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Cited by 5 (0 self)
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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...
On Support Varieties Of AuslanderReiten Components
 Indag. Math
"... 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 AuslanderReiten quiver of u(L; Ø) is of type Z[A1 ], whenever its support variety VL (\Theta) has ..."
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Cited by 4 (4 self)
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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 AuslanderReiten quiver of u(L; Ø) is of type Z[A1 ], whenever its support variety VL (\Theta) has dimension 3. Various applications concerning ARcomponents 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 AuslanderReiten quivers belonging to wild blocks of a modular group algebra are isomorphic to Z[A1 ]. Accordingly, the AuslanderReiten theory of group algebras is now very well understood. By contrast, relatively little is known about the ARquivers 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...
Universal deformation rings and dihedral defect groups
, 2006
"... Abstract. Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group, and B is a block of kG with dihedral defect group D which is Morita equivalent to the principal 2modular block of a finite simple group. We deter ..."
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Cited by 4 (4 self)
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Abstract. Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group, and B is a block of kG with dihedral defect group D which is Morita equivalent to the principal 2modular block of a finite simple group. We determine the universal deformation ring R(G, V) for every kGmodule V which belongs to B and has stable endomorphism ring k. It follows that R(G, V) is always isomorphic to a subquotient ring of WD. Moreover, we obtain an infinite series of examples of universal deformation rings which are not complete intersections. 1.