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17
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
Sˇtovíček, All tilting modules are of finite type
 Proc. Amer. Math. Soc
, 2005
"... Abstract. We prove that any infinitely generated tilting module is of finite type, namely that its associated tilting class is the Extorthogonal of a set of modules possessing a projective resolution consisting of finitely generated projective modules. 1. ..."
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Cited by 12 (3 self)
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Abstract. We prove that any infinitely generated tilting module is of finite type, namely that its associated tilting class is the Extorthogonal of a set of modules possessing a projective resolution consisting of finitely generated projective modules. 1.
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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Cited by 8 (0 self)
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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 ...
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 ; ...
Covers in finitely accessible categories
, 2009
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, 2005
"... Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday Let R be an artin algebra and let modR denote the category of finitely presented right Rmodules. The radical rad = rad(modR) of this category and its finite powers play a major role in the representation theory of R. The i ..."
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Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday Let R be an artin algebra and let modR denote the category of finitely presented right Rmodules. The radical rad = rad(modR) of this category and its finite powers play a major role in the representation theory of R. The intersection of these finite powers is denoted rad ω and the nilpotence of this ideal has been investigated in [13], [7] for instance. In [17] arbitrary transfinite powers, rad α, of rad were defined and linked to the extent to which morphisms in modR may be factorised. In particular it was shown that if R is an artin algebra then the transfinite radical, rad ∞ , the intersection of all ordinal powers of rad, is nonzero if and only if there is a ‘factorisable system ’ of morphisms in rad and, in that case, the KrullGabriel dimension of modR, equals ∞ (that is, is undefined). More precise results concerning the index of nilpotence of rad for artin algebras have been proved in [14], [20] [24], [25], [26]. If R is an artin algebra then any morphism between indecomposable
DEFINABLE ADDITIVE CATEGORIES
, 2008
"... This is essentially the talk I gave on definable additive categories; I define these categories, say where they came from, describe some of what is around them and then point out the 2category which they form. ..."
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This is essentially the talk I gave on definable additive categories; I define these categories, say where they came from, describe some of what is around them and then point out the 2category which they form.