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44
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
Higher dimensional AuslanderReiten theory on maximal orthogonal subcategories
, 2005
"... We introduce the concept of maximal orthogonal subcategories over artin algebras and orders, and develop higher AuslanderReiten theory on them. ..."
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Cited by 82 (21 self)
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We introduce the concept of maximal orthogonal subcategories over artin algebras and orders, and develop higher AuslanderReiten theory on them.
Ascent of Finite CohenMacaulay Type
 J. Algebra
, 1999
"... this paper we prove the other direction. Specifically, we prove the following theorem. ..."
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Cited by 17 (9 self)
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this paper we prove the other direction. Specifically, we prove the following theorem.
The Classification of Homogeneous CohenMacaulay Rings of Finite Representation Type
, 1988
"... In this paper we classify the homogeneous CohenMacaulay rings which are of finite representation type, that is, the CohenMacaulay rings which are positively graded and generated in degree 1, with an algebraically closed field of characteristic 0 in degree 0, and which have, up to isomorphisms and ..."
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Cited by 14 (0 self)
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In this paper we classify the homogeneous CohenMacaulay rings which are of finite representation type, that is, the CohenMacaulay rings which are positively graded and generated in degree 1, with an algebraically closed field of characteristic 0 in degree 0, and which have, up to isomorphisms and shifts in the grading, only a finite number of indecomposable maximal CohenMacaulay modules (MCMmodules). Our main contribution is to show that a homogeneous CohenMacaulay ring of finite representation type and dimension>_ _ 2 must have "minimal multiplicity". Putting this together with previous results of Auslander, AuslanderReiten, BuchweitzGreuelSchreyer, GreuelKn6rrer, and Solberg, we obtain: Theorem. Let R be a homogeneous CohenMacaulay ring. R is of finite representation type, if and only if R is isomorphic to one of the following rings: Arbitrary dimension:
MODULES IN RESOLVING SUBCATEGORIES WHICH ARE FREE ON THE PUNCTURED SPECTRUM
, 901
"... Abstract. Let R be a commutative noetherian local ring, and let X be a resolving subcategory of the category of finitely generated Rmodules. In this paper, we study modules in X by relating them to modules in X which are free on the punctured spectrum of R. We do this by investigating nonfree loci ..."
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Cited by 9 (4 self)
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Abstract. Let R be a commutative noetherian local ring, and let X be a resolving subcategory of the category of finitely generated Rmodules. In this paper, we study modules in X by relating them to modules in X which are free on the punctured spectrum of R. We do this by investigating nonfree loci and establishing an analogue of the notion of a level in a triangulated category which has been introduced by Avramov, Buchweitz, Iyengar and Miller. As an application, we prove a result on the dimension of the nonfree locus of a resolving subcategory having only countably many nonisomorphic indecomposable modules in it, which is a generalization of a theorem of Huneke and Leuschke. 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 9 (6 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 ; ...
Simplicity of rings of differential operators in prime characteristic
 Proc. London Math. Soc
"... Abstract. Let W be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric algebra of W has a simple ring of differential operato ..."
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Abstract. Let W be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric algebra of W has a simple ring of differential operators. In this paper, we show that this is true in prime characteristic. Indeed, if R is a graded subring of a polynomial ring over a perfect field of characteristic p> 0 and if the inclusion R ֒ → S splits, then Dk(R) is a simple ring. In the last section of the paper, we discuss how one might try to deduce the characteristic zero case from this result. As yet, however, this is a subtle problem and the answer to the question of Levasseur and Stafford remains open in characteristic