Results 1  10
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25
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 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
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 38 (12 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 12 (7 self)
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this paper we prove the other direction. Specifically, we prove the following theorem.
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 ; ...
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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Cited by 5 (1 self)
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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
Local rings of countable CohenMacaulay type
 Proc. Amer. Math. Soc
"... Abstract. We prove (the excellent case of) Schreyer’s conjecture that a local ring with countable CM type has at most a onedimensional singular locus. Furthermore we prove that the localization of a CohenMacaulay local ring of countable CM type is again of countable CM type. Let (R,m) be a (commut ..."
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Cited by 4 (0 self)
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Abstract. We prove (the excellent case of) Schreyer’s conjecture that a local ring with countable CM type has at most a onedimensional singular locus. Furthermore we prove that the localization of a CohenMacaulay local ring of countable CM type is again of countable CM type. Let (R,m) be a (commutative Noetherian) local ring of dimension d. Recall that a nonzero Rmodule M is called maximal Cohen–Macaulay (MCM) provided it is finitely generated and there exists an Mregular sequence {x1,...,xd} in the maximal ideal m. We say that R itself is Cohen– Macaulay (CM) if it is MCM as a module over itself. The CM local rings of finite CMrepresentation type (meaning that they have only finitely many nonisomorphic indecomposable MCM modules) have been carefully studied over the last twenty years. The complete equicharacteristic hypersurfaces of finite CM type have been completely classified ([6], [4], [9]), as have the complete equicharacteristic 2dimensional normal domains ([2]). More generally, it is known that a CM local ring of finite CM type has at most an isolated singularity (proved by Auslander [1] in the complete case, LeuschkeWiegand [10] in the excellent case, and HunekeLeuschke [8] in general). Yoshino’s monograph [15] is a comprehensive source for information about rings of finite CM type. The related property of countable CM type has received much less attention. Buchweitz, Greuel, and Schreyer [4] classified the complete hypersurface singularities of countable CM type, but very little more has been learned since then. The open questions and conjectures in Schreyer’s 1987 survey article [12] have inspired work on both finite and countable CMrepresentation type. For example, Conjecture 7.3(a) states that a CM local ring R has finite CM type if and only if the madic completion has finite CM type; this was recently proved in case R is excellent in [10]. This paper is concerned with another of Schreyer’s conjectures:
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 3 (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 2 (1 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.
AuslanderReiten theory revisited
 IN TRENDS IN REPRESENTATION THEORY OF ALGEBRAS AND RELATED TOPICS, EMS SER. CONGR. REP
, 2008
"... We recall several results in AuslanderReiten theory for finitedimensional algebras over fields and orders over complete local rings. Then we introduce ncluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained. ..."
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Cited by 2 (2 self)
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We recall several results in AuslanderReiten theory for finitedimensional algebras over fields and orders over complete local rings. Then we introduce ncluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained.
Representation dimension and Solomon zeta function
, 2003
"... ClineParshallScott introduced the concept of quasihereditary algebras (§2.5) to study highest weight categories in the representation theory of Lie algebras and algebraic groups [CPS1,2]. Quasihereditary algebras were effectively applied in the representation theory of artin algebras as well by ..."
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Cited by 1 (1 self)
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ClineParshallScott introduced the concept of quasihereditary algebras (§2.5) to study highest weight categories in the representation theory of Lie algebras and algebraic groups [CPS1,2]. Quasihereditary algebras were effectively applied in the representation theory of artin algebras as well by DlabRingel [DR1,2,3] and many other authors. On the other hand, in the representation theory of orders, the concept of overorders and overrings (§1.1), a noncommutative analogy of the normalization in the commutative ring theory, plays a crucial role. From an overring Γ of an order Λ, we naturally obtain a full subcategory latΓ of lat Λ. Formulating this correspondence Γ ↦ → latΓ categorically, we obtain the concept of the rejection (§1,§2). Recently it was effectively applied to study orders of finite representation type by the author [I1,2,3] and Rump [Ru1,2,3]. Originally DrozdKirichenkoRoiter found the onepoint rejection (§1.3) in their theory of Bass orders [DKR], and later HijikataNishida applied the fourpoints rejection (§1.5) to local orders of finite representation type and suggested a possibility of generalization [HN1,2,3]. In this paper, we will show that there exists a close relationship between quasihereditary algebras and the rejection from the viewpoint of the approximation theory of AuslanderSmalo [AS2]. As an application, we will solve two open problems [I4,5]. One concerns the representation dimension of artin algebras introduced by M. Auslander about 30 years ago [A1], and another concerns the Solomon zeta functions of orders introduced by L. Solomon about 25 years ago [S1,2]. It will turn out that the rejection relates these two quite different problems with each other closely. orders (Krull dimension one) artin algebras (Krull dimension zero) overrings of an order Λ factor algebras of an artin algebra Λ