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46
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
Tilting theory and cluster combinatorics
 572–618. EQUIVALENCE AND GRADED DERIVED EQUIVALENCE 43
"... of a finitedimensional hereditary algebra H over a field. We show that, in the simplylaced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin–Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin–Zelevinsk ..."
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Cited by 55 (4 self)
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of a finitedimensional hereditary algebra H over a field. We show that, in the simplylaced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin–Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin–Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of selfinjective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APRtilting.
Cluster mutation via quiver representations
 Comment. Math. Helv
"... Abstract. Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretic interpretation of ..."
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Cited by 45 (15 self)
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Abstract. Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretic interpretation of cluster mutation in case of acyclic cluster algebras.
Cluster structures for 2CalabiYau categories and unipotent groups
"... Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This c ..."
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Cited by 32 (6 self)
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Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This class of 2CalabiYau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2CalabiYau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related
Clustertilted algebras
 Trans. Amer. Math. Soc
"... Abstract. We introduce a new class of algebras, which we call clustertilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of th ..."
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Cited by 31 (3 self)
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Abstract. We introduce a new class of algebras, which we call clustertilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of socalled APRtilting.
Smashing Subcategories And The Telescope Conjecture  An Algebraic Approach
 Invent. Math
, 1998
"... . We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a cl ..."
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Cited by 25 (6 self)
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. We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pureinjective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory. Introduction Smashing subcategories naturally arise in the stable homotopy category S from localization functors l : S ! S which induce for every spectrum X a natural isomorphism l(X) ' X l(S) between the localization of X and the smash product of X with the localization of the sphere spectrum S. In fact, a localization functor has this property if and only if it preserv...
Realizability Of Modules Over Tate Cohomology
, 2001
"... Let k be a eld and let G be a nite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology G 2 HH 3; 1 ^ H (G; k) with the following property. Given a graded ^ H (G; k)module X, the image of G in Ext 3; 1 ^ H (G;k) (X; X) vanishes if and only if X is is ..."
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Cited by 20 (1 self)
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Let k be a eld and let G be a nite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology G 2 HH 3; 1 ^ H (G; k) with the following property. Given a graded ^ H (G; k)module X, the image of G in Ext 3; 1 ^ H (G;k) (X; X) vanishes if and only if X is isomorphic to a direct summand of ^ H (G; M) for some kGmodule M . The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a dierential graded algebra A, there is also a canonical element of Hochschild cohomology HH 3; 1 H (A) which is a predecessor for these obstructions.
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.
Covers Induced by Ext
 J. Algebra
"... . We prove a generalization of the Flat Cover Conjecture by showing for any ring R that (1) each (right R) module has a Ker Ext(, C)cover, for any class of pureinjective modules C, and that (2) each module has a Ker Tor(, B)cover, for any class of left Rmodules B. For Dedekind domains ..."
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Cited by 11 (7 self)
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. We prove a generalization of the Flat Cover Conjecture by showing for any ring R that (1) each (right R) module has a Ker Ext(, C)cover, for any class of pureinjective modules C, and that (2) each module has a Ker Tor(, B)cover, for any class of left Rmodules B. For Dedekind domains, we describe Ker Ext(, C) explicitly for any class of cotorsion modules C; in particular, we prove that (1) holds, and that Ker Ext(, C) is a cotilting torsionfree class. For right hereditary rings, we prove the consistency of the existence of special Ker Ext(, G)precovers for any set of modules G. 1. Introduction A classical result of EckmannSchopf says that if I is the class of all injective (right R) modules, then each module has an Ienvelope. Bass proved that if P is the class of all projective modules, then each module has a Pcover i# R is a right perfect ring. Bass' result is often interpreted as a lack of duality for modules over nonright perfect rings. Cal...