Results 1  10
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34
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
Clustertilted algebras are Gorenstein and stably
 CalabiYau, Adv. Math
"... Abstract. We prove that in a 2CalabiYau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its CohenMacaulay modules is 3CalabiYau. We deduce in particular that ..."
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Cited by 56 (12 self)
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Abstract. We prove that in a 2CalabiYau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its CohenMacaulay modules is 3CalabiYau. We deduce in particular that clustertilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [27]. In addition, we prove a general result about relative 3CalabiYau duality over non stable endomorphism rings. This strengthens and generalizes the Extgroup symmetries obtained in [27] for simple modules. Finally, we generalize the results on relative CalabiYau duality from 2CalabiYau to dCalabiYau categories. We show how to produce many examples of dcluster tilted algebras. 1.
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 37 (11 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.
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
Derived categories, resolutions, and Brown representability
, 2004
"... These notes are based on a series of five lectures given during the ..."
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Cited by 20 (2 self)
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These notes are based on a series of five lectures given during the
Exactly Definable Categories
"... this paper is to show that certain properties ofmodules become more transparent if one views them as exact functors. In particular, one can use the machinery of localization theory for locally coherent Grothendieck categories because Ex(C ..."
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Cited by 12 (7 self)
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this paper is to show that certain properties ofmodules become more transparent if one views them as exact functors. In particular, one can use the machinery of localization theory for locally coherent Grothendieck categories because Ex(C
The PopescuGabriel theorem for triangulated categories
, 2008
"... ... Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differential graded categories. An analogous result for topolog ..."
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Cited by 5 (0 self)
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... Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differential graded categories. An analogous result for topological triangulated categories has recently been obtained by A. Heider.
Hochschild cohomology and representationfinite algebras”, preprint. RagnarOlaf Buchweitz
"... Dedicated to Idun Reiten to mark her sixtieth birthday ..."
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Cited by 5 (1 self)
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Dedicated to Idun Reiten to mark her sixtieth birthday
Stable categories of higher preprojective algebras
, 2009
"... Abstract. We show that if an algebra is nrepresentationfinite then its (n + 1)preprojective algebra is selfinjective. In this situation, we show that the stable module category is (n + 1)CalabiYau, and, more precisely, it is the (n+1)Amiot cluster category of the stable nAuslander algebra. F ..."
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Cited by 5 (4 self)
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Abstract. We show that if an algebra is nrepresentationfinite then its (n + 1)preprojective algebra is selfinjective. In this situation, we show that the stable module category is (n + 1)CalabiYau, and, more precisely, it is the (n+1)Amiot cluster category of the stable nAuslander algebra. Finally we show that if the (n + 1)preprojective algebra is not selfinjective, under certain assumptions (which are always satisfied for n ∈ {1, 2}) the result above still holds for