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22
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 ALGEBRAS, QUIVER REPRESENTATIONS AND TRIANGULATED CATEGORIES
"... Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). I ..."
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Cited by 32 (5 self)
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Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences. Contents
Cluster tilting for higher Auslander algebras
"... Abstract. The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the ca ..."
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Cited by 13 (4 self)
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Abstract. The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category Mn of preinjectivelike modules obtained by applying τn to injective modules repeatedly. We call a finite dimensional algebra Λ ncomplete if Mn = add M for an ncluster tilting object M. Our main result asserts that the endomorphism algebra EndΛ(M) is (n + 1)complete. This gives an inductive construction of ncomplete algebras. For example, any representationfinite hereditary algebra Λ (1) is 1complete. Hence the Auslander algebra Λ (2) of Λ (1) is 2complete. Moreover, for any n ≥ 1, we have an ncomplete algebra Λ (n) which has an ncluster tilting object M (n) such that Λ (n+1) = End Λ (n)(M (n)). We give the presentation of Λ (n) by a quiver with relations. We apply our results to construct ncluster tilting subcategories of derived categories of ncomplete algebras. Contents 1. Our results 3 1.1. ncluster tilting in module categories 4
Generic Modules Over Artin Algebras
 Proc. London Math. Soc
, 1995
"... this paper is to develop further the analysis of existence and properties of generic modules. Our approach depends to a large extent on the embedding of a module category into a bigger functor category. These general concepts are explained in the first two sections. We continue in Section 3 with a n ..."
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Cited by 12 (2 self)
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this paper is to develop further the analysis of existence and properties of generic modules. Our approach depends to a large extent on the embedding of a module category into a bigger functor category. These general concepts are explained in the first two sections. We continue in Section 3 with a new characterization of the pureinjective modules which occur as the source of a minimal left almost split morphism. This is of interest in our context because generic modules are pureinjective. Next we consider indecomposable endofinite modules. Recall that a module is endofinite if it is of finite length when regarded in the natural way as a module over its endomorphism ring. Changing slightly the original definition, we say that a module is generic if it is indecomposable endofinite but not finitely presented. Section 4 is devoted to several characterizations of generic modules in order to justify the choice of the nonfinitely presented modules as the generic objects. We prove them for dualizing rings, i.e. a class of rings which includes noetherian algebras and artinian PIrings. Existence results for generic modules over dualizing rings follow in Section 5. Several results in this paper depend on the fact that a functor f : Mod(\Gamma) ! Mod() which commutes with direct limits and products, preserves certain finiteness conditions. For example, if a \Gammamodule M is endofinite then f(M) is endofinite. If in addition End \Gamma (M) is a PIring, then End (N) is a PIring for every indecomposable direct summand N of f(M ). This material is collected in Section 6 and 7. In Section 8 we introduce an effective method to construct generic modules over artin algebras from socalled generalized tubes. The special case of a tube in the AuslanderReiten quiver is discussed in t...
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Cited by 8 (7 self)
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Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
Acyclic CalabiYau categories, with an appendix by Michel Van den Bergh, preprint
, 2006
"... Abstract. We show that an algebraic 2CalabiYau triangulated category over an algebraically closed field is a cluster category if it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As a first application, ..."
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Cited by 7 (1 self)
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Abstract. We show that an algebraic 2CalabiYau triangulated category over an algebraically closed field is a cluster category if it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As a first application, we show that the stable category of maximal CohenMacaulay modules over a certain isolated singularity of dimension three is a cluster category. As a second application, we prove the nonacyclicity of the quivers of endomorphism algebras of clustertilting objects in the stable categories of representationinfinite preprojective algebras. In the appendix, Michel Van den Bergh gives an alternative proof of the main theorem by appealing to the universal property of the triangulated orbit category. 1.
The classification of special Cohen Macaulay modules
, 809
"... Abstract. In this paper we completely classify all the special CohenMacaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In every case we exhibit the specials explicitly in a combinatorial way. Our ..."
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Cited by 4 (4 self)
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Abstract. In this paper we completely classify all the special CohenMacaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In every case we exhibit the specials explicitly in a combinatorial way. Our result relies on realizing the specials as those CM modules whose first Ext group vanishes against the ring R, thus reducing the problem to combinatorics on the AR quiver; such possible AR quivers were classified by Auslander and Reiten. We also give some general homological properties of the special CM modules and their corresponding reconstruction algebras. Contents
From Auslander algebras to tilted algebras
"... For an (n − 1)Auslander algebra Λ with global dimension n ≥ 2, we show that if Λ admits a trivial maximal (n − 1)orthogonal subcategory of mod Λ, then Λ is of finite representation type and the projective dimension or injective dimension of any indecomposable module in mod Λ is at most n − 1. As a ..."
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Cited by 2 (2 self)
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For an (n − 1)Auslander algebra Λ with global dimension n ≥ 2, we show that if Λ admits a trivial maximal (n − 1)orthogonal subcategory of mod Λ, then Λ is of finite representation type and the projective dimension or injective dimension of any indecomposable module in mod Λ is at most n − 1. As a result, we have that for an Auslander algebra Λ with global dimension 2, if Λ admits a trivial maximal 1orthogonal subcategory of mod Λ, then Λ is a tilted algebra of finite representation type; furthermore, in case there exists a unique simple module in mod Λ with projective dimension 2, then the converse also holds true. 1.