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122
Triangulated categories of singularities and Dbranes in LandauGinzburg models
 Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.):240–262
, 2005
"... Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend ..."
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Cited by 202 (7 self)
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Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend
Support Varieties And Cohomology Over Complete Intersections
, 2000
"... this paper we develop geometric methods for the study of nite modules over a ..."
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Cited by 74 (10 self)
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this paper we develop geometric methods for the study of nite modules over a
Complete intersection dimension
"... A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CIdimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CIdimension, providing the first class of modules of (possibly) ..."
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Cited by 63 (13 self)
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A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CIdimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CIdimension, providing the first class of modules of (possibly)
Homological mirror symmetry for the genus two curve
"... The Homological Mirror Symmetry conjecture relates symplectic and algebraic geometry through their associated categorical structures. Kontsevich’s original version [31] concerned ..."
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Cited by 50 (2 self)
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The Homological Mirror Symmetry conjecture relates symplectic and algebraic geometry through their associated categorical structures. Kontsevich’s original version [31] concerned
Compact generators in categories of matrix factorizations
 MR2824483 (2012h:18014), Zbl 1252.18026, arXiv:0904.4713
"... Abstract. We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. We exhibit the stabilized residue field as a compact generator. This implies a quasiequivalence between the category of matrix factorizations and the dg derived category of an ex ..."
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Cited by 49 (1 self)
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Abstract. We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. We exhibit the stabilized residue field as a compact generator. This implies a quasiequivalence between the category of matrix factorizations and the dg derived category of an explicitly computable dg algebra. Building on this quasiequivalence we establish a derived Morita theory which identifies the functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of matrix factorization categories. Finally, we give interpretations of the results of this work in terms of noncommutative geometry modelled on dg categories. Contents
Duality and equivalence of module categories in noncommutative geometry II: Mukai . . .
, 2006
"... This is the second in a series of papers intended to set up a framework to study categories of modules in the context of noncommutative geometries. In [3] we introduced ..."
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Cited by 45 (6 self)
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This is the second in a series of papers intended to set up a framework to study categories of modules in the context of noncommutative geometries. In [3] we introduced
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 38 (15 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.
Support varieties for selfinjective algebras
 KTheory
"... Abstract. Support varieties for any finite dimensional algebra over a field were introduced in [20] using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from ..."
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Cited by 37 (12 self)
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Abstract. Support varieties for any finite dimensional algebra over a field were introduced in [20] using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webb’s theorem is true.
Boundedness versus periodicity over commutative local rings
 Trans. Amer. Math. Soc
, 1990
"... Abstract: Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with constant Betti numbers which are not eventually periodic. They provide counterexamples to a conjecture of D. Eisenbud, that every module with bounded Bett ..."
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Cited by 27 (3 self)
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Abstract: Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with constant Betti numbers which are not eventually periodic. They provide counterexamples to a conjecture of D. Eisenbud, that every module with bounded Betti numbers over a commutative local ring is eventually periodic of period 2. We prove however, that the conjecture holds over rings of small length. 1.