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Derived equivalences from mutations of quivers with potential
 ADVANCES IN MATHEMATICS 226 (2011) 2118–2168
, 2011
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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 46 (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
Notes on A∞algebras, A∞categories and noncommutative geometry, Homological mirror symmetry
 Lecture Notes in Phys
, 2009
"... 1.1 A∞algebras as spaces........................ 2 ..."
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Cited by 43 (0 self)
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1.1 A∞algebras as spaces........................ 2
Hochschild homology and semiorthogonal decompositions
"... Abstract. We investigate Hochschild cohomology and homology of admissible subcategories of derived categories of coherent sheaves on smooth projective varieties. We show that the Hochschild cohomology of an admissible subcategory is isomorphic to the derived endomorphisms of the kernel giving the co ..."
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Abstract. We investigate Hochschild cohomology and homology of admissible subcategories of derived categories of coherent sheaves on smooth projective varieties. We show that the Hochschild cohomology of an admissible subcategory is isomorphic to the derived endomorphisms of the kernel giving the corresponding projection functor, and the Hochschild homology is isomorphic to derived morphisms from this kernel to its convolution with the kernel of the Serre functor. We investigate some basic properties of Hochschild homology and cohomology of admissible subcategories. In particular, we check that the Hochschild homology is additive with respect to semiorthogonal decompositions and construct some long exact sequences relating the Hochschild cohomology of a category and its semiorthogonal components. We also compute Hochschild homology and cohomology of some interesting admissible subcategories, in particular of the nontrivial components of derived categories of some Fano threefolds and of the nontrivial components of the derived categories of conic bundles. 1.
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 24 (9 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
Topological equivalences for differential graded algebras
 Adv. Math
, 2006
"... Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an EilenbergMac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are ..."
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Cited by 22 (8 self)
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Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an EilenbergMac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are topologically equivalent, but we produce explicit counterexamples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting. Contents
Categorical resolution of singularities
"... Abstract. Building on the concept of a smooth DG algebra we define the notion of a smooth derived category. We then propose the definition of a categorical resolution of singularities. Our main examples are concerned with a categorical resolution of the derived category of quasicoherent sheaves on ..."
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Cited by 22 (6 self)
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Abstract. Building on the concept of a smooth DG algebra we define the notion of a smooth derived category. We then propose the definition of a categorical resolution of singularities. Our main examples are concerned with a categorical resolution of the derived category of quasicoherent sheaves on a scheme. We propose two kinds of such resolutions. Contents
DIMENSIONS OF TRIANGULATED CATEGORIES VIA KOSZUL OBJECTS
"... Abstract. Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin alge ..."
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Cited by 20 (10 self)
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Abstract. Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras. 1.
Motivic structures in noncommutative geometry. Available at arXiv:1003.3210
 the Proceedings of the ICM
, 2010
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