Results 1  10
of
56
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 ..."
Abstract

Cited by 28 (4 self)
 Add to MetaCart
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
Derived equivalences from mutations of quivers with potential
 ADVANCES IN MATHEMATICS 226 (2011) 2118–2168
, 2011
"... ..."
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 ..."
Abstract

Cited by 14 (6 self)
 Add to MetaCart
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
Notes on A∞algebras, A∞categories and noncommutative geometry, Homological mirror symmetry
 Lecture Notes in Phys
, 2009
"... 1.1 A∞algebras as spaces........................ 2 ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
1.1 A∞algebras as spaces........................ 2
Noncommutative motives, numerical equivalence, and semisimplicity
, 2011
"... In this article we further the study of the relationship between pure motives and noncommutative motives, initiated in [25]. Making use of Hochschild homology, we introduce the category NNum(k)F of noncommutative numerical motives (over a base ring k and with coefficients in a field F). We prove t ..."
Abstract

Cited by 7 (7 self)
 Add to MetaCart
In this article we further the study of the relationship between pure motives and noncommutative motives, initiated in [25]. Making use of Hochschild homology, we introduce the category NNum(k)F of noncommutative numerical motives (over a base ring k and with coefficients in a field F). We prove that NNum(k)F isabelian semisimpleand that Grothendieck’s category Num(k)Q of numerical motives embeds in NNum(k)Q after being factored out by the action of the Tate object. As an application we obtain an alternative proof of Jannsen’s semisimplicity result, which uses the noncommutative world instead of a Weil cohomology.
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 ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
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
PERVERSE BUNDLES AND CALOGEROMOSER SPACES
, 2007
"... Abstract. We present a simple description of moduli spaces of torsionfree Dmodules (Dbundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with CalogeroMoser quiver varieties. Namely, we show that the moduli of Dbundles form twiste ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Abstract. We present a simple description of moduli spaces of torsionfree Dmodules (Dbundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with CalogeroMoser quiver varieties. Namely, we show that the moduli of Dbundles form twisted cotangent bundles to moduli of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of perverse vector bundles on T ∗ X, which contain as open subsets the moduli of framed torsionfree sheaves (the Hilbert schemes T ∗ X [n] in the rank one case). The proof is based on the description of the derived category of Dmodules on X by a noncommutative version of the Beilinson transform on P 1. 1.
CalabiYau categories and Poincaré duality spaces
, 801
"... Abstract. The singular cochain complex of a topological space is a classical object. It is a Differential Graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory. More recently, the tools of AuslanderReiten theory have also been applied t ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. The singular cochain complex of a topological space is a classical object. It is a Differential Graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory. More recently, the tools of AuslanderReiten theory have also been applied to the singular cochain complex. One of the highlights is that by these methods, each Poincaré duality space gives rise to a CalabiYau category. This paper is a review of the theory.