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292
Symplectic reflection algebras, CalogeroMoser space, and deformed HarishChandra homomorphism
 Invent. Math
"... To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multiparameter deformation, Hκ, of the algebra C[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of CP r, where r = number of conjugacy classes of symplectic ..."
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Cited by 179 (36 self)
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To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multiparameter deformation, Hκ, of the algebra C[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of CP r, where r = number of conjugacy classes of symplectic reflections in Γ. The algebra Hκ, called a symplectic reflection algebra, is expected to be related to the coordinate ring of a universal Poisson deformation of the quotient singularity V/Γ. If Γ is the Weyl group of a root system in a vector space h and V = h ⊕ h ∗ , then the algebras Hκ are certain ‘rational ’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let Γ = Sn, the Weyl group of g = gl n. We construct a 1parameter deformation of the HarishChandra homomorphism from D(g) g, the algebra of invariant polynomial differential operators on the Lie algebra g = gl n, to the algebra of Sninvariant differential operators on the Cartan subalgebra C n with rational coefficients. The second order Laplacian on g goes, under our deformed homomorphism, to the CalogeroMoser
Categorical homotopy theory
 Homology, Homotopy Appl
"... This paper is an exposition of the ideas and methods of Cisinksi, in the context of Apresheaves on a small ..."
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Cited by 164 (7 self)
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This paper is an exposition of the ideas and methods of Cisinksi, in the context of Apresheaves on a small
Isomorphism conjectures in algebraic Ktheory
 J. Amer. Math. Soc
, 1993
"... 1.1 The Isomorphism Conjecture in algebraic Ktheory........ 2 1.2 Main Results and Corollaries.................... 4 1.3 A brief outline............................ 6 ..."
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Cited by 109 (12 self)
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1.1 The Isomorphism Conjecture in algebraic Ktheory........ 2 1.2 Main Results and Corollaries.................... 4 1.3 A brief outline............................ 6
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 94 (4 self)
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Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend
Stability conditions on triangulated categories
, 212
"... This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas’s notion of Πstability. From a mathematical point of view, the most interesting feature of the definition ..."
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Cited by 78 (6 self)
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This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas’s notion of Πstability. From a mathematical point of view, the most interesting feature of the definition is that the set of stability
Deriving Dg Categories
, 1993
"... We investigate the (unbounded) derived category of a differential Zgraded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categories. After adapt ..."
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Cited by 74 (8 self)
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We investigate the (unbounded) derived category of a differential Zgraded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a 'Morita theorem` (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories. Summary We give an account of the contents of this paper for the special case of DG algebras. Let k be a commutative ring and A a DG (k)algebra, i.e. a Zgraded kalgebra A = a p2Z A p endowed with a differential d of degree 1 such that d(ab) = (da)b + (\Gamma1) p a(db) for all a 2 A p , b 2 A. A DG (right) Amodule is a Zgraded Amodule M = ` p2Z M p endowed with a differential d of degree 1 such that d(ma) = (dm)a + (\Gamma1) p m(da) for all m 2 M p , a 2 A. A morphism of DG Amodules is a homogeneous morphism of degree 0 of the underlying graded Amodules commuting with the differentials. The DG Amodules form an abelian category CA. A morphism f : M ! N of CA is nullhomotopic if f = dr + rd for some homogeneous morphism r : M ! N of degree1 of the underlying graded Amodules.
CHAIN COMPLEXES AND STABLE CATEGORIES
 MANUS. MATH.
, 1990
"... Under suitable assumptions, we extend the inclusion of an additive ... complexes concentrated in positive degrees. We thereby obtain a new proof for the key result of J. Rickard’s ’Morita theory for Derived categories ‘ [17] and a sharpening of a theorem of Happel [12, 10.10] on the ’moduletheoreti ..."
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Cited by 52 (8 self)
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Under suitable assumptions, we extend the inclusion of an additive ... complexes concentrated in positive degrees. We thereby obtain a new proof for the key result of J. Rickard’s ’Morita theory for Derived categories ‘ [17] and a sharpening of a theorem of Happel [12, 10.10] on the ’moduletheoretic description ‘ of the derived
Derived categories of coherent sheaves and triangulated categories of singularities
, 2005
"... ..."
Higher topos theory
, 2006
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain com ..."
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Cited by 47 (0 self)
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Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of Gvalued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category
THE SPECTRAL SEQUENCE RELATING ALGEBRAIC KTHEORY TO MOTIVIC COHOMOLOGY
"... The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic Ktheory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilins ..."
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Cited by 45 (5 self)
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The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic Ktheory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilinson [Be] as a natural analogue of the AtiyahHirzebruch spectral sequence from the singular cohomology to the topological Ktheory of a topological space. The expectation of such a spectral sequence has provided much of the impetus for the development of motivic cohomology (e.g., [B1], [V2]) and should facilitate many computations in algebraic Ktheory. In the special case in which X equals SpecF, this spectral sequence was established by S. Bloch and S. Lichtenbaum [BL]. Our construction depends crucially upon the main result of [BL], the existence of an exact couple relating the motivic cohomology of the field F to the multirelative Ktheory of coherent sheaves on standard simplices over F (recalled as Theorem 5.5 below). A major step in generalizing the work of Bloch and Lichtenbaum is our reinterpretation of their spectral sequence in terms of the “topological filtration ” on the Ktheory of the standard cosimplicial scheme ∆ • over F. We find that the spectral sequence arises from a tower of Ωprespectra K( ∆ • ) = K 0 ( ∆ • ) ← − K 1 ( ∆ • ) ← − K 2 ( ∆ • ) ← − · · · Thus, even in the special case in which X equals SpecF, we obtain a much clearer understanding of the BlochLichtenbaum spectral sequence which is essential for purposes of generalization. Following this reinterpretation, we proceed using techniques introduced by V. Voevodsky in his study of motivic cohomology. In order to do this, we provide an equivalent formulation of Ktheory spectra associated to coherent sheaves on X with conditions on their supports K q ( ∆ • × X) which is functorial in X. We then Partially supported by the N.S.F. and the N.S.A.