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Categorical homotopy theory
 HOMOLOGY, HOMOTOPY APPL
, 2006
"... 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 322 (9 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
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 292 (46 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
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 206 (7 self)
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Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend
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 183 (1 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
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 166 (13 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
The homotopy theory of fusion systems
"... The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like pcompleted classifying spaces of finite groups. These spaces occur as the “classifying spaces ” of certain algebraic objects, which we call plocal finite groups. A plocal finite group con ..."
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Cited by 132 (19 self)
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The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like pcompleted classifying spaces of finite groups. These spaces occur as the “classifying spaces ” of certain algebraic objects, which we call plocal finite groups. A plocal finite group consists, roughly speaking, of a finite pgroup S and fusion data on subgroups of S, encoded in a way explained below. Our starting point is our earlier paper [BLO] on pcompleted classifying spaces of finite groups, together with the axiomatic treatment by Lluís Puig [Pu], [Pu2] of systems of fusion among subgroups of a given pgroup. The pcompletion of a space X is a space X ∧ p which isolates the properties of X at the prime p, and more precisely the properties which determine its mod p cohomology. For example, a map of spaces X f −− → Y induces a homotopy equivalence
Derived categories of coherent sheaves and triangulated categories of singularities
, 2005
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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 categori ..."
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Cited by 124 (9 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 107 (9 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