Results 11  20
of
144
Cyclic Cohomology of Étale Groupoids; The General Case
 Ktheory
, 1999
"... We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the FeiginTsyganNistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution alge ..."
Abstract

Cited by 21 (1 self)
 Add to MetaCart
We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the FeiginTsyganNistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution algebra C 1 c (G) of an 'etale groupoid, removing the Hausdorffness condition and including the computation of hyperbolic components. Examples like group actions on manifolds and foliations are considered. Keywords: cyclic cohomology, groupoids, crossed products, duality, foliations. Contents 1 Introduction 3 2 Homology and Cohomology of Sheaves on ' Etale Groupoids 4 2.1 ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 \Gamma c in the nonHausdorff case : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3 Homology and Cohomology of ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : 8 3 Cyclic Homologies of Sheaves ...
CastelnuovoMumford Regularity in Biprojective Spaces
, 2002
"... We define the concept of regularity for bigraded modules over a bigraded polynomial ring. In this setting we prove analogs of some of the classical results on mregularity for graded modules over polynomial algebras. ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
We define the concept of regularity for bigraded modules over a bigraded polynomial ring. In this setting we prove analogs of some of the classical results on mregularity for graded modules over polynomial algebras.
Deformation theory and Lie algebra homology
, 1997
"... 1.1. Let X be a smooth proper scheme X over a field k of characteristic 0, G an algebraic group over k and p: P − → X a Gtorsor over X. Consider the following deformation problems. Problem 1. Flat deformations of X. Problem 2. Flat deformations of the pair (X,P). ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
1.1. Let X be a smooth proper scheme X over a field k of characteristic 0, G an algebraic group over k and p: P − → X a Gtorsor over X. Consider the following deformation problems. Problem 1. Flat deformations of X. Problem 2. Flat deformations of the pair (X,P).
Krichever correspondence for algebraic varieties
 english translation in Izv. Math. 65 (2001
"... In the work is constructed new acyclic resolutions of quasicoherent sheaves. These resolutions is connected with multidimensional local fields. Then the obtained resolutions is applied for a construction of generalization of the Krichever map to algebraic varieties of any dimension. This map gives i ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
In the work is constructed new acyclic resolutions of quasicoherent sheaves. These resolutions is connected with multidimensional local fields. Then the obtained resolutions is applied for a construction of generalization of the Krichever map to algebraic varieties of any dimension. This map gives in the canonical way two ksubspaces B ⊂ k((z1))... ((zn)) and W ⊂ k((z1))... ((zn)) ⊕r from arbitrary algebraic ndimensional CohenMacaulay projective integral scheme X over a field k, a flag of closed integral subschemes X = Y0 ⊃ Y1 ⊃...Yn (such that Yi is an ample Cartier divisor on Yi−1, and Yn is a smooth kpoint on all Yi), formal local parameters of this flag in the point Yn, a rank r vector bundle F on X, and a trivialization F in the formal neighbourhood of the point Yn, where the ndimensional local field k((z1))... ((zn)) is associated with the flag Y0 ⊃... ⊃ Yn. In addition, the constructed map is injective, i. e., it is possible to reconstruct uniquely all the original geometrical data. Besides, from the subspace B is written explicitly a complex, which calculates cohomology of the sheaf OX on X; and from the subspace W is written explicitly a complex, which calculates cohomology of F on X. 1
Virtual Betti numbers of real algebraic varieties
 Comptes Rendus Acad. Sci. Paris, Ser. I
"... Abstract. The weak factorization theorem for birational maps is used to prove that for all i ≥ 0 the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti number βi defined for all real algebraic varieties, such that if Y is a closed subvari ..."
Abstract

Cited by 14 (4 self)
 Add to MetaCart
Abstract. The weak factorization theorem for birational maps is used to prove that for all i ≥ 0 the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti number βi defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then βi(X) = βi(X \ Y) + βi(Y). We define an invariant of the Grothendieck ring K0(VR) of real algebraic varieties, the virtual Poincaré polynomial β(X,t), which is a ring homomorphism K0(VR) → Z[t]. For X nonsingular and compact, β(X,t) is the classical Poincaré polynomial for cohomology with Z2 coefficients. The coefficients of the virtual Poincaré polynomial are the virtual Betti numbers. By the weak factorization theorem for birational morphisms (Abramovich et al. [1]), the existence of β(X,t) follows from a simple formula for the Betti numbers of the blowup of a compact nonsingular variety along a closed nonsingular center. The existence of the virtual Betti numbers for certain real analytic spaces, including real algebraic varieties, has also been announced by Totaro [23]. Kontsevich’s motivic measure on the arc space of a complex algebraic variety takes values in the completion of the localized Grothendieck ring (cf. [20], [11], [7]). Completion with
βnbcbases for cohomology of local systems on hyperplane complements
 TRANS. AMER. MATH. SOC
, 1997
"... We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes A. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes A. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only nonvanishing cohomology group, under some nonresonance conditions on the local system, for any arrangement A. The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain “nobrokencircuits” bases of A. The basic forms depend on the local system, but any two bases constructed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.
Holonomy on Poisson manifolds and the modular class
"... Abstract. We introduce linear holonomy on Poisson manifolds. The linear holonomy of a Poisson structure generalizes the linearized holonomy on a regular symplectic foliation. However, for singular Poisson structures the linear holonomy is defined for the lifts of tangential path to the cotangent bun ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
Abstract. We introduce linear holonomy on Poisson manifolds. The linear holonomy of a Poisson structure generalizes the linearized holonomy on a regular symplectic foliation. However, for singular Poisson structures the linear holonomy is defined for the lifts of tangential path to the cotangent bundle (cotangent paths). The linear holonomy is closely related to the modular class studied by A. Weinstein. Namely, the logarithm of the determinant of the linear holonomy is equal to the integral of the modular vector field along such a lift. This assertion relies on the notion of the integral of a vector field along a cotangent path on a Poisson manifold, which is also introduced in the paper. In the second part of the paper we prove that for locally unimodular Poisson manifolds the modular class is an invariant of Morita equivalence. 1.
New Model Categories From Old
 J. Pure Appl. Algebra
, 1995
"... . We review Quillen's concept of a model category as the proper setting for defining derived functors in nonabelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categor ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
. We review Quillen's concept of a model category as the proper setting for defining derived functors in nonabelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categories of cosimplicial coalgebras. 1. Introduction Model categories, first introduced by Quillen in [Q1], have proved useful in a number of areas  most notably in his treatment of rational homotopy in [Q2], and in defining homology and other derived functors in nonabelian categories (see [Q3]; also [BoF, BlS, DwHK, DwK, DwS, Goe, ScV]). From a homotopy theorist's point of view, one interesting example of such nonabelian derived functors is the E 2 term of the mod p unstable Adams spectral sequence of Bousfield and Kan. They identify this E 2 term as a sort of Ext in the category CA of unstable coalgebras over the mod p Steenrod algebra (see x7.4). The original purpose of this note w...
Theory of valuations on manifolds, I. Linear spaces
 Israel J. Math
"... This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of an affine space. In this article we still work only with linear spaces. We introduce a space of smooth (nont ..."
Abstract

Cited by 12 (8 self)
 Add to MetaCart
This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of an affine space. In this article we still work only with linear spaces. We introduce a space of smooth (nontranslation invariant) valuations on a linear space V. We present three descriptions of this space. We describe the canonical multiplicative structure on this space generalizing the results from [3] obtained for polynomial valuations. 0 Introduction. This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of an affine space. In this article we still work only with linear spaces. In the subsequent parts of this series we are going to generalize constructions of this article to arbitrary smooth