Annulation et pureté des groupes de cohomologie rigide associés à des sommes exponentielles

We show a purity property for the rigid cohomology groups associated to exponential sums on a smooth and affine variety. For this purpose, we compare these groups with the p-adic Banach spaces of Dwork's theory constructed by Adolphson and Sperber [1].     

Syntomic cohomology as a p-adic absolute Hodge cohomology

The purpose of this paper is to interpret rigid syntomic cohomology, defined by Amnon Besser [Bes], as a p-adic absolute Hodge cohomology. This is a p-adic analogue of a work of Beilinson [Be1] which interprets Beilinson-Deligne cohomology in terms of absolute Hodge cohomology. In the process, we will define a theory of p-adic absolute Hodge cohomology with coefficients, which may be interpreted as a generalization of rigid syntomic cohomology to the case with coefficients. Received: 25 September 2000 / In final form: 23 March 2001 / Published online: 28 February 2002     

Germs of de Rham cohomology classes which vanish at the generic point

We show that hypergeometric differential equations, unitary and Gauß-Manin connections give rise to de Rham cohomology sheaves which do not admit a Bloch-Ogus resolution [1]. The latter is in contrast to Panin's theorem [8] asserting that corresponding étale cohomology sheaves do fulfill Bloch-Ogus theory.     

Syntomic regulators andp-adic integration I: Rigid syntomic regulators

We construct a new version of syntomic cohomology, called rigid syntomic cohomology, for smooth schemes over the ring of integers of ap-adic field. This version is more refined than previous constructions and naturally maps to most of them. We construct regulators fromK-theory into rigid syntomic cohomology. We also define a “modified” syntomic cohomology, which is better behaved in explicit computations yet is isomorphic to rigid syntomic cohomology in most cases of interest.     

The Geisser-Levine method revisited and algebraic cycles over a finite field

We reformulate part of the arguments of T. Geisser and M. Levine relating motivic cohomology with finite coefficients to truncated étale cohomology with finite coefficients [9,10]. This reformulation amounts to a uniqueness theorem for motivic cohomology, and shows that the Geisser-Levine method can be applied generally to compare motivic cohomology with other types of cohomology theories. We apply this to prove an equivalence between conjectures of Tate and Beilinson on cycles in characteristic p and a vanishing conjecture for continuous étale cohomology. Received: 23 November 2000 / Published online: 5 September 2002     

Koszul cohomology and singular curves

We investigate Koszul cohomology on irreducible nodal curves following the lines of [2]. In particular, we prove both Green and Green-Lazarsfeld conjectures for any k-gonal nodal curve which is general in the sense of [4].     

The cohomology of monogenic extensions in the noncommutative setting

We extend the notion of monogenic extension to the noncommutative setting, and we study the Hochschild cohomology ring of such an extension. As an application we complete the computation of the cohomology ring of the rank one Hopf algebras begun in [S.M. Burciu, S.J. Witherspoon, Hochschild cohomology of smash products and rank one Hopf algebras, math.RA/0608762, 2006].     

Hodge modules on Shimura varieties and their higher direct images in the Baily-Borel compactification

We study the degeneration in the Baily-Borel compactification of variations of Hodge structure on Shimura varieties. Our main result, Theorem 2.6, expresses the degeneration of variations given by algebraic representations in terms of Hochschild, and abstract group cohomology. It is the Hodge theoretic analogue of Pink's theorem on degeneration of étale and ?-adic sheaves [Math. Ann. 292 (1992) 197], and completes results by Harder and Looijenga-Rapoport [Eisenstein-Kohomologie arithmetischer Gruppen: Allgemeine Aspekte, Preprint, 1986; Proc. of Symp. in Pure Math., vol. 53, 1991, pp. 223-260]. The induced formula on the level of singular cohomology is equivalent to the theorem of Harris-Zucker on the Hodge structure of deleted neighbourhood cohomology of strata in toroidal compactifications [Inv. Math. 116 (1994) 243].     

Cohomology of osp(2|2) acting on spaces of linear differential operators on the superspace ℝ1|2

We compute the first differential cohomology of the orthosymplectic Lie superalgebra osp(2|2) with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the (1, 2)-dimensional real superspace. We also compute the same, but osp(1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomologies. This work is the simplest generalization of a result from [1].     

Algebraic cohomology of compact monoids

We outline the development of a cohomology theory for compact monoids which extends the existing theory for finite monoids. This research was inspired by the work of K. H. Hofmann and P. S. Mostert [2], which defines the cohomology of a compact group in terms of the Čech cohomology of its classifying space. We use the classifying space construction of A. Dold and R. Lashof [1]. These results are contained in the author’s doctoral dissertation, submitted to Tulane University and written under the direction of Professor K. H. Hofmann. This research was partially supported by a National Aeronautics and Space Administration Graduate Traineeship.     

Exponential sums on An. III

We give two applications of our earlier work [4]. We compute the p-adic cohomology of certain exponential sums on A ⁿ involving a polynomial whose homogeneous component of highest degree defines a projective hypersurface with at worst weighted homogeneous isolated singularities. This study was motivated by recent work of García [9]. We also compute the p-adic cohomology of certain exponential sums on A ⁿ whose degree is divisible by the characteristic. Received: 12 October 1999     

Pentes en cohomologie rigide et F-isocristaux unipotents

We study the slopes of Frobenius on the rigid cohomology and the rigid cohomology with compact support of an algebraic variety over a perfect field of positive characteristic. We then prove that any unipotent overconvergent F-isocrystal on a smooth variety has a slope filtration whose graded parts are pure. Received: 23 December 1998 / Revised version: 5 July 1999     

A trace formula approach to control theorems for overconvergent automorphic forms

We present an approach to proving control theorems for overconvergent automorphic forms on Harris–Taylor unitary Shimura varieties based on a comparison between the rigid cohomology of the multiplicative ordinary locus and the rigid cohomology of the overlying Igusa tower, the latter which may be computed using the Harris–Taylor version of the Langlands–Kottwitz method. We also prove a higher level version, generalizing work of Coleman.     

On the Representation Theory of Lie Triple Systems

In this paper, we take a new look at the representation theory of Lie triple systems. We consider both ordinary Lie triple systems and restricted Lie triple systems in the sense of [14]. In a final section, we begin a study of the cohomology of Lie triple systems.

     

Exponential sums and rigid cohomology

In this article, we prove a comparison theorem between the Dwork cohomology introduced by Adolphson and Sperber and the rigid cohomology. As a corollary, we can calculate the rigid cohomology of Dwork isocrystal on torus.     

Suite spectrale d'Adams et invariants cohomologiques des formes quadratiques

For any field k of characteristic 0 the Adams spectral sequence for the sphere spectrum based on Suslin-Voevodsky modulo 2 motivic cohomology [8] converges to the graded ring associated to the filtration of the Grothendieck-Witt ring of quadratic forms over k by powers of the ideal generated by even-dimensional forms. Moreover, some property of the modulo 2 motivic cohomology of k, which is a consequence of Voevodsky 's proof of Milnor's conjecture on modulo 2 Galois cohomology of k [9], implies that the spectral sequence degenerates in the critical area. This allows us to give a new proof of the Milnor conjecture on the graded ring of the Witt ring of k [4] which differs from [11].     

Hochschild cohomology of ? n -Galois coverings of an algebra

We consider the ? _n -Galois covering ?? _n of the algebra A introduced by F. Xu [Adv. Math., 2008, 219: 1872?C1893]. We calculate the dimensions of all Hochschild cohomology groups of ?? _n and give the ring structure of the Hochschild cohomology ring modulo nilpotence. As a conclusion, we provide a class of counterexamples to Snashall-Solberg??s conjecture.     

Chen-Ruan Cohomology and Stringy Orbifold K-Theory for Stable Almost Complex Orbifolds

Comparing to the construction of stringy cohomology ring of equivariant stable almost complex manifolds and its relation with the Chen-Ruan cohomology ring of the quotient almost complex orbifolds, the authors construct in this note a Chen-Ruan cohomology ring for a stable almost complex orbifold. The authors show that for a finite group G and a G-equivariant stable almost complex manifold X, the G-invariant part of the stringy cohomology ring of (X, G) is isomorphic to the Chen-Ruan cohomology ring of the global quotient stable almost complex orbifold [X/G]. Similar result holds when G is a torus and the action is locally free. Moreover, for a compact presentable stable almost complex orbifold, they study the stringy orbifold K-theory and its relation with Chen-Ruan cohomology ring.     

Blow-up formulae for SO(3)-Donaldson polynomials

We calculate first through seventh order terms of a blow-up formula for -Donaldson polynomials, using equivariant cohomology and techniques similar to those of [Oz]. These formulas are used to extend the results of [B2] to any rank one negative definite four manifold and to describe the extension of the equivalence proved in [M] of the Donaldson polynomial and O'Grady's algebro-geometric analogues to t he four dimensional classes. Received: 12 April 1994; in final form 15 November 1995     

Local cohomology with support in a parameter ideal

Motivated in part by an attempt to understand better the notion of a parameter-like sequence introduced in Hochster (2003) [7, (2.2)], we study results concerning the heights of the annihilators and the finiteness of the dimension of the socle in certain local cohomology modules with support in a parameter ideal. We obtain positive results under certain hypotheses of low dimension or codimension, but we also find examples that show, in the general case, that support in a parameter ideal does not restrict the behavior of local cohomology much more than support in an arbitrary ideal. The results obtained here strongly suggest that it would be worthwhile to seek a modification of the notion of a parameter-like sequence introduced in [7].