Graded Witt ring and unramified cohomology

It is proved that for a smooth affine curveX over a local ring or global field, the graded Witt ring ofX is isomorphic to the graded unramified cohomology ring ofX. IfX is projective and has a rational point, the same result holds if and only if every quadratic space defined on the complement of a rational point extends toX. Such an extension is possible, for instance, if the canonical line bundle onX is a square in PicX.     

Two-primary algebraic -theory of rings of integers in number fields

We relate the algebraic -theory of the ring of integers in a number field to its étale cohomology. We also relate it to the zeta-function of when is totally real and Abelian. This establishes the -primary part of the ``Lichtenbaum conjectures.' To do this we compute the -primary -groups of and of its ring of integers, using recent results of Voevodsky and the Bloch-Lichtenbaum spectral sequence, modified for finite coefficients in an appendix. A second appendix, by M. Kolster, explains the connection to the zeta-function and Iwasawa theory.

     

First cohomology group of rank two Witt algebra to its Larsson modules

Let U = ℂ², Γ = ℤ², and let ℂ[x ₁^±1, x ₂^±1] be the ring of Laurent polynomials. The Witt algebra L is the Lie algebra of derivations over ℂ[x ₁^±1, x ₂^±1], which is spanned by elements of the form D(u, r) = x ^r (u ₁ d ₁ + u ₂ d ₂), u = (u ₁, u ₂) ∈ U, r ∈ Γ, where d ₁ and d ₂ are the degree derivations of ℂ[x ₁^±1, x ₂^±1]. The image of gl ₂-module V under Larsson functor F ^α , denoted by W = F ^α (V), gives a class of L-modules, often called the Larsson-modules of L. In this paper, we study the derivations from the Witt algebra L to its Larsson-modules W, and we determine the first cohomology group H ¹(L,W).     

Galois cohomology of completed link groups

In this paper we compute the Galois cohomology of the pro- completion of primitive link groups. Here, a primitive link group is the fundamental group of a tame link in whose linking number diagram is irreducible modulo (e.g. none of the linking numbers is divisible by ).

The result is that (with -coefficients) the Galois cohomology is naturally isomorphic to the -cohomology of the discrete link group.

The main application of this result is that for such groups the Baum-Connes conjecture or the Atiyah conjecture are true for every finite extension (or even every elementary amenable extension), if they are true for the group itself.

     

A conjecture on the Eichler cohomology of automorphic forms

We prove partially a conjecture of Knopp about the Eichler cohomology of automorphic forms on H-groups.     

Twisted Weyl groups of Lie groups and nonabelian cohomology

For a cyclic group A and a connected Lie group G with an A-module structure (with the additional assumptions that G is compact and the A-module structure on G is 1-semisimple if ), we define the twisted Weyl group W = W(G,A,T), which acts on T and H ¹(A,T), where T is a maximal compact torus of , the identity component of the group of invariants G ^A . We then prove that the natural map is a bijection, reducing the calculation of H ¹(A,G) to the calculation of the action of W on T. We also prove some properties of the twisted Weyl group W, one of which is that W is a finite group. A new proof of a known result concerning the ranks of groups of invariants with respect to automorphisms of a compact Lie group is also given.      

The integral cohomology of the Bianchi groups

We calculate the integral cohomology ring structure for various members of the Bianchi group family. The main tools we use are the Bockstein spectral sequence and a long exact sequence derived from Bass-Serre theory.

     

Motivic cohomology and unramified cohomology of quadrics

This is the last of a series of three papers where we compute the unramified cohomology of quadrics in degree up to 4. Complete results were obtained in the two previous papers for quadrics of dimension ≤4 and ≥11. Here we deal with the remaining dimensions between 5 and 10. We also prove that the unramified cohomology of Pfister quadrics with divisible coefficients always comes from the ground field, and that the same holds for their unramified Witt rings. We apply these results to real quadrics. For most of the paper we have to assume that the ground field has characteristic 0, because we use Voevodsky’s motivic cohomology. Received August 18, 1999 / final version received December 10, 1999?Published online April 19, 2000     

Witt groups of function fields of hyperelliptic curves

The local cohomology modules H^J _I(M) of a Matlis reflexive module are shown to be I-cofinite when j >= 1 and have finite Bass numbers when j >= 0, where I is an ideal satisfying any one of a list of properties. In addition, we show that the completion of a Matlis reflexive module is finitely generated over the completion of the ring and we classify Matlis reflexive modules over a one dimensional ring.     

Some Weyl modules and cohomology for algebraic groups

Abstract

In this article, we give a generalization of O’Halloran’s theorem that gives rise to families of Weyl modules with simple radicals. This generalization is then applied to simply connected, semisimple algebraic groups of types B, C, and D. We also construct families of Weyl modules with formal characters of length 4, and compute the extensions and cohomology of simple modules associated with these Weyl modules.     

Motives and Etale Motives with Finite Coefficients

We prove that for suitable base fields, inverting the Bott element in Voevodsky’s category of motives with finite coefficients yields the category of étale motives with finite coefficients. Mathematics Subject Classifications (2000): 19E15, 14F42, 14F20. The first author was partially supported by the Clay Mathematics Institute.     

The second bounded cohomology of an amalgamated free product of groups

We study the second bounded cohomology of an amalgamated free product of groups, and an HNN extension of a group. As an application, we show that a group with infinitely many ends has infinite dimensional second bounded cohomology.

     

Comparison between overconvergent de Rham‐Witt and crystalline cohomology for projective and smooth varieties

We prove that for a projective smooth scheme X the hypercohomology of the overconvergent de Rham‐Witt complex is canonically isomorphic to crystalline cohomology.     

Vanishing of some cohomology groups and bounds for the Shafarevich-Tate groups of elliptic curves

Let E be an elliptic curve over Q and ? be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at ?. Under certain assumptions, we prove the vanishing of the Galois cohomology group H¹(Gal(K(E[?ⁱ])/K),E[?ⁱ]) for all i?1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyvagin, which finds a bound for the order of the ?-primary part of Shafarevich-Tate groups of E over K. This bound is consistent with the prediction of Birch and Swinnerton-Dyer conjecture.     

Dirac cohomology of unitary representations of equal rank exceptional groups

In this paper, we consider the unitary representations of equal rank exceptional groups of type E with a regular lambda-lowest K-type and classify those unitary representations with the nonzero Dirac cohomology.