Bruno Kahn

The Arason invariant and mod 2 algebraic cycles

Introduction 73 1. Review of the Arason invariant 75 2. The special Clifford group 76 3. K-cohomology of split reductive algebraic groups 78 4. K-cohomology of BG 86 5. GL(N) and Cliff(n, n) 92 6. Two invariants for Clifford bundles 95 7. Snaking a Bloch-Ogus differential 100 8. Proof of Theorem 1 101 9. Application to quadratic forms 102 Appendix A. Toral descent 104 Appendix B. The Rost invariant 108 Appendix C. An amusing example 113 Acknowledgements 116 References 116

Formes quadratiques de hauteur et de degré 2

Abstract We classify quadratic forms of height 2 and degree 2 in the sense of Knebusch [15, 16] over any field of characteristic ≠ 2, completing the earlier results of Knebusch and Fitzgerald. Such an anisotropic form is either excellent, or a 6-dimensional ‘Albert form’ or an 8-dimensional form of a certain type. We also give partial results for forms of degree > 2. We then show that evendimensional forms of height 2 and degree ≤ 2 (at least the non-excellent ones) can be parametrised, up to similitude, by cohomological invariants. Does this phenomenon extend to higher heights and degrees?

MOTIVES OF AZUMAYA ALGEBRAS

We study the slice filtration for the K -theory of a sheaf of Azumaya algebras A , and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson–Lichtenbaum conjecture, we apply our results to show the vanishing of SK 2 ( A ) for a central simple algebra A of square-free index (prime to the characteristic). This proves a conjecture of Merkurjev.

ON THE LICHTENBAUM-QUILLEN CONJECTURE

The Lichtenbaum-Quillen conjecture, relating the algebraic K-theory of rings of integers in number fields to their etale cohomology, has been one of the main factors of development of algebraic K-theory in the beginning of the 1980s. Soule’ and Dwyer-Friedlander mapped algebraic K-theory of a ring of integers to its l-adic cohomology by means of a ‘Chern character’, that they proved surjective. Here, on the contrary, we map etale cohomology to algebraic K-theory, providing a right inverse to these Chern characters. This gives a different proof of surjectivity, which avoids Dwyer-Friedlander’s use of ‘secondary transfer’. The constructions and results of this paper concern a much wider class of rings than rings of integers in number fields.

Equivariant Stiefel-Whitney classes

Abstract We develop a theory of equivariant Stiefel-Whitney classes for equivariant orthogonal vector bundles over schemes, parallel to Grothendiecks theory of equivariant Chern classes for equivariant linear vector bundles. We apply it to generalise earlier results on characteristic classes of ‘trace forms’, due to Serre, Frohlich, Snaith, Jardine, Esnault, Viehweg and the author.

Some finiteness results for étale cohomology

Abstract We prove some finiteness theorems for the etale cohomology, Borel–Moore homology and cohomology with proper supports with divisible coefficients of schemes of finite type over a finite or p -adic field. This yields vanishing results for their l -adic cohomology, proving part of a conjecture of Jannsen.

Some Conjectures on the Algebraic K-Theory of Fields, I: K-Theory with Coefficients and Étale K-Theory

We investigate relationships between K-theory with coefficients and etale cohomology. Classically, such relationships are given by a) homomorphisms from the former towards the latter (Chern classes) and b) comparison with etale K-theory, which is the abutment of a spectral sequence starting with etale cohomology groups. The main theme of this paper is to explain that, locally for the Zariski topology, there should be homomorphisms in the opposite direction to a), and that these homomorphisms should split the etale K-theory spectral sequence in a very strong sense. As a consequence, etale K-theory groups of a nice semi-local ring should be isomorphic to a direct sum of etale cohomology groups, and a part of this sum should map to ordinary K-groups (with coefficients) as a direct summand. We further conjecture that these split injections should actually be isomorphisms: this is equivalent to conjecturing that ordinary K-theory injects into etale K-theory, or that Bott elements are nonzero divisors in the former. Given the conjectural homomorphisms above, this would also be a formal consequence of the existence of a spectral sequence from etale cohomology to ordinary K-theory, as conjectured by Beilinson.

Résultats de “Pureté” pour les Variétés Lisses sur un Corps Fini

In this note, we extend the main results of [CT] to more general coefficients than μ n ⨂d . For a constant-twisted sheaf A, with geometric fibre ℤ/l n , coming from the ground field (e.g. A = μ n ⨂i . ), we still prove that, with the notation of [CT], H i (X Zar , H X d+1 (A)) for i = d − 1 and d − 2. (If l = 2, a technical hypothesis on A is necessary; it holds for A = μ n ⨂i .) For an ind-constant-twisted sheaf B, with geometric fibre ℚ l /ℤ l , not isomorphic to ℚ l /ℤ l (d), we prove (under a small technical hypothesis when l = 2) that the sheaf H X d+1 (B) is itself 0, as well as all the terms of its Gersten resolution. The latter result in fact holds for a smooth variety defined over an arbitrary (not necessarily finite) finitely generated field; its proof is much easier than the one for the former result and does not rely on the results of [CT], while the proof of the first result does.

A description of the second mod p cohomology group of a p-group

Abstract We show that any finite mixed Lie algebra in the sense of Lazard, of characteristic p and length 3, is the associated graded of some finite p -filtered group. This amounts to calculating the second mod p cohomology group of a p -group of p -class 2 in terms of the mixed Lie algebra associated to its Frattini filtration.