This work is motivated by the search for an "explicit" proof of the Bloch-Kato conjecture in Galois cohomology, proved by Voevodsky. Our concern here is to lay the foundation for a theory that, we believe, will lead to such a proof- and to further applications. Let p be a prime number. Let k be a perfect field of characteristic p. Let m be a positive integer. Our first goal is to provide a canonical process for "lifting" a module M, over the ring of Witt vectors Wm(k) (of length m), to a Wm+1(k)-module, in a way that deeply respects Pontryagin duality. These are our big, medium and small Omega powers, each of which naturally occurs as a direct factor of the previous one. In the case where M is a k-vector space, they come equipped with Verschiebung and Frobenius operations. If moreover the field k is finite, Omega powers are endowed with a striking extra operation: the Transfer, to shifted Omega powers of finite-codimensional linear subspaces. To show how this formalism fits into Galois theory, we first offer an axiomatized approach to Hilbert's Theorem 90 (or more precisely, to its consequence for cohomology with finite coefficients: Kummer theory). In the context of profinite group cohomology, we thus define the notions a cyclotomic G-module, and of a smooth profinite group. We bear in mind that the fundamental example is that of an absolute Galois group, together with the Tate module of roots of unity. We then define the notion of exact sequences of G-modules of Kummer type. To finish, we give applications of this formalism. The first ones are the Stable Lifting Theorems, enabling the lifting to higher torsion in the cohomology of smooth profinite groups, with p-primary coefficients. We finish by an application to p-adic deformations. We state and prove a general descent statement, for the quotient map Z/p2Z -> Z/pZ.
Motivic equivalence for algebraic groups was recently introduced by first author, where a characterization of motivic equivalent groups in terms of higher Tits indices is given. As a consequence, if the quadrics associated to two quadratic forms have the same Chow motives with coefficients in $\mf_2$, this remains true for any two projective homogeneous varieties of the same type under the orthogonal groups of those two quadratic forms. In the first part of the paper, we extend this result to all groups of classical type, introducing a notion of critical variety. On the way, we prove that motivic equivalence of the automorphism groups of two involutions on a given algebra can be checked after extending scalars to an index reduction field, which depends on the type of the involutions. The second part of the paper describes conditions on the base field which guarantee that motivic equivalent involutions actually are isomorphic, extending a result of Hoffmann on quadratic forms.
Publications
On the Tits p-indexes of semisimple algebraic groups (avec Skip Garibaldi)
The first author has recently shown that semisimple algebraic groups are classified up to motivic equivalence by the local versions of the classical Tits indexes over field extensions, known as Tits p-indexes. We provide in this article the complete description of the values of the Tits p-indexes over fields. From this exhaustive study, we also deduce criteria for motivic equivalence of semisimple groups of many types, hence giving a dictionary between classic algebraic structures, representation theory, cohomological invariants, and Chow motives of the twisted flag varieties for those groups.
Équivalence motivique des groupes algébriques semisimples
We prove that the standard motives of a semisimple algebraic group G with coefficients in a field of order p are determined by the upper motives of the group G. As a consequence of this result, we obtain a partial version of the motivic rigidity conjecture of special linear groups. The result is then used to construct the higher indexes which characterize motivic equivalence of semisimple algebraic groups. The criteria of motivic equivalence derived from the expressions of these indexes produce a dictionnary between motives, algebraic structures and the birational geometry of twisted flag varieties. This correspondence is then described for special linear groups and orthogonal groups (the criteria associated to other groups being obtained in [14]). The proofs rely on the Levi-type motivic decompositions of isotropic twisted flag varieties due to Chernousov, Gille and Merkurjev, and on the notion of pondered field extensions.
Upper motives of products of projective linear groups
This paper is devoted to the classification of the upper motives of all the products of projective linear groups with coefficients in a finite field. We first give a complete classification of the upper motives of all the projective linear groups and derive from it the motivic dichotomy of PGL_1. We then provide several classification results as well as counterexamples for arbitrary products of projective linear groups, showing that the situation is less rigid. The proofs involve a neat study of rational maps between generalized Severi-Brauer varieties which is certainly of independent interest.
Let D be a central division algebra over a field F. We study in this note the rigidity of the motivic decompositions of the Severi-Brauer varieties of D, with respect to the ring of coefficients and to the base field. We first show that if the ring of coefficient is a field, these decompositions only depend on its characteristic. In a second part we show that if D remains division over a field extension E/F, the motivic decompositions of several Severi-Brauer varieties of D remain the same when extending the scalars to E.
A going down theorem for Grothendieck Chow motives
Let X be a geometrically split, geometrically irreducible variety over a field F satisfying Rost
nilpotence principle. Consider a field extension E/F and a finite field k. We provide in this note
a motivic tool giving sufficient conditions for so-called outer motives of direct summands of the
Chow motive of X_E with coefficients in k to be lifted to the base field. This going down result
has been used by S. Garibaldi, V. Petrov and N. Semenov to give a complete classification of the
motivic decompositions of projective homogeneous varieties of inner type E6, and to answer a
conjecture of Rost and Springer.
Classification of upper motives of algebraic groups of inner type A_n
Let A, A' be two central simple algebras over a field F and k be a finite field of characteristic p. We prove that the upper indecomposable direct summands of the motives of two anisotropic varieties of flags of right ideals X(d_1,...,d_s;A) and X(d'_1,...,d'_s';A') with coefficients in k are isomorphic if and only if the p-adic valuations of gcd(d_1,...,d_s) and gcd(d'_1,..,d'_s') are equal and the classes of the p-primary components of A and A' generate the same group in the Brauer group of F. This result leads to a surprising dichotomy between upper motives of absolutely simple adjoint algebraic groups of inner type A_n.
Motivic decompositions of projective homogeneous varieties and change of coefficients
We prove that under some assumptions on an algebraic group G, indecomposable direct summands of the motive of a projective G-homogeneous variety with coefficients in F_p remain indecomposable if the ring of coefficients is any field of characteristic p. In particular for any projective G-homogeneous variety X, the decomposition of the motive of X in a direct sum of indecomposable motives with coefficients in any finite field of characteristic p corresponds to the decomposition of the motive of X with coeficients in F_p. We also construct a counterexample to this result in the case where G is arbitrary.
Projets interdisciplinaires
Surgery of the recurrent dislocation of low jaw, after Girard and Leclerc (with Professor Natsuki Segami)
In a joint work with Prof. Natsuki Segami of the Kanazawa Medical University Oral and Maxillofacial Surgery department, we investigated the new techniques provided by Girard and Leclerc for the surgery of the recurrent dislocation of low jaw. These results were presented at the 61rst congress of the Japan Oral Surgery association.