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 S. Garibaldi, V. Petrov and N. Semenov to give a complete classification of the motivic decompositions of projective homogeneous varieties of inner type E_6 and to answer a conjecture of Rost and Springer.Comment: Final version of the manuscrip

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 \mathbb{F} 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_k;A) and X(d'_1,...,d'_s;A') with coefficients in \mathbb{F} are isomorphic if and only if the p-adic valuations of gcd(d_1,...,d_k) and gcd(d'_1,..,d'_s) are equal and the classes of the p-primary components A_p and A'_p 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_

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 $\mathbb{F}_p$ 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 coefficients in $\mathbb{F}_p$. We also construct a counterexample to this result in the case where $G$ is arbitrary

Lifting vector bundles to Witt vector bundles

Let $p$ be a prime, and let $S$ be a scheme of characteristic $p$. Let $n \geq 2$ be an integer. Denote by $\mathbf{W}_n(S)$ the scheme of Witt vectors of length $n$, built out of $S$. The main objective of this paper concerns the question of extending (=lifting) vector bundles on $S$ to vector bundles on $\mathbf{W}_n(S)$. After introducing the formalism of Witt-Frobenius Modules and Witt vector bundles, we study two significant particular cases, for which the answer is positive: that of line bundles, and that of the tautological vector bundle of a projective space. We give several applications of our point of view to classical questions in deformation theory---see the Introduction for details. In particular, we show that the tautological vector bundle of the Grassmannian $Gr_{\mathbb{F}_p}(m,n)$ does not extend to $\mathbf{W}_2(Gr_{\mathbb{F}_p}(m,n))$, if $2 \leq m \leq n-2$. In the Appendix, we give algebraic details on our (new) approach to Witt vectors, using polynomial laws and divided powers. It is, we believe, very convenient to tackle lifting questions.Comment: Enriched version, with an appendi

Motivic rigidity of Severi-Brauer varieties

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.Comment: more details, ~10p, final versio

Smooth profinite groups, III: the Smoothness Theorem

Let $p$ be a prime. The goal of this article is to prove the Smoothness Theorem 5.1 (Theorem D), which notably asserts that a $(1,\infty)$-cyclotomic pair is $(n,1)$-cyclotomic, for all $n \geq 1$. In the particular case of Galois cohomology, the Smoothness Theorem provides a new proof of the Norm Residue Isomorphism Theorem. Using the formalism of smooth profinite groups, this proof presents it as a consequence of Kummer theory for fields.Comment: Added index for reader's convenience. Comments welcom