A local-global principle for twisted flag varieties

We prove a local-global principle for twisted flag varieties over a semiglobal field

Singularities of generic characteristic polynomials and smooth finite splittings of Azumaya algebras over surfaces

Let k be an algebraically closed field. Let P(X 11, . . . , X nn , T) be the characteristic polynomial of the generic matrix (X ij ) over k. We determine its singular locus as well as the singular locus of its Galois splitting. If X is a smooth quasi-projective surface over k and A an Azumaya algebra on X of degree n, using a method suggested by M. Artin, we construct finite smooth splittings for A of degree n over X whose Galois closures are smoot

Hasse principles for multinorm equations

Let $k$ be a global field and let $L_0$,...,$L_m$ be finite separable field extensions of $k$. In this paper, we are interested in the Hasse principle for the multinorm equation $\underset{i=0}{\overset{m}{\prod}}N_{L_i/k}(t_i)=c$. Under the assumption that $L_0$ is a cyclic extension, we give an explicit description of the Brauer-Manin obstruction to the Hasse principle. We also give a complete criterion for the Hasse principle for multinorm equations to hold when $L_0$ is a meta-cyclic extension.Comment: minor change

On the Grothendieck-Serre Conjecture for Classical Groups

We prove some new cases of the Grothendieck-Serre conjecture for classical groups. This is based on a new construction of the Gersten-Witt complex for Witt groups of Azumaya algebras with involution on regular semilocal rings, with explicit second residue maps; the complex is shown to be exact when the ring is of dimension $\le 2$ (or $\le 4$, with additional hypotheses on the algebra with involution). Note that we do not assume that the ring contains a field.Comment: 37 pages. Changes from previous version include many improvements to section 2. Comments are welcom

Singularities of generic characteristic polynomials and smooth finite splittings of Azumaya algebras over surfaces

Let k be an algebraically closed field. Let P(X-11,...,X-nn, T) be the characteristic polynomial of the generic matrix (X-ij) over k. We determine its singular locus as well as the singular locus of its Galois splitting. If X is a smooth quasi-projective surface over k and A an Azumaya algebra on X of degree n, using a method suggested by M. Artin, we construct finite smooth splittings for A of degree n over X whose Galois closures are smooth

GALOIS ALGEBRAS, HASSE PRINCIPLE, AND INDUCTION RESTRICTION METHODS (vol 16, pg 677, 2011)

This is a correction to [BP 11] E. Bayer-Fluckiger, R. Parimala, Galois algebras, Hasse principle and induction-restriction methods, Documenta Math. 16 (2011), 677-707