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Alexandre Lourdeaux

Publications and source records attributed to Alexandre Lourdeaux.

6 recordsLinked to original sources
arXiv2026

Conjecture I for unirational algebraic groups over imperfect fields

Serre's Conjecture I states that the first Galois cohomology set of any smooth connected linear algebraic group is trivial over a perfect field of cohomological dimension at most 1. We prove that this result remains valid for any unirational algebraic group, dropping the perfection assumption on the field. To do so, we rely on the theory of pseudo-reductive groups, combined with the structure of unirational wound unipotent groups and the recent theory of permawound unipotent groups. Finally, we extend several related results on Galois cohomology associated with the Conjecture.

math.AG↗
arXiv2026

Cohomological invariants of hermitian forms that detect hyperbolicity

By using unramified cohomology groups, we construct a full sequence of cohomological invariants for hermitian forms of any (orthogonal, symplectic or unitary) type that can be used to detect hyperbolicity. The base central simple algebra can have arbitrary degree and the base field can have arbitrary characteristic. In the orthogonal case, we work with hermitian pairs, and we apply our construction to show that over fields of separable dimension 3, hermitian pairs over quaternion algebras with trivial classical invariants are hyperbolic. This last result extends a result of Berhuy to arbitrary characteristic.

math.RA↗
arXiv2022

On geometry of some pseudo-reductive groups

Based on the work of Conrad-Gabber-Prasad, the paper deals with the geometry of particular pseudo-semisimple groups, namely those which can be written as quotient of Weil restriction of semisimple groups. We establish that these groups are retract rational when their are split, and give results on their Picard groups.

math.GR↗
arXiv2020

Degree 2 cohomological invariants of linear algebraic groups

The paper deals with the cohomological invariants of smooth and connected linear algebraic groups over an arbitrary field. More precisely, we study degree $2$ invariants with coefficients $\mathbb{Q}/\mathbb{Z}(1)$, that is invariants taking values in the Brauer group. Our main tool is the \'etale cohomology of sheaves on simplicial schemes. We get a description of these invariants for \emph{every} smooth and connected linear groups, in particular for non reductive groups over an imperfect field.

math.AG↗
arXiv2020

Faisceaux Q/Z(j) et conjecture de Gersten sur un corps imparfait

On revoit explicitement la construction ainsi que certaines propriétés des complexes de faisceaux étales $\mathbb{Q}/\mathbb{Z}(j)$ sur certains schémas. Le but de ces notes est d'avoir une référence précise pour la conjecture de Gersten pour les faisceaux $\mathbb{Q}/\mathbb{Z}(j)$ sur un corps imparfait ainsi que pour la comparaison entre les groupes de cohomologie $\mathrm{H}^2(\ast,\mathbb{Q}/\mathbb{Z}(1))$ et $\mathrm{H}^2(\ast,\mathbb{G}_m)$. We review explicitly the definition and some properties of étale sheaf complexes $\mathbb{Q}/\mathbb{Z}(j)$ on some schemes. The purpose of these notes is to be a precise reference for the Gersten conjecture of $\mathbb{Q}/\mathbb{Z}(j)$ over an imperfect field and also for the comparison between the cohomology groups $\mathrm{H}^2(\ast,\mathbb{Q}/\mathbb{Z}(1))$ and $\mathrm{H}^2(\ast,\mathbb{G}_m)$.

math.AG↗