L'invariant de Suslin en caract\'eristique positive

TL;DR

This paper extends Suslin's cohomological invariant for SK_1 of central simple algebras from characteristic zero to positive characteristic by utilizing Kato's cohomology of logarithmic differentials.

Contribution

It introduces a method to define Suslin's invariant in positive characteristic via lifting to characteristic zero and employing Kato's cohomology theory.

Findings

Successfully generalizes Suslin's invariant to positive characteristic

Uses Kato's cohomology of logarithmic differentials for the construction

Provides a framework for invariants in broader algebraic settings

Abstract

Pour une k-alg\`ebre simple centrale A d'indice inversible dans k, Suslin a d\'efini un invariant cohomologique de SK_1(A). Dans ce texte, nous g\'en\'eralisons cet invariant \`a toute k-alg\`ebre simple centrale par un rel\`evement de la caract\'eristique positive \`a la caract\'eristique 0. Pour pouvoir d\'efinir cet invariant, on a besoin des groupes de cohomologie des diff\'erentielles logarithmiques de Kato. For a central simple k-algebra A with index invertible in k, Suslin defined a cohomological invariant for SK_1(A). In this text, we generalise his invariant to any central simple k-algebra using a lift from positive characteristic to characteristic 0. To be able to define the invariant, we use Kato's cohomology of logarithmic differentials.