Extensions of algebraic groups with finite quotient and nonabelian 2-cohomology

Abstract

For a finite smooth algebraic group $F$ over a field $k$ and a smooth algebraic group $\bar{G}$ over the separable closure of $k$, we define the notion of $F$-kernel in $\bar{G}$ and we associate to it a set of nonabelian 2-cohomology. We use this to study extensions of $F$ by an arbitrary smooth $k$-group $G$. We show in particular that any such extension comes from an extension of finite $k$-groups when $k$ is perfect and we give explicit bounds on the order of these finite groups when $G$ is linear. We prove moreover some finiteness results on these sets.

27 pages. Final version

References in corpus (2)

• Groupe de Brauer non ramifié algébrique des espaces homogènes (The unramified algebraic Brauer group of homogeneous spaces)
• Essential p-dimension of algebraic groups whose connected component is a torus

Cited by in corpus (3)

• Jordan property for algebraic groups and automorphism groups of projective varieties in arbitrary characteristic
• Homogeneous vector bundles over abelian varieties via representation theory
• Nonabelian $H^2$ with coefficients in a group and with coefficients in a crossed module