Extensions and biextensions of locally constant group schemes, tori and abelian schemes

TL;DR

This paper explicitly computes homomorphisms and extensions among locally constant group schemes, tori, and abelian schemes over a scheme, and studies biextensions involving these objects, revealing categorical equivalences.

Contribution

It provides explicit calculations of homomorphisms and extensions for these group schemes and establishes a key equivalence in biextension categories involving extensions of abelian schemes by tori.

Findings

Explicit descriptions of homomorphisms and extensions.

Categorical equivalence of biextensions involving extensions of abelian schemes by tori.

Foundations for understanding biextensions in algebraic geometry.

Abstract

Let S be a scheme. We compute explicitly the group of homomorphisms, the S-sheaf of homomorphisms, the group of extensions, and the S-sheaf of extensions involving locally constant S-group schemes, abelian S-schemes, and S-tori. Using the obtained results, we study the categories of biextensions involving these geometrical objets. In particular, we prove that if G_i (for i=1,2,3) is an extension of an abelian S-scheme A_i by an S-torus T_i, the category of biextensions of (G_1,G_2) by G_3 is equivalent to the category of biextensions of the underlying abelian S-schemes (A_1,A_2) by the underlying S-torus T_3.