Tensor products of affine and formal abelian groups

TL;DR

This paper investigates the tensor products of affine abelian group schemes over perfect fields, establishing their existence and describing their multiplicative and unipotent components using Galois modules and Dieudonné theory.

Contribution

It proves the existence of tensor products of affine abelian group schemes over perfect fields and characterizes their structure in terms of Galois modules and Dieudonné theory.

Findings

Tensor products of affine abelian group schemes exist over perfect fields.

The multiplicative part is described via Galois modules.

The unipotent part is explicitly characterized using Dieudonné theory.

Abstract

In this paper we study tensor products of affine abelian group schemes over a perfect field $k.$ We first prove that the tensor product $G_1 \otimes G_2$ of two affine abelian group schemes $G_1,G_2$ over a perfect field $k$ exists. We then describe the multiplicative and unipotent part of the group scheme $G_1 \otimes G_2$. The multiplicative part is described in terms of Galois modules over the absolute Galois group of $k.$ We describe the unipotent part of $G_1 \otimes G_2$ explicitly, using Dieudonn\'e theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.