# Tensor Products of Affine and Formal Abelian Groups

### Tilman Bauer

Department of Mathematics, Kungliga Tekniska Högskolan, Lindstedtsvägen 25, 10044 Stockholm, Sweden### Magnus Carlson

Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

## 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 Dieudonné theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.

## Cite this article

Tilman Bauer, Magnus Carlson, Tensor Products of Affine and Formal Abelian Groups. Doc. Math. 24 (2019), pp. 2525–2582

DOI 10.4171/DM/733