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.
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