Tell me more ×  Is the category of finite flat group schemes
equipped with "tensor products" and Hom-objects, encoding
bilinear maps? I'm aware that the Cartier dual is
$Hom(\mathbb{G}, \mathbb{G}_m)$, and want to know if this is
part of a systematic collection of objects. For example, is
there a "free ring scheme on $\mathbb{G}$"?  If so, given two
affine group schemes whose underlying rings are free over the
base, are there explicit descriptions of the tensor product and
Hom objects in terms of the multiplication and comultiplication
rules on the original rings?  Over a field, is there a
description in terms of the Dieudonne correspondence?
(References, if they exist, would be very much appreciated.
Thank you.)  share|improve this question Well, the category of
group schemes sits inside the category of fppf sheaves of groups
over your base. If you restrict to fppf sheaves of abelian
groups, then the latter category has Homs and tensor products
(just sheaf Hom and sheaf tensor product). In general, these op