Sur le produit tensoriel des groupes affines

Let A and B be two commutative affine group schemes over a field. There exists an affine group A?B such that Hom(A?B,C)?Bil(A×B,C) for any affine group C. We use technics of the commutative algebraic groups theory, in order to compute these tensor products and to characterize “flat” groups in the unipotent case. The tensor product of commutative affine groups has most properties of the usual tensor product but it is not always associative. As an application we prove a structure theorem of the category of modules over some affine connected prosmooth rings.