Abstract

In this lecture we study the multilinear morphisms between group schemes and associated constructions. We will also do some explicit calculations and give examples that show that this theory behaves in a way that one would naturally expect. Let G₁,...,G_r and H be commutative group schemes over a base scheme S. A multilinear morphism f:G₁×...×G_r® H is a morphism of schemes over S that is linear in each G_i. The group of all such multilinear morphismsl is denoted by Mult(G₁×...×G_r,H). We define in the same fashion the symmetric and alternating morphisms and the group of all such morphisms. Dually, we define the tensor product of G₁,..., G_r to be a commutative group scheme G₁ؤ...ؤG_r together with a üniversal" multilinear morphism f:G₁×...×G_r®G₁ؤ...ؤG_r that yields an isomorphism