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
G_{1},...,G_{r} and H be commutative group schemes over a base
scheme S. A multilinear morphism f:G_{1}×...×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(G_{1}×...×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 G_{1},..., G_{r} to be a commutative group scheme
G_{1}ؤ...ؤG_{r} together with a üniversal"
multilinear morphism f:G_{1}×...×G_{r}®G_{1}ؤ...ؤG_{r} that yields an isomorphism
Hom(G_{1}ؤ...ؤG_{r},H) ~ Mult(G_{1}×...×G_{r},H), y® y°f. 

Similarly, we define the symmetric
resp. alternating power of a commutative group scheme G
replacing multilinear morphims by symmetric resp. alternating
morphisms. Finally, we give some results that show the analogy of
this theory and multilinear algebra.
Information:
Date:  Thursday, July 13, 2006, 13:3015:00 
Place:  Niavaran Bldg., Niavaran Square, Tehran, Iran 
