Schubert Calculus on a Grassmann Algebra

The ({\em classical}, {\em small quantum}, {\em equivariant}) cohomology ring of the grassmannian $G(k,n)$ is generated by certain derivations operating on an exterior algebra of a free module of rank $n$ ({\em Schubert Calculus on a Grassmann Algebra)}. Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. It also provides, by results of Laksov and Thorup, a presentation of the universal splitting algebra of a monic polynomial of degree $n$ into the product of two monic polynomials, one of degree $k$.