A STUDY IN THE FREENESS OF FINITELY GENERATED Anp-MODULES UPON RESTRICTION TO PRINCIPAL SUBALGEBRAS

Abstract

We are interested in quantitative information on the freeness of modules over a truncated polynomial ring when restricting to subalgebras generated by a linear form. After investigating the structure of the truncated polynomial ring, subalgebras generated by a linear form, and corresponding vector spaces, we construct a generic representation and discuss its connection to a certain affine space. We quantify the abundance of freeness of modules using a certain variety called the rank variety. For any possible dimension we construct a module whose rank variety has that dimension. Finally, we define another variety, called the module variety, and show that the dimension of this variety is invariant under a change of subalgebra.