Krull dimension and integrality of symmetric algebras

Abstract

For a finitely generated module E over the Noetherian ring R we consider formulas for the Krull dimension of the symmetric algebra S(E). A result of Huneke and Rossi is re-proved and an eiffective formulais derived that reads the dimension of S(E) from a presentation of E. They provide a first line of obstructions for S(E) to be an integral domain. For algebras of codimension at most four we give methods, including computer-assisted ones, to ascertain whether S(E) is a Cohen-Macaulay domain.