The Buchsbaum–Rim Polynomial of a Module

For a Noetherian local ring (R, ), the Hilbert functions of -primary ideals provide considerable information about constructions and singularities coded by the ideal. For a module E which is the analogue of such ideals (E is a proper submodule of a free module so that the quotient has finite length), Buchsbaum and Rim defined a similar function and introduced the notion of Buchsbaum–Rim multiplicity br(E) of the module. Unlike in the ideal case, the information about these functions is still scant. When R is Cohen–Macaulay of dimension d > 1 and E has rank r, we develop some of their properties sufficiently enough to establish the following bound for the reduction number of the module E, r(E) ≤ (d + r − 1) • (br(E) − 2) + 1. The same techniques lead to bounds on the number of generators of the integral closure of the powers of certain ideals and modules.