The homological degree of a module

Abstract

A homological degree of a graded module M M is an extension of the usual notion of multiplicity tailored to provide a numerical signature for the module even when M M is not Cohen–Macaulay. We construct a degree, hdeg ⁡ ( M ) \operatorname {hdeg}(M) , that behaves well under hyperplane sections and the modding out of elements of finite support. When carried out in a local algebra this degree gives a simulacrum of complexity à la Castelnuovo–Mumford’s regularity. Several applications for estimating reduction numbers of ideals and predictions on the outcome of Noether normalizations are given.