The homological degree of a module
Wolmer V. Vasconcelos
Department of Mathematics - Hill Center, Rutgers University, 
110 Frelinghuysen RD, Piscataway, New Jersey 08854-8019 
Trans. Amer. Math. Soc. 350 (1998), 1167-1179.

Abstract. A homological degree of a graded module M is an extension of the 
usual notion of multiplicity tailored to provide a numerical signature for the 
module even when M is not Cohen-Macaulay. We construct a degree, 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 \`a la Castelnuovo-Mumford's regularity. Several 
applications for estimating reduction numbers of ideals and predictions on the 
outcome of Noether normalizations are given.