The multiplicity of a set of homogeneous polynomials over a commutative ring

Abstract

Some thirty years ago Buchsbaum and Rim [1] extended the notion of multiplicity e(a₁, …, a_n; E) for elements a₁, …, a_n of a commutative ring R with identity and a Noetherian R-module E (≠0) with length_R (E/[sum ]ⁿ_t=1a_iE) finite to give a multiplicity e((a_ij); E) associated with E and an m×n matrix (a_ij) over R satisfying a certain extended finiteness condition. One of their results states that for each of a set of m complexes depending on E, (a_ij) the Euler–Poincaré characteristic is a certain integer multiple of e((a_ij); E), at least when R is a local ring.

Some of these ideas were taken up in [4] where it is shown that when n[ges ]m−1, each of the complexes K((a_ij); E; t) with t∈ℤ introduced in [3] also have e((a_ij); E) as their Euler–Poincaré characteristic. With a slight change in viewpoint (a_ij) can be replaced by linear forms a_j=[sum ]^m_i=1a_ijx_i (j=1, …, n) of the graded polynomial ring R[x₁, …, x_m]; the complex K((a_ij); E; t) then becomes the component of degree t in a certain graded double complex

formula here

where K(a₁, …, a_n; F) is the standard Koszul complex (see [4; section 2]). From this point of view the construction can be extended to allow the homogeneous polynomials a₁, …, a_n to have any (possibly unequal) positive degrees [6]. The main aim of the present note is to extend similarly the results of [4] and to strengthen those results to give information on the vanishing of the multiplicity.