Buchsbaum-Eisenbud complexes in a Koszul-Cech approach

Abstract

In this work we present an study of the known family of Buchsbaum-Eisenbud complexes via the approach of Koszul-Cech spectral sequences given by Bouça and Hassanzadeh. We first construct this family of complexes using the Koszul-Cech structure and give new proofs for the basic facts as acyclicity and support of the homologies. Second, via convergence of spectral sequences, we give a formula of the Buchsbaum-Rim multiplicity as the arithmetic genus (Euler-Poincaré characteristic) of Koszul homology sheaves on a projective space over an arbitrary Noetherian base scheme. This formula is a generalization of Serre, the formula for the Hilbert-Samuel multiplicity of a system of parameters to the case of Buchsbaum-Rim multiplicity. In order to obtain this formula, we introduce a notion of Hilbert function of a graded ring over an arbitrary Noetherian base ring.