CABRAL, T. F. C.; http://lattes.cnpq.br/6928955373251872; CABRAL, Thiago Fiel da Costa.
Abstract:
In this work we present an study of the known family of Buchsbaum-Eisenbud complexes
via the approach of Koszul-Čech spectral sequences given by Bouça and Hassanzadeh.
We first construct this family of complexes using the Koszul-Čech 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.