Mathematical Proceedings of the Cambridge Philosophical Society



Codimension, multiplicity and integral extensions


ARON SIMIS a1 1 , BERND ULRICH a2 2 and WOLMER V. VASCONCELOS a3 3
a1 Departamento de Matemática, Universidade Federal de Pernambuco, 50740-540 Recife, PE, Brazil; e-mail: aron@dmat.ufpe.br
a2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A. e-mail: ulrich@math.msu.edu
a3 Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, U.S.A. e-mail: vasconce@math.rutgers.edu

Abstract

Let A [subset or is implied by] B be a homogeneous inclusion of standard graded algebras with A0 = B0. To relate properties of A and B we intermediate with another algebra, the associated graded ring G = grA1B(B). We give criteria as to when the extension A [subset or is implied by] B is integral or birational in terms of the codimension of certain modules associated to G. We also introduce a series of multiplicities associated to the extension A [subset or is implied by] B. There are applications to the extension of two Rees algebras of modules and to estimating the (ordinary) multiplicity of A in terms of that of B and of related rings. Many earlier results by several authors are recovered quickly.

(Received July 20 1999)
(Revised October 4 1999)



Footnotes

1 Partially supported by CNPq, Brazil.

2 Partially supported by the NSF.

3 Partially supported by the NSF.