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Codimension, multiplicity and integral extensions

Published online by Cambridge University Press:  26 March 2001

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

Let A ⊂ 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 ⊂ 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 ⊂ 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 130 , Issue 2 , March 2001 , pp. 237 - 257
Copyright
2001 Cambridge Philosophical Society

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