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Rees Algebras of Modules

Published online by Cambridge University Press:  23 October 2003

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: [email protected]
Bernd Ulrich
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395, USA. E-mail: [email protected]
Wolmer V. Vasconcelos
Affiliation:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA. E-mail: [email protected]
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Abstract

We study Rees algebras of modules within a fairly general framework. We introduce an approach through the notion of Bourbaki ideals that allows the use of deformation theory. One can talk about the (essentially unique) generic Bourbaki ideal I(E) of a module E which, in many situations, allows one to reduce the nature of the Rees algebra of E to that of its Bourbaki ideal I(E). Properties such as Cohen–Macaulayness, normality and being of linear type are viewed from this perspective. The known numerical invariants, such as the analytic spread, the reduction number and the analytic deviation, of an ideal and its associated algebras are considered in the case of modules. Corresponding notions of complete intersection, almost complete intersection and equimultiple modules are examined in some detail. Special consideration is given to certain modules which are fairly ubiquitous because interesting vector bundles appear in this way. For these modules one is able to estimate the reduction number and other invariants in terms of the Buchsbaum–Rim multiplicity.

Keywords

analytic spreadBourbaki idealBuchsbaum–Rim multiplicityCohen–Macaulay ringminimal reductionreduction numberRees algebra13H1013A3013H15
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 87 , Issue 3 , November 2003 , pp. 610 - 646
Copyright
2003 London Mathematical Society

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Footnotes

The research of A.S. was partially supported by CNPq, Brazil. That of B.U. and W.V.V. was partially supported by the NSF.