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Proceedings of the American Mathematical Society

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What is the Rees algebra of a module?


Authors: David Eisenbud, Craig Huneke and Bernd Ulrich
Journal: Proc. Amer. Math. Soc. 131 (2003), 701-708
MSC (2000): Primary 13A30, 13B21; Secondary 13C12
DOI: https://doi.org/10.1090/S0002-9939-02-06575-9
Published electronically: September 17, 2002
MathSciNet review: 1937406
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Abstract: In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module $M$ may be computed in terms of a “maximal” map $f$ from $M$ to a free module as the image of the map induced by $f$ on symmetric algebras. We show that the analytic spread and reductions of $M$ can be determined from any embedding of $M$ into a free module, and in characteristic 0—but not in positive characteristic!—the Rees algebra itself can be computed from any such embedding.


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Additional Information

David Eisenbud
Affiliation: Mathematical Sciences Research Institute, 1000 Centennial Dr., Berkeley, California 94720
MR Author ID: 62330
ORCID: 0000-0002-5418-5579
Email: de@msri.org

Craig Huneke
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
MR Author ID: 89875
Email: huneke@math.ukans.edu

Bernd Ulrich
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
MR Author ID: 175910
Email: ulrich@math.purdue.edu

Keywords: Rees algebra, module, integral dependence
Received by editor(s): May 2, 2001
Received by editor(s) in revised form: October 19, 2001
Published electronically: September 17, 2002
Additional Notes: All three authors were partially supported by the NSF
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2002 American Mathematical Society