What is the Rees algebra of a module?
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- by David Eisenbud, Craig Huneke and Bernd Ulrich PDF
- Proc. Amer. Math. Soc. 131 (2003), 701-708 Request permission
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.References
- D. Eisenbud, C. Huneke and B. Ulrich, Order ideals and a generalized Krull height theorem, to appear in Math. Ann.
- —, Heights of ideals of minors, preprint, 2001.
- Terence Gaffney and Steven L. Kleiman, Specialization of integral dependence for modules, Invent. Math. 137 (1999), no. 3, 541–574. MR 1709870, DOI 10.1007/s002220050335
- D. Katz, Reduction criteria for modules, Comm. Algebra 23 (1995), no. 12, 4543–4548. MR 1352554, DOI 10.1080/00927879508825485
- Daniel Katz and Vijay Kodiyalam, Symmetric powers of complete modules over a two-dimensional regular local ring, Trans. Amer. Math. Soc. 349 (1997), no. 2, 747–762. MR 1401523, DOI 10.1090/S0002-9947-97-01819-9
- Steven Kleiman and Anders Thorup, Conormal geometry of maximal minors, J. Algebra 230 (2000), no. 1, 204–221. MR 1774764, DOI 10.1006/jabr.1999.7972
- Vijay Kodiyalam, Integrally closed modules over two-dimensional regular local rings, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3551–3573. MR 1308016, DOI 10.1090/S0002-9947-1995-1308016-0
- Jung-Chen Liu, Rees algebras of finitely generated torsion-free modules over a two-dimensional regular local ring, Comm. Algebra 26 (1998), no. 12, 4015–4039. MR 1661272, DOI 10.1080/00927879808826392
- D. Rees, Reduction of modules, Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 3, 431–449. MR 878892, DOI 10.1017/S0305004100066810
- Aron Simis, Bernd Ulrich, and Wolmer V. Vasconcelos, Codimension, multiplicity and integral extensions, Math. Proc. Cambridge Philos. Soc. 130 (2001), no. 2, 237–257. MR 1806775, DOI 10.1017/S0305004100004667
- —, Rees algebras of modules, to appear in Proc. London Math. Soc.
- Wolmer V. Vasconcelos, Arithmetic of blowup algebras, London Mathematical Society Lecture Note Series, vol. 195, Cambridge University Press, Cambridge, 1994. MR 1275840, DOI 10.1017/CBO9780511574726
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
- 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
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1937406