Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

What is the Rees algebra of a module?

Author(s): David Eisenbud; Craig Huneke; Bernd Ulrich
Journal: Proc. Amer. Math. Soc. 131 (2003), 701-708.
MSC (2000): Primary 13A30, 13B21; Secondary 13C12
Posted: September 17, 2002
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

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:

[EHU1]
D. Eisenbud, C. Huneke and B. Ulrich, Order ideals and a generalized Krull height theorem, to appear in Math. Ann.

[EHU2]
-, Heights of ideals of minors, preprint, 2001.

[GK]
T. Gaffney and S. Kleiman, Specialization of integral dependence for modules, Invent. Math. 137 (1999), 541-574. MR 2000k:32025

[K]
D. Katz, Reduction criteria for modules, Comm. in Algebra 23 (1995), 4543-4548. MR 96j:13022

[KK]
D. Katz and V. Kodiyalam, Symmetric powers of complete modules over a two-dimensional regular local ring, Trans. Amer. Math. Soc. 349 (1997), 747-762. MR 97g:13041

[KT]
S. Kleiman and A. Thorup, Conormal geometry of maximal minors, J. Algebra 230 (2000), 204-221. MR 2001h:13006

[Ko]
V. Kodiyalam, Integrally closed modules over two-dimensional regular local rings, Trans. Amer. Math. Soc. 347 (1995), 3551-3573. MR 95m:13015

[L]
J.-C. Liu, Rees algebras of finitely generated torsion-free modules over a two-dimensional regular local ring, Comm. in Algebra 26 (1998), 4015-4039. MR 99k:13003

[R]
D. Rees, Reduction of modules, Math. Proc. Camb. Phil. Soc. 101 (1987), 431-449. MR 88a:13001

[SUV1]
A. Simis, B. Ulrich and W. V. Vasconcelos, Codimension, multiplicity and integral extensions, Math. Proc. Camb. Phil. Soc. 130 (2001), 237-257. MR 2002c:13017

[SUV2]
-, Rees algebras of modules, to appear in Proc. London Math. Soc.

[V]
W. V. Vasconcelos, Arithmetic of Blowup Algebras, London Math. Soc. Lect. Notes, vol. 195, Cambridge University Press, Cambridge, 1994. MR 95g:13005

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13A30, 13B21, 13C12

Retrieve articles in all Journals with MSC (2000): 13A30, 13B21, 13C12

Additional Information:

David Eisenbud
Affiliation: Mathematical Sciences Research Institute, 1000 Centennial Dr., Berkeley, California 94720
Email: de@msri.org

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

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

DOI: 10.1090/S0002-9939-02-06575-9
PII: S 0002-9939(02)06575-9
Keywords: Rees algebra, module, integral dependence
Received by editor(s): May 2, 2001
Received by editor(s) in revised form: October 19, 2001
Posted: September 17, 2002
Additional Notes: All three authors were partially supported by the NSF
Communicated by: Wolmer V. Vasconcelos
Copyright of article: Copyright 2002, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google