A simple proof of some generalized principal ideal theorems

Authors:
David Eisenbud, Craig Huneke and Bernd Ulrich

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2535-2540

MSC (2000):
Primary 13C15, 13C40; Secondary 13D10

DOI:
https://doi.org/10.1090/S0002-9939-01-05877-4

Published electronically:
February 22, 2001

MathSciNet review:
1838374

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Using symmetric algebras we simplify and slightly strengthen the Bruns-Eisenbud-Evans ``generalized principal ideal theorem'' on the height of order ideals of nonminimal generators in a module. We also obtain a simple proof and an extension of a result by Kwiecinski, which estimates the height of certain Fitting ideals of modules having an equidimensional symmetric algebra.

**1.**W. Bruns: The Eisenbud-Evans generalized principal ideal theorem and determinantal ideals. Proc. Amer. Math. Soc.**83**(1981), 19-24. MR**82k:13010****2.**D. Eisenbud:*Commutative Algebra with a View Toward Algebraic Geometry.*Springer Verlag, 1995.**3.**D. Eisenbud and E. G. Evans: A generalized principal ideal theorem. Nagoya Math. J.**62**(1976), 41-53. MR**53:13195****4.**D. Eisenbud, C. Huneke, and B. Ulrich: Order ideals and a generalized Krull height theorem. Preprint.**5.**D. Eisenbud, C. Huneke, and B. Ulrich: Heights of ideals of minors. In preparation.**6.**C. Huneke and M. Rossi: The dimension and components of symmetric algebras. J. Algebra**98**(1986), 200-210. MR**87d:13010****7.**M. Johnson: Equidimensional symmetric algebras and residual intersections. Preprint.**8.**M. Kwiecinski: Bounds for codimensions of Fitting ideals. J. Algebra**194**(1997), 378-382. MR**98m:13018****9.**J.-P. Serre:*Algèbre locale, multiplicités.*Springer Lect. Notes in Math.**11**, 1965. MR**34:1352****10.**W. Vasconcelos:*Arithmetic of blowup algebras.*London Math. Soc. Lect. Notes**195**, Cambridge Univ. Press, 1994. MR**95g:13005**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
13C15,
13C40,
13D10

Retrieve articles in all journals with MSC (2000): 13C15, 13C40, 13D10

Additional Information

**David Eisenbud**

Affiliation:
Mathematical Sciences Research Institute, 1000 Centennial Drive, 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, Michigan State University, E. Lansing, Michigan 48824

Email:
ulrich@math.msu.edu

DOI:
https://doi.org/10.1090/S0002-9939-01-05877-4

Keywords:
Height,
order ideals,
determinantal ideals,
symmetric algebras,
equidimensionality

Received by editor(s):
September 21, 1999

Received by editor(s) in revised form:
January 14, 2000

Published electronically:
February 22, 2001

Additional Notes:
The authors are grateful to the NSF and to MSRI for support.

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 2001
American Mathematical Society