On the number of generators of modules over polynomial rings

Author:
Hongnian Li

Journal:
Proc. Amer. Math. Soc. **121** (1994), 347-351

MSC:
Primary 13E15; Secondary 13F20

DOI:
https://doi.org/10.1090/S0002-9939-1994-1213864-6

MathSciNet review:
1213864

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove the following

**Theorem.** *Let* , *where A is a universally catenary equidimensional ring. Let M be a finitely generated B-module of rank r. Denote by d the Krull dimension of A, by* *the minimal number of generators of M, and by* *the* (*radical*) *ideal which defines the set of primes of B at which M is not locally free. Assume that*

*where*

*is a positive integer. Then*.

This improves a result of R. G. López, *On the number of generators of modules over polynomial affine rings*, Math. Z. **208** (1991), 11-21.

**[BR]**s. M. Bhatwadekar and A. Roy,*Some theorems about projective modules over polynomial rings*, J. Algebra**86**(1986), 150-158. MR**727374 (86m:13016)****[L]**R. G. López,*On the number of generators of modules over polynomial affine rings*, Math. Z.**208**(1991), 11-21. MR**1125729 (93b:13033)****[M]**H. Matsumura,*Commutative ring theory*, Cambridge Stud. Adv. Math., vol. 8, Cambridge Univ. Press, Cambridge and New York, 1986. MR**879273 (88h:13001)****[Mi]**J. Milnor,*Introduction to algebraic K-theory*, Ann. of Math. Stud., no. 72, Princeton Univ. Press, Princeton, NJ, 1971. MR**0349811 (50:2304)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1213864-6

Keywords:
Number of generators,
modules,
polynomial rings,
universal catenary rings

Article copyright:
© Copyright 1994
American Mathematical Society