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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
MathSciNet review: 1213864
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Abstract: In this paper we prove the following

Theorem. Let $ B = A[{X_1}, \cdot ,{X_n}]$, 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 $ \mu (M)$ the minimal number of generators of M, and by $ {I_M}$ the (radical) ideal which defines the set of primes of B at which M is not locally free. Assume that

$\displaystyle \mu (M/{I_M}M) \leq \eta \;and\;\eta \geq \max \{ d + r,\dim B/{I_M} + r + 1\} ,$

where $ \eta $ is a positive integer. Then $ \mu (M) \leq \eta $.

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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1213864-6
PII: S 0002-9939(1994)1213864-6
Keywords: Number of generators, modules, polynomial rings, universal catenary rings
Article copyright: © Copyright 1994 American Mathematical Society