On the number of generators of modules over polynomial rings

Li, Hongnian

doi:10.1090/S0002-9939-1994-1213864-6

On the number of generators of modules over polynomial rings
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by Hongnian Li

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

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

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