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The Number of Generators of Modules over Polynomial Rings

Gennady Lyubeznik
Proceedings of the American Mathematical Society
Vol. 103, No. 4 (Aug., 1988), pp. 1037-1040
DOI: 10.2307/2047081
Stable URL: http://www.jstor.org/stable/2047081
Page Count: 4
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The Number of Generators of Modules over Polynomial Rings
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Abstract

Let $k$ be an infinite field and $B = k\lbrack X_1,\ldots, X_n \rbrack$ a polynomial ring over $k$. Let $M$ be a finitely generated module over $B$. For every prime ideal $P \subset B$ let $\mu(M_P)$ be the minimum number of generators of $M_P$, i.e., $\mu(M_P) = \dim_{B_P/P_P}(M_P \otimes _{B_P} (B_P/P_P))$. Set $\eta(M) = \max\{\mu(M_P) + \dim(B/P)|P \in \operatorname{Spec} B\quad\text{such that}\quad M_P\quad\text{is not free}\quad \}$. Then $M$ can be generated by $\eta(M)$ elements. This improves earlier results of A. Sathaye and N. Mohan Kumar on a conjecture of Eisenbud-Evans.

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