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

 

The number of generators of modules over polynomial rings


Author: Gennady Lyubeznik
Journal: Proc. Amer. Math. Soc. 103 (1988), 1037-1040
MSC: Primary 13C10
MathSciNet review: 954979
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Abstract: Let $ k$ be an infinite field and $ B = k[{X_1}, \ldots ,{X_n}]$ 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)\left\vert {P \in \operatorname{Spe... ...ext{such}}\;{\text{that}}\;{M_{P\;}}{\text{is}}\;{\text{not}}\;{\text{free}}\} $. 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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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0954979-4
PII: S 0002-9939(1988)0954979-4
Article copyright: © Copyright 1988 American Mathematical Society