The number of generators of modules over polynomial rings

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 $\eta (M) = \max \{ \mu ({M_P}) + \dim (B/P)\left | {P \in \operatorname {Spec} } \right .B\;{\text {such}}\;{\text {that}}\;{M_{P\;}}{\text {is}}\;{\text {not}}\;{\text {free}}\}$. 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.