Lemma 15.8.6. Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \geq 0$. Let $\mathfrak p \subset R$ be a prime ideal. The following are equivalent

$\text{Fit}_ k(M) \not\subset \mathfrak p$,

$\dim _{\kappa (\mathfrak p)} M \otimes _ R \kappa (\mathfrak p) \leq k$,

$M_\mathfrak p$ can be generated by $k$ elements over $R_\mathfrak p$, and

$M_ f$ can be generated by $k$ elements over $R_ f$ for some $f \in R$, $f \not\in \mathfrak p$.

## Comments (2)

Comment #5107 by typo_bot on

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