Lemma 15.8.4. Let $R$ be a ring. Let $M$ be a finite $R$-module.

If $M$ can be generated by $n$ elements, then $\text{Fit}_ n(M) = R$.

Given a second finite $R$-module $M'$ we have

\[ \text{Fit}_ l(M \oplus M') = \sum \nolimits _{k + k' = l} \text{Fit}_ k(M)\text{Fit}_{k'}(M') \]If $R \to R'$ is a ring map, then $\text{Fit}_ k(M \otimes _ R R')$ is the ideal of $R'$ generated by the image of $\text{Fit}_ k(M)$.

If $M$ is of finite presentation, then $\text{Fit}_ k(M)$ is a finitely generated ideal.

If $M \to M'$ is a surjection, then $\text{Fit}_ k(M) \subset \text{Fit}_ k(M')$.

We have $\text{Fit}_0(M) \subset \text{Ann}_ R(M)$.

We have $V(\text{Fit}_0(M)) = \text{Supp}(M)$.

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## Comments (2)

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