The Stacks project

Lemma 51.17.1. Let $p$ be a prime number. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring with $p = 0$ in $A$. Let $M$ be a finite $A$-module such that $M \otimes _{A, F} A \cong M$. Then $M$ is finite free.

Proof. Choose a presentation $A^{\oplus m} \to A^{\oplus n} \to M$ which induces an isomorphism $\kappa ^{\oplus n} \to M/\mathfrak m M$. Let $T = (a_{ij})$ be the matrix of the map $A^{\oplus m} \to A^{\oplus n}$. Observe that $a_{ij} \in \mathfrak m$. Applying base change by $F$, using right exactness of base change, we get a presentation $A^{\oplus m} \to A^{\oplus n} \to M$ where the matrix is $T = (a_{ij}^ p)$. Thus we have a presentation with $a_{ij} \in \mathfrak m^ p$. Repeating this construction we find that for each $e \geq 1$ there exists a presentation with $a_{ij} \in \mathfrak m^ e$. This implies the fitting ideals (More on Algebra, Definition 15.8.3) $\text{Fit}_ k(M)$ for $k < n$ are contained in $\bigcap _{e \geq 1} \mathfrak m^ e$. Since this is zero by Krull's intersection theorem (Algebra, Lemma 10.51.4) we conclude that $M$ is free of rank $n$ by More on Algebra, Lemma 15.8.7. $\square$

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