Lemma 38.19.5. Let $A \to B$ be a local ring map of local rings which is essentially of finite type. Let $N$ be a finite $B$-module which is flat as an $A$-module. If $A$ is henselian, then $N$ is a filtered colimit

$N = \mathop{\mathrm{colim}}\nolimits _ i F_ i$

of free $A$-modules $F_ i$ such that all transition maps $u_ i : F_ i \to F_{i'}$ of the system induce injective maps $\overline{u}_ i : F_ i/\mathfrak m_ AF_ i \to F_{i'}/\mathfrak m_ AF_{i'}$. Also, $N$ is a Mittag-Leffler $A$-module.

Proof. We can find a morphism of finite type $X \to S = \mathop{\mathrm{Spec}}(A)$ and a point $x \in X$ lying over the closed point $s$ of $S$ and a finite type quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ such that $\mathcal{F}_ x \cong N$ as an $A$-module. After shrinking $X$ we may assume that each point of $\text{Ass}_{X_ s}(\mathcal{F}_ s)$ specializes to $x$. By Lemma 38.19.4 we see that there exists a fundamental system of affine open neighbourhoods $U_ i \subset X$ of $x$ such that $\Gamma (U_ i, \mathcal{F})$ is a free $A$-module $F_ i$. Note that if $U_{i'} \subset U_ i$, then

$F_ i/\mathfrak m_ AF_ i = \Gamma (U_{i, s}, \mathcal{F}_ s) \longrightarrow \Gamma (U_{i', s}, \mathcal{F}_ s) = F_{i'}/\mathfrak m_ AF_{i'}$

is injective because a section of the kernel would be supported at a closed subset of $X_ s$ not meeting $x$ which is a contradiction to our choice of $X$ above. Since the maps $F_ i \to F_{i'}$ are $A$-universally injective (Lemma 38.7.5) it follows that $N$ is Mittag-Leffler by Algebra, Lemma 10.89.9. $\square$

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