Lemma 30.23.4. The functor (30.23.3.1) is exact.

**Proof.**
It suffices to check this locally on $X$. Hence we may assume $X$ is affine, i.e., we have a situation as in Lemma 30.23.1. The functor is the functor $\text{Mod}^{fg}_ A \to \text{Mod}^{fg}_{A^\wedge }$ which associates to a finite $A$-module $M$ the completion $M^\wedge $. Thus the result follows from Algebra, Lemma 10.97.2.
$\square$

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