The Stacks project

Proof. By Lemma 10.78.2 being finite projective over a local ring is the same thing as being finite free. Suppose that $M \otimes _ R S$ is a finite free $S$-module. Pick $x_1, \ldots , x_ r \in M$ whose images in $M/\mathfrak m_ RM$ form a basis over $\kappa (\mathfrak m)$. Then we see that $x_1 \otimes 1, \ldots , x_ r \otimes 1$ are a basis for $M \otimes _ R S$. This implies that the map $R^{\oplus r} \to M, (a_ i) \mapsto \sum a_ i x_ i$ becomes an isomorphism after tensoring with $S$. By faithful flatness of $R \to S$, see Lemma 10.39.17 we see that it is an isomorphism. $\square$