Let $R\to S$ be a faithfully flat ring map between commutative rings and $M$ any projective $S$-module. Is it true that $M$ is projective $R$-module?
Here, is my attempt: Let $M\oplus X\cong S^{I}$, for some $S$-module $X$ and indexing set $I$. Since it also an isomorphism as $R$-modules. Then it is enough to prove that $S$ is projective over $R$.
Note that the map $h:S\to S\otimes_R R$, $s\mapsto s\otimes 1_R$ is an $R$-module isomorphism. Then $h$ is also an $S$-module homomoprphism as follows: $h(ts)=ts\otimes 1_R=t(s\otimes 1_R)=th(s)$. Hence, $S\cong S\otimes_R R$ as $S$-modules. So, $S\otimes_R R$ is projective over $S$. Then the result follows from Theorem 9.6 in the following link: https://arxiv.org/pdf/1011.0038v1.pdf
My above attempt is correct or there is some mistake. Please guide me.