Lemma 2

Let A be a ring and M a projective A-module. The $A_M$-module M is projective.

Let $(x_i)_{i\in I}$ be a generating family of the A-module M. There exists a family $(x^*_i)_{i\in I}$ of linear forms on the A-module M such that for every $x\in M$, the family $(\langle x, x^*_i\rangle )_{i\in I}$ has finite support and $x=\sum_{i\in I}\langle x, x^*_i\rangle x_i$ (II, §2, No. 6, p. 238, Proposition 12). For every $i\in I$, the mapping $x\mapsto \langle x, x^*_i\rangle_M$ is a linear form on the $A_M$-module M, and we have $x=\sum_{i\in I}\langle x, x^*_i\rangle_Mx_i$ for every $x\in M$. By loc. cit., M is a projective $A_M$-module.