Module projective

Question

Is this relation true?

• $M$ is projective module if and only if $M_m$ is projective module for every maximal ideal $m$.
• $M$ is finitely generated free module if and only if $M_m$ is finitely generated free module for every maximal $w$-ideal $m$.

Thank you so much

Answer 1

Over a commutative Noetherian ring a finitely generated module is projective if and only if it's localization at every maximal ideal is projective (in fact free). Over a non-Noetherian commutative ring it can happen that a finitely generated module is not projective yet every localization is free. So in general the answer to both of your questions is no.