Endlich erzeugter Modul/Lokal freie Garbe/Projektiv und flach/en/Fakt

Let ${}R$ denote a commutative noetherian ring and let ${}M$ denote a finitely generated ${}R$-module. Let ${}r\in \mathbb {N}$. Then the following conditions are equivalent.

1. ${}{\tilde {M}}$ is locally free.
2. ${}M$ is a projective module.
3. ${}M$ is a (faithfully) flat module.

Einen Beweis erstellen