Endlich erzeugter Modul/Lokal freie Garbe/Charakterisierungen von lokal/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. The localizations ${}M_{\mathfrak {p}}$ are free of rank ${}r$ for every prime ideal ${}{\mathfrak {p}}\in \operatorname {Spek} {\left(R\right)}$.
2. The localizations ${}M_{\mathfrak {m}}$ are free of rank ${}r$ for every maximal ideal ${}{\mathfrak {m}}$ of ${}R$.
3. There exists elements ${}f_{1},\ldots ,f_{k}\in R$ which generate the unit ideal and such that the localizations ${}M_{f_{j}}$ are free of rank ${}r$ for every ${}j=1,\ldots ,k$.
4. The coherent sheaf ${}{\tilde {M}}$ on ${}\operatorname {Spek} {\left(R\right)}$ associated to ${}M$ is locally free.

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