Endlich erzeugter Modul/Lokal freie Garbe/Charakterisierungen von lokal/en/Fakt/Beweis

Beweis

. This is a specialization.
. We fix a maximal ideal . By assumtion there exists an -isomorphism

We may write the image of the th standard vector as

with and . Let and consider the situation on . The above given isomorphism is defined over , that is we have a homomorphism

which induces in the localization at , but may not be an isomorphism over . Let be a generating system for the module . Since induces a surjection, there exist mapping to . Since the denominators do not belong to , we may replace by and then we may assume that is surjective. Finally, let be the kernel of (this new) . Since is injective, we know that . Since is noetherian, is finitely generated and so again there will be a , , such that . So further shrinking shows that there is an isomorphism for some , .

So far we have shown that for every maximal ideal there exists an open neighborhood such that is free of rank . Hence

contains all maximal ideals and therefore all prime ideals, so it is an open cover of . This means that is the unit ideal, and so finitely many of these generate already the unit ideal.
. Since the elements generate the unit ideal, the corresponding open subsets , , cover . That is a free -module of rank means that . So is locally free.
. Let be a prime ideal. The local freeness means that we have an open covering such that is free of rank . So there exists an index such that . We may shrink to a smaller open neighborhood of and then assume that and that is free of rank . But then also the localization is free of rank .