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Results tagged with projective-modules
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user 450
For questions about projective modules over a ring and projective objects in related categories.
35
votes
Accepted
A ring on which all finitely generated projectives modules are free but not all projectives ...
Cher Michel, these rings are uncommon.
Over a local ring ALL projective modules are free : this is a celebrated theorem due to Kaplansky.
If $R$ is commutative noetherian and $Spec(R)$ is connected …
- 2,821
12
votes
2
answers
1k
views
Is every locally free module of rank $1$ over a commutative ring concretely invertible?
Since this subject is full of misunderstandings (see here, here, here, and here) let us fix a precise terminology.
Let $A$ be a commutative ring and $P$ an $A$-module.
I) We'll say that $P$ is a lo …
- 4,985
8
votes
What is the right definition of the Picard group of a commutative ring?
1) About the second definition:
$\alpha$) It is not true that for an arbitrary ring a) is equivalent to c):
Indeed Bourbaki in Algèbre commutative, Chapitre II, Exercices §5, 12) c) exhibits a ring …
- 47.5k
11
votes
Nonfree projective module over a regular UFD?
If Pete or someone else is still interested despite the fine answers already given, here is an analysis of what might be the simplest situation. Let $k$ be a field of characteristic $\neq 2$ and defin …
- 47.5k