Hello!
This is a very short question:
Given a local graded Noetherian ring $R_{\bullet}$, is it true that any graded projective module over $R_{\bullet}$ is free?
In the ungraded case, this is true, but I do not know where the graded case is considered. Are there any references?
Thank you!
Hanno
Best Answer
There is a proof of the ungraded result on page 10 of Matsumura's "Commutative Ring Theory", and you can insert gradings everywhere in a straightforward way to prove the graded result.
The main reason why you cannot just appeal to the ungraded result is as follows: a local graded ring has (by definition) precisely one homogeneous ideal that is maximal among proper homogeneous ideals, but typically there will be many maximal inhomogeneous ideals, so the underlying ungraded ring will not be local.