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Results tagged with coherent-sheaves
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user 2083
12
votes
Are schemes that "have enough locally frees" necessarily separated
Actually the implication should be reversed: a separated regular Noatherian scheme has enough locally frees (this is Exercise 6.8, Chapter III Hartshorne). So the hypothesis is certainly needed for t …
- 30.6k
9
votes
Is it true that if the pushforward of a coherent sheaf is locally free, then the original sh...
Since you asked for reference, here are some references that may be helpful (the question is local):
1) (EDITED: Thanks to BCnrd for keeping me honest here!) If $(R,m)\to (S,n)$ is a map of Cohen-M …
- 30.6k
3
votes
Accepted
Algebraic vector bundles on the punctured spectrum: an exact reference for a result
One modern account can be found in this thesis of Majidi-Zolbani, especially Chapter 1 and Appendix A. The point is that if $E$ is a vector bundle on $U$, then the global sections $\Gamma_U(E)$ is a f …
- 30.6k
1
vote
Accepted
On locally 3-syzygy sheaves
Locally, the depth increases along syzygies, so there are always counter examples as long as there is a closed point whose local ring has depth at least 3.
For instance, let $R=k[x_1,...,x_n]$ for $n\ …
- 30.6k