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Let $\mathcal{F}$ be a locally free sheaf of finite rank of scheme $X$, is $\mathcal{F}$ coherent?

By the definition of locally free sheaf, there exists an open cover {$U_i$} of $X$ such that $\mathcal{F}|_{U_i}$ is isomorphic to the sheaf $\widetilde{\mathcal{O}(U_i)^n}$. But we don't know each $U_i$ affine or not!

So, it that true or not?

How about $X$ being locally noetherian? It $X$ is, we can find $V_{ij} \subset U_i$ s.t. $V_{ji} = Spec(A_{ji})$. And $\mathcal{F}|_{V_{ij}} = \mathcal{F}|_{U_i}|_{V_{ij}}$...?

For example, $X(\Delta)$ is a toric varity with $\Delta$ consists of strongly convex polyhedral cones.

Thank you very much!!

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Being coherent involves a finiteness condition, whereas being locally free does not. – Zhen Lin Jun 30 '12 at 11:15
If we only consider locally free sheaf of finite rank? – Peter Hu Jun 30 '12 at 11:17
What do you mean by $\widetilde{\mathcal{O}(U_i)^n}$? As you said $U_i$ need not be affine, So $\widetilde{\mathcal{O}(U_i)^n}$ doesn't make any sense. You should write $\bigoplus_{i=1}^n\mathcal{O}|_{U_i}$. – Babai Feb 20 '15 at 7:04
up vote 9 down vote accepted

The exact condition for locally free sheaves on a ringed space $(X,\mathcal O_X)$ to be coherent is exactly that $\mathcal O_X$ be coherent.

a) The condition is clearly necessary since $\mathcal O_X$ is locally free.
b) It is sufficient because if the structure shaf is coherent, then coherence is a local property and because a direct sum of coherent sheaves is coherent: apply to $\mathcal F \mid U_i \cong (\mathcal O\mid U_i)^{\oplus r} $

And when is $\mathcal O_X$ coherent?
There is, to my knowledge, no very good non-tautological criterion.
However, for locally noetherian schemes, it is the case that $\mathcal O_X$ is coherent, so for these schemes, yes, locally free sheaves are coherent.

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Oh! I see. But for the noetherian (or locally noetherian) case, how do we show that locally free sheaves are coherent? Thank you very much! – Peter Hu Jun 30 '12 at 13:12
Dear Peter, from what I wrote, you need only check that the structural sheaf is coherent. You might look that up in Qing Liu's Algebraic Geometry and Arithmetic Curves, Chapter 5, Proposition 1.11, page 161. – Georges Elencwajg Jun 30 '12 at 13:28

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