Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!!

share|improve this question
1  
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

1 Answer 1

up vote 7 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.

share|improve this answer
    
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
1  
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.