I am trying to compute the vanishing of $\operatorname{Ext}^1$ for two sheaves of $\mathcal{O}_X$-Modules and I was wondering if it was possible to use some local argument to reduce the problem to the category of $\mathcal{O}_{X,x}$-modules.

So my question is: Are there any conditions on $\operatorname{Ext}^1(\mathcal{F}_x,\mathcal{G}_x)$ that are enough to show that $\operatorname{Ext}^1(\mathcal{F},\mathcal{G})=0$?

I am working in the case $X= \mathbb{P}^2$.


1 Answer 1


No. This is basically the point of sheaf cohomology -- global Ext is a global computation that relies on how the different pieces of the sheaf patch together.

For example, suppose $F$ and $G$ are coherent and locally free. Then $\mathrm{Ext}^i(F,G) = H^i(F^* \otimes G)$, whereas over any local ring $\mathcal{O}_{X,x}$, $Ext^i(F_x,G_x)$ vanishes for $i>0$ since $F_x$ is free.

That said, a statement of this form does hold for the sheaf version of Ext, namely

$\mathcal{E}xt(F,G)_x = Ext_{\mathcal{O}_{X,x}}(F_x,G_x)$

holds under the right finiteness hypotheses (coherent sheaves on a Noetherian scheme is enough, I think). So a necessary and sufficient for sheaf Ext to vanish is for each of these stalks to vanish.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .