Ext between two coherent sheaves

Question

Let $X$ be a smooth projective variety over a field $k = \overline k$. From Hartshorne we know, that $\textrm{dim} \, H^i (X,F)<\infty$ for any coherent sheaf $F$.

How to show, that all $Ext^i (F,G)$ are finite-dimensional for coherent $F$ and $G$? And how to show, that for $F,G \in D^b (Coh(X))$ : $Ext^i (F,G)$ are finite-dimensional?

May be, we should use a spectral sequence $E^{p,q}_2 = Ext^p (F, H^q(G)) \Rightarrow Ext^{p+q} (F,G)$?

Answer 1

I'll try to do the non-derived category case.

Let $X$ be a projective variety over $k$. We will do an induction on $i$. The result is clear for $i=0$, since in this case

$Ext^{0}(\mathcal{F}, \mathcal{G}) = Hom(\mathcal{F}, \mathcal{G}) \simeq \Gamma(X, \mathcal{H}om(\mathcal{F},\mathcal{G}))$

which is finite-dimensional by Hartshorne, II.5.19, because $\mathcal{H}om(\mathcal{F}, \mathcal{G})$ is coherent.

Moreover, I claim that the result is also true for all $i$ and when $\mathcal{F} = \bigoplus \mathcal{L}_{j}$ is a direct sum of line bundles. Indeed, in this case

$Ext^{i}(\mathcal{F}, \mathcal{G}) = Ext^{i}(\bigoplus \mathcal{L} _{j}, \mathcal{G}) = \bigoplus Ext^{i}(\mathcal{L} _{j}, \mathcal{G}) \simeq \bigoplus Ext^{i}(\mathcal{O}_{X}, \mathcal{G} \otimes \mathcal{L} _{j} ^{\vee})$

where the last isomorphism is III.6.7 and then

$\bigoplus Ext^{i}(\mathcal{O}_{X}, \mathcal{G} \otimes \mathcal{L} _{j} ^{\vee}) \simeq \bigoplus H^{i}(\mathcal{G} \otimes \mathcal{L} _{j} ^{\vee})$

by III.6.3. (I am secretly using that cohomology of coherent sheaves is finite-dimensional, which is a part of the question and also III.5.2).

Now let's do the general case. By Hartshorne, Corollary II.5.18, any coherent sheaf $\mathcal{F}$ on $X$ is a quotient of some sum of line bundles $\mathcal{E}$. Let $\mathcal{R}$ be the kernel, ie. we have an exact sequence of sheaves

$0 \rightarrow \mathcal{R} \rightarrow \mathcal{E} \rightarrow \mathcal{F} \rightarrow 0$.

By the universal property of $Ext$ we have a long exact sequence

$0 \rightarrow Ext^{0}(\mathcal{F}, \mathcal{G}) \rightarrow Ext^{0}(\mathcal{E}, \mathcal{G}) \rightarrow Ext^{0}(\mathcal{R}, \mathcal{G}) \rightarrow Ext^{1}(\mathcal{F}, \mathcal{G}) \rightarrow \ldots$

and the interesting part is

$\ldots \rightarrow Ext^{i}(\mathcal{R}, \mathcal{G}) \rightarrow Ext^{i+1}(\mathcal{F}, \mathcal{G}) \rightarrow Ext^{i+1}(\mathcal{E}, \mathcal{G}) \rightarrow \ldots$.

Since both $Ext^{i}(\mathcal{R}, \mathcal{G})$ (by inductive assumption) and $Ext^{i+1}(\mathcal{E}, \mathcal{G})$ (by proof above) are finite-dimensional, it follows that $Ext^{i+1}(\mathcal{F}, \mathcal{G})$ is also finite-dimensional.