A theorem conjectured by Lichtenbaum, due to Grothendieck, states the following. Let $X$ be a quasiprojective variety over a field $k$ of dimension $n$, and let $\mathcal{F}$ be a coherent sheaf on $X$. Then $H^n(X, \mathcal{F})$ is f.d. over $k$. Hartshorne (in "Local cohomology") proves this by embedding $X$ as a dense subset of a projective variety $\overline{X}$ and extending $\mathcal{F}$ to $\overline{X}$. The exact sequence $H^n(\overline{X}, \mathcal{F}) \to H^n(X, \mathcal{F}) \to H^{n+1}_{\overline{X} - X}(\overline{X}, \mathcal{F})$ shows that the middle term is f.d., because the first term is (by the proper mapping theorem), and the last vanishes (by dimensional vanishing). If I am not mistaken, this works for any separated variety over $k$ by either appealing to Nagata compactification to embed $X$ as a dense open subset of a proper variety and using the same argument. In a more elementary manner, I believe that one may "bootstrap" to arbitrary separated varieties if one uses Chow's lemma to find a quasiprojective $k$-variety $Y$ and a projective, surjective, and birational morphism $Y \to X$, and then uses the same spectral sequence argument as in EGA III.3 to get finiteness of $H^n(X, \mathcal{F})$ because we have the analogous result on $Y$.

Is there a relative form of this? Here's what I'm curious about: Let $f: X \to Y$ be a separated morphism of finite type, between noetherian schemes. Let $\mathcal{F}$ be coherent on $X$, and suppose the fibers of $f$ have dimension at most $n$. Then $R^n f_*(\mathcal{F})$ is coherent on $Y$. That is, if $X$ is a separated scheme of finite type over $Spec(A)$ for $A$ noetherian, with fibers of dimension $\leq n$, then $H^n(X, \mathcal{F})$ is a finite $A$-module for any coherent sheaf $\mathcal{F}$.

I think *most,* but maybe not all, of the above argument works (assume for starters $f$ quasiprojective -- if it's true here, I'm pretty sure we can bootstrap as above). Indeed, fix the case of $Y$ affine, equal to $Spec(A)$; then we can embed $X$ as a dense open subset of a projective $A$-scheme $\overline{X}$, to which $\mathcal{F}$ extends. Then we know that $H^n(\overline{X}, \mathcal{F})$ is finitely generated, but the problem is to see that $H^{n+1}_{\overline{X} - X}(\overline{X}, \mathcal{F})$ vanishes by some kind of dimensional argument because of the condition on the fibers.

There is a result of this kind for normal cohomology: if $g: T \to S$ is a proper morphism whose fibers have dimension at most $r$, then $R^i g_*()$ is identically zero on coherent sheaves for $i>r$. This is a direct application of the formal function theorem. However, I don't know what to do for local cohomology.

So: **Is there a local cohomology version of the formal function theorem, or the dimension-vanishing result for $R^i$ alluded to above?** In addition, is the relative form of Lichtenbaum's theorem even true?

finite typequasicoherent subsheaf $\mathcal{G}'$ of $\mathcal{G}$ that restricts to $\mathcal{F}$ on $U$ (EGA I, new ed., 6.9.7). This is what I meant by "extend $\mathcal{F}$.") – Akhil Mathew Jan 21 '11 at 4:27