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?

  • $\begingroup$ Just a little comment: if you are quasi-projective over an affine scheme, then by using the fact that appropriate twists of coherent sheaves by an ample line bundle are generated by global sections, you are probably reduced to showing the result for twists of a particular ample line bundle. The reason I say this is because in your original argument, you say 'extend $\mathcal{F}$ to $\bar{X}$', but it's not clear to me what such an extension is or why it should be coherent. But, if $\mathcal{F}$ is the restriction of an ample line bundle on $\bar{X}$, then there is no issue. $\endgroup$ – Keerthi Madapusi Pera Jan 21 '11 at 4:12
  • $\begingroup$ Dear Keerthi: I agree with you that it is enough to work with ample line bundles. (However, if $X$ is a quasicompact and quasiseparated scheme, $U \subset X$ a quasicompact open subset, and $\mathcal{F}$ a quasicoherent subsheaf of finite type of $\mathcal{G}|_U$ (for $\mathcal{G}$ some q-c sheaf on $X$; for example the push-forward of $\mathcal{F}$), then there is a finite type quasicoherent 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 $\mathcal{F}$.") $\endgroup$ – Akhil Mathew Jan 21 '11 at 4:27
  • $\begingroup$ Dear Akhil: Thanks for the clarification, and the reference. $\endgroup$ – Keerthi Madapusi Pera Jan 21 '11 at 7:13
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    $\begingroup$ Dear Akhil: the question in the second paragraph has a negative answer, even for open immersions. Let (R,m) be a noetherian local ring of dimension d > 1, let Y = Spec(R), and let X = Y - {m}. The map X -> Y is an open immersion, and so the fibre dimension is at most n for any n >= 0. However, the R-module H^{d-1}(U,O_U) =~ H^d_m(R) is never finitely generated: if R is Gorenstein, this module is the injective hull R/m over R. Also, the problem here is local at {m}, so you can spread out Y (and X) to obtain examples where X and Y are f.t over a field. $\endgroup$ – B.B. Jun 15 '11 at 19:45
  • $\begingroup$ Dear Bhargav: Thanks. I would be happy to accept that as an answer. $\endgroup$ – Akhil Mathew Jun 16 '11 at 1:46

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