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Hi Folks,

I just came across a few lines where a (sheaf-)cohomology group of a scheme is treated as a sheaf. I've never seen this in Hartshorne. Could you give any reference for this?



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I think this question is too vague to get a good answer; there are several situations where one can think of a cohomology group as a sheaf in some sense. But take a look at Dimca's book "Sheaves in topology" and I bet your question will be answered. – Dan Petersen Jul 8 '12 at 9:22
Hey Dan, thanks for your answer. To specify my question: The text I am reading is: On page 6 Bertram is proving the existence of the hilbert scheme and defines a grassmannian $G(P'(d_0),H^0 (\mathbb{P}^{m}_{A}, \mathcal{O}^{n}_{\mathbb{P}^{m}_{A}} (l + d_0)))$. I think that he is using $H^0 (\mathbb{P}^{m}_{A}, \mathcal{O}^{n}_{\mathbb{P}^{m}_{A}} (l + d_0)))$ as a sheaf, otherwise this notation wouldn't fit his definition of the grassmannian from the beginning of the document. Thanks in advance to all of you! – Steven Gro Jul 8 '12 at 10:10
up vote 3 down vote accepted

I guess you have seen sheaf cohomology as being the right derived functor of the global section functor, taking a sheaf $\mathcal{F}$ on a space $X$ to the abelian group $\Gamma(X,\mathcal{F})$. Suppose $X$ is a $k$-scheme, where $k$ is any field, with structural morphism $f:X\to\mathrm{Spec}(k)$. Then you can consider, on $\mathrm{Spec}(k)$, the sheaf $f_* \mathcal{F}$. Since sheaves on the spectrum of a field are not terribly sexy, you see that this guy is defined by its global sections, which by definition coincide with global sections of $\mathcal{F}$ over $X$: in other words, the functor $\Gamma(X,-)$ "coincides" with the functor $f_*$ (the reason for my quotes is that the first functor takes values in Ab while the second takes values in Sh($\mathrm{Spec}(k)$) but you can figure out the point, I guess).

Then, in general, given any map of schemes $f:X\to Y$ you can define for any sheaf $\mathcal{F}$ on $X$ its direct image $f_* \mathcal{F}$ getting a functor from Sh($X$) to Sh($Y$) who is left exact. Its right derived functors $R^if_*$ now produce sheaves on $Y$ and the $R^if_*\mathcal{F}$ can be thought of as the relative cohomology of $\mathcal{F}$, precisely as before. This is indeed done in Hartshorne, see Section 8 of chapter $III$ and self-references therein. You can also find something on this point of view in Weibel's Homological Algebra. Note that what I have said above does not need $f$ to really be a map between schemes, it works in a more general setting once you have a formalism taking "sheaves over somebody to sheaves over somebody else" – and this is the starting point of many cohomology theories you might encounter, like étale cohomology.

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Thanks for your wonderful answer. Just a small correction: It's Section 8 :-) – Steven Gro Jul 8 '12 at 11:16
Oh yes, right... I corrected it! – Filippo Alberto Edoardo Jul 8 '12 at 11:39

Just as an addition:

In many settings you can think about the higher direct image of sheaves as the $\mathcal{O}_X$-module associated to the cohomology group.

Proposition 8.5 Hartshorne:

Let $X$ be a noetherian scheme, and let $f:X \rightarrow Y$ be a morphism of $X$ to an affine Scheme $Y=Spec\; A$. For any quasi-coherent sheaf $\mathcal{F}$ on $X$ we have: $R^i f_{*}( \mathcal{F}) \cong H^ i(X,\mathcal{F})^{\sim}$

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