Welcome to MathOverflow

Question

0

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?

Thanks

Steven

flag	asked Jul 8 at 8:57 Steven Gro 3●1

	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 at 9:22
	Hey Dan, thanks for your answer. To specify my question: The text I am reading is: math.utah.edu/~bertram/courses/hilbert/ps/… 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 at 10:10

Filippo Alberto Edoardo · Accepted Answer · 2012-07-08 11:39:58Z UTC

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.

Daniel Sommerhoff · Answer 2 · 2012-07-08 11:46:19Z UTC

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: $R^i f_{*}( \mathcal{F}) \cong H^ i(X,\mathcal{F})^{\sim}$

Cohomology groups interpreted as sheafs

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.