Cohomology groups interpreted as sheafs

2012-07-08T08:57:20Z

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

Answer by Filippo Alberto Edoardo for Cohomology groups interpreted as sheafs

2012-07-08T10:22:30Z

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.

Answer by Daniel Sommerhoff for Cohomology groups interpreted as sheafs

2012-07-08T11:46:19Z

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 - MathOverflow

Cohomology groups interpreted as sheafs

Answer by Filippo Alberto Edoardo for Cohomology groups interpreted as sheafs

Answer by Daniel Sommerhoff for Cohomology groups interpreted as sheafs