Cohomology groups interpreted as sheafs - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:34:43Z http://mathoverflow.net/feeds/question/101630 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101630/cohomology-groups-interpreted-as-sheafs Cohomology groups interpreted as sheafs Steven Gro 2012-07-08T08:57:20Z 2012-07-08T11:46:19Z <p>Hi Folks,</p> <p>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?</p> <p>Thanks</p> <p>Steven</p> http://mathoverflow.net/questions/101630/cohomology-groups-interpreted-as-sheafs/101636#101636 Answer by Filippo Alberto Edoardo for Cohomology groups interpreted as sheafs Filippo Alberto Edoardo 2012-07-08T10:22:30Z 2012-07-08T11:39:58Z <p>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 <code>$f_* \mathcal{F}$</code>. 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 <code>$f_*$</code> (the reason for my quotes is that the first functor takes values in <strong>Ab</strong> while the second takes values in <strong>Sh($\mathrm{Spec}(k)$)</strong> but you can figure out the point, I guess).</p> <p>Then, in general, given any map of schemes <code>$f:X\to Y$</code> you can define for any sheaf $\mathcal{F}$ on $X$ its direct image $f_* \mathcal{F}$ getting a functor from <strong>Sh($X$)</strong> to <strong>Sh($Y$)</strong> who is left exact. Its right derived functors $R^if_*$ now produce sheaves on $Y$ and the <code>$R^if_*\mathcal{F}$</code> 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 <em>Homological Algebra</em>. 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.</p> http://mathoverflow.net/questions/101630/cohomology-groups-interpreted-as-sheafs/101646#101646 Answer by Daniel Sommerhoff for Cohomology groups interpreted as sheafs Daniel Sommerhoff 2012-07-08T11:46:19Z 2012-07-08T11:46:19Z <p>Just as an addition:</p> <p>In many settings you can think about the higher direct image of sheaves as the $\mathcal{O}_X$-module associated to the cohomology group.</p> <p>Proposition 8.5 Hartshorne:</p> <p>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: $R^i f_{*}( \mathcal{F}) \cong H^ i(X,\mathcal{F})^{\sim}$ </p>