User daniel sommerhoff - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T09:24:40Z http://mathoverflow.net/feeds/user/24980 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111821/algebraic-closure-and-git/111835#111835 Answer by Daniel Sommerhoff for Algebraic closure and GIT Daniel Sommerhoff 2012-11-08T17:02:59Z 2012-11-08T17:02:59Z <p>Hey MBeasy,</p> <p>you can construct the quotients over an arbitrary "not necessarily algebraically closed field", as Mumford states at the beginning of chapter 1.</p> <p>Greetings</p> <p>Daniel</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>