Euler Characteristic of a Variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:05:29Z http://mathoverflow.net/feeds/question/70303 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70303/euler-characteristic-of-a-variety Euler Characteristic of a Variety Jesko Hüttenhain 2011-07-14T08:31:12Z 2011-07-14T08:31:12Z <p>Let $Y$ be a &quot;nice&quot; scheme. I am thinking projective varieties over an algebraically closed field, for now, but I am open to more general results. </p> <p>In terms of singular homology (coefficients in $\mathbb{Z}$), I can define the Euler characteristic $\chi(Y)$. My question is: Can I express $\chi(Y)$ in terms of the Euler characteristic of certain coherent sheaves on $Y$, in terms of sheaf cohomology? Most preferably, I would like $\chi(Y)=\chi(Y,\mathcal{F})$ for some particular sheaf $\mathcal{F}$. </p> <p>I am sorry if this is really trivial or widely known, my searching and asking (in the real world) has led me nowhere so far.</p>