Timeline for Euler Characteristic of simple sheaves

6 events

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Nov 25, 2015 at 15:09	comment	added	Mohan		$M=K^1$ only if the point $x$ is $K$-rational. If $K$ is algebraically closed, all closed points are $K$-rational. $M$ is simple as an $\mathcal{O}_X$ module is not the same as being simple as a $K$-vector space.
Nov 25, 2015 at 5:25	review	Close votes
Nov 25, 2015 at 22:08
Nov 25, 2015 at 1:56	comment	added	Francisco Portillo		Thank you for your help. I believe I am starting to understand: the sheaf $\mathcal{F}$ in my context must be a skycraper sheaf of $X$ at a point $x$ determined by a simple module $M$. In such a case, it is known for skycraper sheaves that $H^1(X,\mathcal{F})=0$. But, since $X$ is projective $\mathcal{O}_X=K$, hence $M$ is a $K$ vector space, therefore $M\cong K^1$ because $M$ is simple. But, $H^0(X,\mathcal{F})=M$, hence $dim_K (H^0(X,\mathcal{F}))=1$. Is this correct? I am not very sure!
Nov 24, 2015 at 18:03	comment	added	Mohan		The only simple sheaves with your definition are skyscraper sheaves $K(x)$ for a closed point $x\in X$. Thus $\chi(\mathcal{F})$ for such a sheaf would be $[K(x):K]$. If you want it to be one, you must assume that $K$ is algebraically closed.
Nov 24, 2015 at 17:54	review	First posts
Nov 24, 2015 at 17:57
Nov 24, 2015 at 17:51	history	asked	Francisco Portillo	CC BY-SA 3.0