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Let $X$ be a scheme (or complex manifold if you like) and $E$ be a sheaf on $X$. I would like to know the definition of so-called trace map $Ext^i(E,E)\rightarrow H^i(X,O_X)$ for $1\le i\le \dim X$.

Edit I am also interested in noncommutative case; is there any traca map which looks like $Ext_R^i(E,E)\rightarrow ?,$ where $R$ is a noncommutative algebra and $E$ is a noetherian right $R$-module.

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Proceeding formally, we have $Ext^i_X(E,E)\cong Ext^i_{X\times X}(\mathcal O_\Delta,\hom(E,E))$, where $\Delta\subset X\times X$ is the diagonal. Now there is a trace map $\hom(E,E)\to\mathcal O_X$ (if $E$ is finitely generated projective over affines, for example), and this map is $\mathcal O_\Delta$ linear, so induces a map $Ext^i_{X\times X}(\mathcal O_\Delta,\hom(E,E))\to Ext^i_{X\times X}(\mathcal O_\Delta,\mathcal O_\Delta)=H^i(X,\mathcal O_X)$.

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    Well, determinant of module over noncommutative ring does not make much sense, but I want to know if there is any good interpretation.2012-11-04