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.