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I have read that for a locally free sheaf $F$ on a scheme $X$ one has a canonical trace map

$\underline{Hom}_{O_X}(F,F)\rightarrow O_X$

where on the left side I mean the inner hom.

Can someone explain how I get this map?

Thanks a lot!

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1 Answer 1

There's a canonical section of Hom(F,F) given by the identity map locally. This corresponds to a morphism $O_X \to Hom(F,F) = F^* \otimes F$. Now dualize and use that $F^{**} = F$ since you assumed $F$ is locally free.

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