Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.