I've never understood how one would actually go about computing a trace map associated with the canonical sheaf on a smooth projective variety, if it's even possible. Hartshorne proves that the canonical sheaf represents the relevant functor through some very non-explicit homological algebra gymnastics. The name of the map seems to suggest that I ought to be able to be able to write down some kind of matrix, hopefully in terms of homogeneous coordinates, and take its trace. Is this true (if not, why is it called a trace map)? How does one do this?

(I also posted this here: https://math.stackexchange.com/questions/3400374/can-one-determine-the-trace-map-for-a-nonsingular-projective-variety-explicitly)

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    $\begingroup$ Just to be clear, you're referring to the canonical isomorphism $H^n(X, \omega_X)\to k$, where $n$ is the dimension of $X$? If $k=\mathbb{C}$, this is just integration of top forms, as you probably know... $\endgroup$ Commented Oct 19, 2019 at 18:24
  • $\begingroup$ Yes, exactly that. $\endgroup$ Commented Oct 19, 2019 at 18:33
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    $\begingroup$ The only situation where I know how to do this over general fields (not $\mathbb{C}$) is the case $X$ is a curve, where indeed this is given by some kind of (infinite-dimensional) trace, as explained by Tate here: numdam.org/article/ASENS_1968_4_1_1_149_0.pdf $\endgroup$ Commented Oct 19, 2019 at 18:36

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This is a good question. It's one that my colleague Joe Lipman spent a lot of time thinking about. You can look at some of his papers for a more explicit answer for computing the trace. Probably you should start with his book "Dualizing sheaves, differentials and residues on algebraic varieties." As for name, the Grothendieck trace really is a classical trace in some cases, e.g. for finite flat maps. I don't know if that's the reason it's called that, however.

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    $\begingroup$ Great. The idea is that the trace is explicit in the case of projective space via the Euler sequence, and for a protective variety you have to use a projective Noether normalization and the trace of the corresponding finite morphism followed by the one for projective space does the trick. Everything is explained in Lipman's book. Also, the general proper case is treated there. $\endgroup$
    – Leo Alonso
    Commented Oct 19, 2019 at 21:35

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