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)