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?
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.