I'm trying to solve the exercise problem iii.7.4 in Hartshorne's AG book. It says that
"first chern class of the line bindle associated to a smooth subvarirety of codimen 1" = "cohomology class of that subvariety"
Suddenly it comes to me that I have NO! ideas about what Serre dualty looks like! There are no differential forms and integration. Everything is just so abstract, very hard to manipulate. Moreover, there are no exponential sequence, which makes the problem much easier in complex analytic case.
So my question is, is there any "explicit" formulation of the Serre duality? Of course, over arbitary (algebraically closed) field.