Traces and residues

Let f : X → Y be a separated morphism of Noetherian schemes, and let W ⊂ X be a union of closed subsets such that the restriction of f to each of them is proper. In duality theory, one considers trace maps Rf∗R⌈wf! !OY → OY. In a recent paper, we gave a new construction of such a trace map, using a certain natural transformation ψ(f) : fx -→ f!. In this note, we show how to compute it. In duality theory, there are abstract, functorial definitions, and there are computationally useful formulas, but they are rarely the same. This makes the new approach remarkable.