Duality for Cousin Complexes

We relate the variance theory for Cousin complexes − developed by Lipman, Nayak and the author to Grothendieck duality for Cousin complexes. Specifically for a Cousin complex F on (Y, ∆)—with ∆ a codimension function on a formal scheme Y (noetherian, universally catenary)—and a pseudo-finite type map f : (X, ∆) → (Y, ∆) of such pairs of schemes with codimension functions, we show there is a derived category map γ! f (F) : f ♯F → f !F , functorial in F ∈ Coz∆(Y), inducing a functorial isomorphism f ♯F ≃ E(f♯F) −→ ∼ E(f !F) (where E is the Cousin functor on (X, ∆′)). The map γ! f (F) is itself an isomorphism if (and clearly only if) f !F is Cohen-Macaulay on (X, ∆′)—which will be so, for example, whenever the complex F is injective or whenever the map f is flat. For a fixed Cousin complex F on (Y, ∆), γ! f (F) is an isomorphism for every map f with target (Y, ∆) if and only if F is a complex of (appropriate) injectives. For a fixed map f , the functorial map γ! f is an isomorphism of functors if and only if f is flat. We also generalize the Residue Theorem of Grothendieck for residual complexes to Cousin complexes by defining a functorial Trace Map of graded OY–modules Trf (F) : f∗f F → F (a sum of local residues) such that when f is pseudo-proper, Trf (F) is a map of complexes and the pair (f ♯F , Trf (F)) represents the functor Hom(f∗G, F) of Cousin complexes G on (X, ∆′).