Residues and Duality for Cousin Complexes

We construct a canonical pseudofunctor ( ) on the category of finite-type maps of (say) connected noetherian universally catenary finite-dimensional separated schemes, taking values in the category of Cousin complexes. This pseudofunctor is a concrete approximation to the restriction of the Grothendieck Duality pseudofunctor( ) to the full subcategory of the derived category having Cohen-Macaulay complexes as objects (a subcategory equivalent to the category of Cousin complexes, once a codimension function has been fixed). Specifically, for Cousin complexes M and any scheme map f : X → Y as above, there is a functorial derived-category map γ f : f M → f M inducing a functorial isomorphism in the category of Cousin complexes f M ∼= E(f M) ∼ −→ E(f M) (where E is the Cousin functor of [3], p. 241). The map γ f is itself an isomorphism if the complex f M is Cohen-Macaulay—which will be so whenever the map f is Cohen-Macaulay or whenever the complex M is injective. Also, f ♯ takes residual (resp. injective) complexes on Y to residual (resp. injective) complexes on X; and so ( ) generalises—and makes canonical—the “variance theory” of residual complexes developed in [3], Chap.VI. Moreover, we generalise the Residue Theorem of [3], p. 369 by defining a functorial Trace map of graded modules Trf (M ) : f∗f M → M (a sum of local residues) such that when f is proper, Trf (M ) is a map of complexes and the pair (f M, T rf (M )) represents the functor Hom(f∗C ,M) of Cousin complexes C.