Residues and duality for Cousin complexes

Abstract

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 \gamma: f^# M -> f^! M inducing a functorial isomorphism in the category of Cousin complexes f^# M \iso E(f^! M) (where E is the Cousin functor). \gamma itself is an isomorphism if the complex f^! M is Cohen-Macaulay--which will be so whenever the map f is flat 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 the pseudofunctor ^# generalises--and makes canonical--the "variance theory" of residual complexes developed in Chapter VI of Hartshorne's "Residues and Duality." Moreover, we generalise the Residue Theorem of loc.cit., p.369 by defining a functorial Trace map of graded modules Tr_f(M): f_*f^# M -> M (a sum of local residues) such that whenever f is proper, Tr_f(M) is a map of complexes and the pair (f^# M, Tr_f(M)) represents the functor Hom(f_*C, M) of Cousin complexes C.