Duality for Cousin Complexes

Abstract

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, \Delta)--with \Delta a codimension function on a formal scheme Y (noetherian, universally catenary)--and a pseudo-finite type map f:(X,\Delta') --> (Y,\Delta) of such pairs of schemes with codimension functions, we show there is a derived category map \gamma^!_f(F):f^#F --> f^!F, which is functorial as F varies over Cousin complexes on (Y,\Delta), and induces an isomorphism f^#F = E(f^#F) --> E(f^!F). E here is the Cousin functor for the codimension function \Delta. Further, we give conditions under which \gamma^!_f is an isomorphism. We also generalize the Residue Theorem of Grothendieck for residual complexes to Cousin complexes by defining trace as a sum of local residues when the map f is pseudo-proper.