  Grothendieck duality theory assigns to essentially-finite-type maps f of
noetherian schemes a pseudofunctor f^\times right-adjoint to Rf_*, and a
pseudofunctor f^! agreeing with f^\times when f is proper, but equal to the
usual inverse image f^* when f is etale. We define and study a canonical map
from the first pseudofunctor to the second. This map behaves well with respect
to flat base change, and is taken to an isomorphism by "compactly supported"
versions of standard derived functors. Concrete realizations are described, for
instance for maps of affine schemes. Applications include proofs of reduction
theorems for Hochschild homology and cohomology, and of a remarkable formula
for the fundamental class of a flat map of affine schemes.
