  There appeared not long ago a Reduction Formula for derived Hochschild
cohomology, that has been useful e.g., in the study of Gorenstein maps and of
rigidity w.r.t. semidualizing complexes. The formula involves the relative
dualizing complex of a ring homomorphism, so brings out a connection between
Hochschild homology and Grothendieck duality. The proof, somewhat ad hoc, uses
homotopical considerations via a number of noncanonical projective and
injective resolutions of differential graded objects. Recent efforts aim at
more intrinsic approaches, hopefully upgradable to "higher" contexts--like
bimodules over algebras in infinity-categories. This would lead to wider
applicability, for example to ring spectra; and the methods might be
globalizable, revealing some homotopical generalizations of aspects of
Grothendieck duality. (The original formula has a geometric version, proved by
completely different methods coming from duality theory.) A first step is to
extend Hom-Tensor adjunction--adjoint associativity--to the infinity-category
setting.
