Seminar

There appeared not long ago a Reduction Formula for derived Hochschild (co)homology, 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 map, and so brings out a connection between Hochschild homology and Grothendieck duality. The proof, somewhat ad hoc, uses homotopical considerations via numerous non canonical resolutions, both projective and injective, of differential graded objects. Recent efforts are producing more intrinsic approaches, which one hopes to upgrade to "higher" contexts, for example 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 Grothendieck duality. (The original formula has a geometric version, proved by completely different methods coming from duality theory.)

After reviewing the formula, I will just try to explain a few elementary things about infinity categories, and how one can express therein the basic relation (adjoint associativity) between Hom and tensor products that underlies any proof.