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Published online by Cambridge University Press: 29 May 2025
There appeared not long ago a reduction formula for derived Hochschild cohomology, that has been useful, for example, in the study of Gorenstein maps and of rigidity with respect to 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 upgradableto “higher” contexts—like bimodules over algebras in ∞- categories. This would lead to wider applicability, for example to ring spectra; and the methodsmight 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 toextend Hom-Tensor adjunction—adjoint associativity—to the ∞- category setting.
There are substantial overlaps between algebra and homotopy theory, making for mutual enrichment—better understanding of some topics, and wider applicability of results from both areas. In this vein, works of Quillen, Neeman, Avramov– Halperin, Schwede–Shipley, Dwyer–Iyengar–Greenlees, to mention just a few, come to mind. See also [Greenlees 2007]. In recent years, homotopy theorists like May, Toën, Joyal, Lurie (again to mention just a few) have been developing a huge theory of algebra categories, dubbed by Lurie “higher algebra”, familiarity with which could be of significant benefit to (lower?) algebraists.
This little sales pitch will be illustrated here by one specific topic that arose algebraically, but can likely be illuminated by homotopical ideas.
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