In a 2008 paper, Avramov and Iyengar studied Cohen-Macaulay and Gorenstein morphisms and characterized them in terms of invariants from Hochschild(co)homology.
The invariants in question turned out to be related to Grothendieck duality, and in a 2010 paper Avramov, Iyengar, Lipman and Nayak pursued the ideas to prove reduction formulas for the Hochschild (co)homology groups in question. In this talk, we will discuss a new approach to the problem, based on Brown representability.