Ambidexterity in stable infinity categories
If G is a finite group and M is a G-module, there is a norm map from the homology of G with coefficients in M to the cohomology. This map arises from a morphism in the derived category from the derived co-invariants to the derived invariants of G.
The resulting map is always an isomorphism over the rational numbers but rarely an isomorphism in mod p representation. In stable homotopy theory, there are many "intermediate" characteristics (p.n) associated with the so called "Morava K-theories". It turns out that the norm map is an isomorphism in all those intermediate characteristics and a vast generalization to this fact was discovered by Hopkins and Lurie. They call this generalization Ambidexterity. In my talk I will explain the notion of ambidexterity in stable infinity categories, present Hopkins and Lurie's result of the ambidexterity in characteristic (p,n) and discuss a recent work on the subject by Tomer Schlank, Lior Yanovski and myself.