Homology of strict polynomial functors over ${\mathbb{F}}_p$-linear additive categories

Abstract

We generalize the strong comparison theorem of Franjou, Friedlander, Scorichenko, and Suslin to the setting of ${\mathbb{F}}_p$-linear additive categories. Our results have a strong impact in terms of explicit computations of functor homology, and they open the way to new applications to stable homology of groups or to $K$-theory. As an illustration, we prove comparison theorems between cohomologies of classical algebraic groups over infinite perfect fields, in the spirit of a celebrated result of Cline, Parshall, Scott, and van der Kallen for finite fields.