Functor homology over an additive category

TL;DR

This paper explores general phenomena in functor homology over additive categories, extending key theorems to Fp-linear categories, enabling explicit computations and applications in algebraic K-theory and group cohomology.

Contribution

It generalizes the strong comparison theorem to Fp-linear additive categories, facilitating new computational techniques and applications in algebraic topology and K-theory.

Findings

Generalized the comparison theorem to Fp-linear categories

Provided explicit computations of functor homology

Established comparison theorems for algebraic group cohomology

Abstract

We uncover several general phenomenas governing functor homology over additive categories. In particular, we generalize the strong comparison theorem of Franjou Friedlander Scorichenko and Suslin to the setting of Fp-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 et van der Kallen for finite fields.