The variety of subadditive functions for finite group schemes

TL;DR

This paper explores the structure of subadditive functions on finite group schemes and demonstrates how the cohomology variety can be reconstructed from these functions, linking them to module theory and pi-points.

Contribution

It establishes a connection between subadditive functions, endofinite modules, and pi-points, providing a new perspective on the cohomology variety of finite group schemes.

Findings

Cohomology variety can be recovered from subadditive functions.

An equivalence relation on point modules aligns with pi-points.

Subadditive functions relate to endofinite modules via Crawley-Boevey's correspondence.

Abstract

For a finite group scheme, the subadditive functions on finite dimensional representations are studied. It is shown that the projective variety of the cohomology ring can be recovered from the equivalence classes of subadditive functions. Using Crawley-Boevey's correspondence between subadditive functions and endofinite modules, we obtain an equivalence relation on the set of point modules introduced in our joint work with Iyengar and Pevtsova. This corresponds to the equivalence relation on pi-points introduced by Friedlander and Pevtsova.