Sur la torsion de Frobenius de la cat\'egorie des modules instables

TL;DR

This paper investigates the Frobenius twist in the category of unstable modules, demonstrating its injectivity on extension groups through explicit computations and minimal injective resolutions.

Contribution

It extends the Frobenius twist to unstable modules and explicitly computes extension groups, revealing injectivity properties in this broader context.

Findings

Frobenius twist induces injective morphisms on certain extension groups

Explicit minimal injective resolution of the free unstable module F(1) is constructed

Results extend known properties from polynomial functors to unstable modules

Abstract

In the category $\mathcal{P}_{d}$ of strict polynomial functors, the morphisms between extension groups induced by the Frobenius twist are injective. In \cite{Cuo14a}, the category $\mathcal{P}_{d}$ is proved to be a full sub-category of the category $\mathcal{U}$ of unstable modules \textit{via} Hai's functor. The Frobenius twist is extended to the category $\mathcal{U}$ but remains mysterious there. This article aims to study the Frobenius twist and its effects on the extension groups of unstable modules. We compute explicitly several extension groups and show that in these cases, the morphisms induced by the Frobenius twist are injective. These results are obtained by constructing the minimal injective resolution of the free unstable module $F(1)$.