The categorical form of Fargues' conjecture for tori

TL;DR

This paper proves Fargues' conjecture for tori with integral coefficients, demonstrating t-exactness and compatibility of the spectral action with the excursion algebra, while developing a new approach to condensed group (co)homology.

Contribution

It establishes the conjecture for tori with integral coefficients, introduces a non-solidified condensed group (co)homology, and shows spectral action compatibility.

Findings

Proved Fargues' conjecture for tori with integral coefficients.

Demonstrated t-exactness of the conjecture.

Established compatibility of spectral action with the excursion algebra.

Abstract

We prove the main conjecture of arXiv:2102.13459 for integral coefficients in the case of tori and prove that it is t-exact. Along the way we prove that the spectral action as constructed in that manuscript is compatible with the action of the excursion algebra and preserves the grading by $\pi_1(G)_Q$ on both sides. We additionally develop a (non-solidified) version of condensed group (co)homology and show that many constructions from classical group (co)homology extend to that case.