A note on mod-$p$ local-global compatibility via Scholze's functor

TL;DR

This paper extends mod-$p$ local-global compatibility results by removing the semisimple condition, showing that certain étale cohomology groups uniquely determine Galois representations under specific assumptions.

Contribution

It removes the semisimple assumption in mod-$p$ local-global compatibility, establishing a more general uniqueness result for Galois representations.

Findings

Étale cohomology determines Galois representations uniquely.

Removed the semisimple condition in previous compatibility results.

Provided remarks on assumptions and their implications.

Abstract

We remove the semisimple condition in the mod-$p$ local-global compatibility result of arXiv:2106.10674. Namely, assuming flatness of $\pi_{\mathfrak{m}}^\vee$ and $\overline{\sigma}_{\mathfrak{m}}|_{\mathrm{Gal}_{F^+_{\mathfrak{p}}}}$ being multiplicity free, we prove that $H^{n-1}_{\text{\'et}}(\mathbb{P}_{\mathbb{C}_p}^{n-1}, \mathcal{F}_{\pi_{\mathfrak{m}}[\mathfrak{m}]})$ determines $\overline{\sigma}_{\mathfrak{m}}|_{\mathrm{Gal}_{F^+_{\mathfrak{p}}}}$ uniquely. We give remarks on our assumptions at the end of this note.