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

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 $H_{\text{eˊt}}^{n-1}({\mathbb{P}}_{{\mathbb{C}}_p}^{n-1},{\mathcal{F}}_{{\pi }_{\mathfrak{m}}[\mathfrak{m}]})$ determines ${\overline{\sigma }}_{\mathfrak{m}}{|}_{{\mathrm{Gal}}_{F_{\mathfrak{p}}^{+}}}$ uniquely. We give remarks on our assumptions at the end of this note.

arXiv comment8 pages, comments welcome