$p$-adic vanishing cycles as Frobenius-fixed points

TL;DR

This paper investigates $p$-adic vanishing cycles of smooth formal schemes over mixed-characteristic perfectoid fields using de Rham--Witt and $q$-de Rham complexes, revealing new insights into their structure as Frobenius-fixed points.

Contribution

It introduces a novel approach to understanding $p$-adic vanishing cycles through the lens of Frobenius-fixed points using advanced cohomological complexes.

Findings

Identification of $p$-adic vanishing cycles as Frobenius-fixed points

Application of de Rham--Witt and $q$-de Rham complexes to study these cycles

New structural insights into the nature of vanishing cycles in mixed characteristic

Abstract

Given a smooth formal scheme over the ring of integers of a mixed-characteristic perfectoid field, we study its $p$-adic vanishing cycles via de Rham--Witt and $q$-de Rham complexes.