Prismatic cohomology and $p$-adic homotopy theory

TL;DR

This paper observes that the prismatic cohomology framework, combined with Mandell's functor, reproduces the p-adic homotopy type for certain complex varieties, linking prismatic theory with p-adic homotopy theory.

Contribution

It demonstrates how the étale comparison theorem in prismatic cohomology recovers the p-adic homotopy type for smooth proper varieties with good reduction.

Findings

Reproduces p-adic homotopy type using prismatic cohomology

Connects prismatic cohomology with p-adic homotopy theory

Shows the étale comparison theorem's role in this reproduction

Abstract

This short note regards an observation about the recent theory of prismatic cohomology developed by Bhatt and Scholze. In particular, by applying a functor of Mandell, we see that the \'etale comparison theorem in the prismatic theory reproduces the $p$-adic homotopy type for a smooth proper complex variety with good reduction mod $p$.