Prismatic cohomology and p-adic homotopy theory

Historically, it was known by the work of Artin and Mazur that the \(\ell \)-adic homotopy type of a smooth complex variety with good reduction mod p can be recovered from the reduction mod p, where \(\ell \) is not p. This short note removes this last constraint, with 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 étale comparison theorem in the prismatic theory reproduces the p-adic homotopy type for a smooth complex variety with good reduction mod p.