Crystalline comparison of $\mathrm{A}_\mathrm{inf}$-cohomology

Abstract: A major goal of $p$-adic Hodge theory is to relate arithmetic structures coming from various cohomology theories of $p$-adic varieties. Such comparisons are usually achieved by constructing intermediate cohomology theories. A somewhat recent successful theory, namely the $\mathrm{A}_\mathrm{inf}$-cohomology, has been invented by Bhatt--Morrow--Scholze, originally via perfectoid spaces. In this talk, I will describe a simpler approach to prove the comparison between $\mathrm{A}_\mathrm{inf}$-cohomology and the crystalline cohomology over Fontaine's period ring $\mathrm{A}_\mathrm{cris}$, using flat descent of cotangent complexes. This approach also allows us to prove compatibilities of certain $p$-adic filtrations. Time permitting, I will discuss some work in progress (partially joint with Hansheng Diao) in the semistable/logarithmic case.

number theory

Audience: researchers in the topic

UCLA Number Theory Seminar