Logarithmic de Rham comparison for open rigid spaces

TL;DR

This paper establishes a logarithmic p-adic comparison theorem for open rigid analytic varieties with normal crossing divisors, extending p-adic Hodge theory to this setting using Scholze's methods.

Contribution

It introduces a pro-version of the Faltings site for open rigid spaces and proves a primitive comparison theorem in this context.

Findings

Proves the logarithmic p-adic comparison theorem for open rigid spaces.

Shows that smooth rigid varieties with normal crossing divisors are locally K(π,1).

Develops period sheaves and establishes a primitive comparison theorem.

Abstract

In this note, we prove the logarithmic $p$-adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally $K(\pi,1)$ (in a certain sense) with respect to $\mathbb{F}_p$-local systems and ramified coverings along the divisor. We follow Scholze's method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.