Hirsch weight-filtered log crystalline complex and Hirsch weight-filtered log crystalline dga of a proper SNCL scheme in characteristic p>0

TL;DR

This paper develops a new theory of filtered differential graded algebras and complexes for log crystalline schemes in characteristic p, resolving compatibility issues with existing p-adic Steenbrink complexes.

Contribution

It introduces the derived PD-Hirsch extension and constructs fundamental filtered dga and complexes for log schemes, linking them to Kim and Hain's structures in characteristic p.

Findings

Constructed fundamental filtered dga and complex for log schemes.

Proved isomorphism with Kim and Hain's filtered structures in characteristic p.

Overcame obstacles related to cup product incompatibility in log crystalline complexes.

Abstract

We construct a theory of the derived PD-Hirsch extension of the log crystalline complex of a log smooth scheme and we construct a fundamental filtered dga $(H_{{\rm zar},{\rm TW}},P)$ and a fundamental filtered complex $(H_{\rm zar},P)$ for a simple normal crossing log scheme $X$ over a family of log points by using the log crystalline method in order to overcome obstacles arising from the incompatibility of the p-adic Steenbrink complexes in [M] and [Nak4] with the cup product of the log crystalline complex of $X$. When the base log scheme is the log point of a perfect field of characteristic $p>0$, we prove that $(H_{{\rm zar},{\rm TW}},P)$ and $(H_{\rm zar},P)$ is canonically isomorphic to Kim and Hain's filtered dga and their filtered complex in [KH], respectively.