Theory of weights for log convergent cohomologies II: the case of a proper SNCL scheme in characteristic $p>0$

TL;DR

This paper develops a theory of weights for log convergent cohomology in the case of proper SNCL schemes over a base of mixed characteristic, establishing fundamental filtered complexes and their properties.

Contribution

It constructs canonical filtered complexes in the convergent topos for SNCL schemes over mixed characteristic bases, defining a weight filtration on log convergent cohomology.

Findings

Constructed two fundamental filtered complexes in the convergent topos.

Proved the complexes are canonically isomorphic.

Established a comparison theorem with previous weight-filtered complexes.

Abstract

For a flat $p$-adic formal family $S$ of log points over a complete discrete valuation ring with perfect residue field of mixed characteristics $(0,p)$ and for a simple normal crossing log scheme $X$ over an exact closed log subscheme of $S$ defined by an element of the maximal ideal of the dvr, we construct two fundamental filtered complexes in the convergent topos the underlying scheme of $X$ over the underlying scheme of $S$. We prove that they are canonically isomorphisc. Because one of the complex is shown to calculate the log convergent cohomology sheaf of $X/S$, the filtered complex produces the weight filtration on the log convergent cohomology sheaf if the underlying scheme of $X$ is proper over the underlying scheme of $S$. We give a comparison theorem between the projection of the filtered complex in the Zariski topos of the underlying scheme of $X$ and the isozariskian weight-filtered complex constructed in the author's previous work.