Theory of weights for log convergent cohomologies I: the case of a proper smooth scheme with an SNCD in characteristic p>0

TL;DR

This paper develops a theory of weights for log convergent cohomologies of proper smooth schemes with SNCD in characteristic p>0, establishing fundamental filtered complexes, p-adic purity, and comparison theorems.

Contribution

It introduces two fundamental filtered complexes in log convergent cohomology, proves p-adic purity, and establishes canonical isomorphisms and comparison theorems for these complexes.

Findings

Defined fundamental filtered complexes $(E_{conv},P)$ and $(C_{conv},P)$.

Proved p-adic purity for the log scheme.

Established canonical isomorphisms and comparison theorems.

Abstract

Using log convergent topoi, %In the derived category of filtered complexes of %sheaves of modules over %an isostructure we define two fundamental filtered complexes $(E_{conv},P)$ and $(C_{conv},P)$ for the log scheme obtained by a smooth scheme with a relative simple normal crossing divisor over a scheme of characteristic $p>0$. Using $(C_{conv},P)$, we prove the $p$-adic purity. As a corollary of it, we prove that $(E_{conv},P)$ and $(C_{conv},P)$ are canonically isomorphic. These filtered complexes produce the weight spectral sequence of the log convergent cohomology sheaf of the log scheme. We also give the comparison theorem between the projections of $(E_{conv},P)$ and $(C_{conv},P)$ to the derived category of bounded below filtered complexes of sheaves of modules in the Zariski topos of the log scheme and the weight-filtered isozariskian filtered complex $(E_{zar},P)_{Q}$ of the log scheme defined in our previous book.