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

Yukiyoshi Nakkajima, Atsushi Shiho

TL;DR

The paper develops a comprehensive log convergent cohomology framework for a proper smooth scheme with a relative SNCD in characteristic $p>0$, defining filtered complexes $(E_{ m conv},P)$ and $(C_{ m conv},P)$ within the log convergent topos. It proves a $p$-adic purity result for the weight-filtered convergent complex and shows a canonical isomorphism between $(E_{ m conv},P)$ and $(C_{ m conv},P)$, from which a weight spectral sequence for the log convergent cohomology arises. The authors establish functoriality and base-change properties, construct a weight-filtered theory with a Gysin boundary mechanism, and prove comparison theorems with zariskian and isozariskian frameworks, linking convergent, crystalline, and zariskian viewpoints. These results extend weight filtration and purity phenomena to log-convergent cohomology in positive characteristic, enabling refined structural and functorial analyses, including truncated and simplicial settings.

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.

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

TL;DR

The paper develops a comprehensive log convergent cohomology framework for a proper smooth scheme with a relative SNCD in characteristic , defining filtered complexes and within the log convergent topos. It proves a -adic purity result for the weight-filtered convergent complex and shows a canonical isomorphism between and , from which a weight spectral sequence for the log convergent cohomology arises. The authors establish functoriality and base-change properties, construct a weight-filtered theory with a Gysin boundary mechanism, and prove comparison theorems with zariskian and isozariskian frameworks, linking convergent, crystalline, and zariskian viewpoints. These results extend weight filtration and purity phenomena to log-convergent cohomology in positive characteristic, enabling refined structural and functorial analyses, including truncated and simplicial settings.

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 and for the log scheme obtained by a smooth scheme with a relative simple normal crossing divisor over a scheme of characteristic . Using , we prove the -adic purity. As a corollary of it, we prove that and 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 and 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 of the log scheme defined in our previous book.
Paper Structure (12 sections, 89 theorems, 442 equations)

This paper contains 12 sections, 89 theorems, 442 equations.

Table of Contents

  1. Introduction
  2. Log convergent topoi and the systems of log universal enlargements
  3. Log convergent linearization functors. I
  4. Log convergent linearization functors. II
  5. Vanishing cycle sheaves in log convergent topoi
  6. Log convergent linearization functors of smooth schemes with relative SNCD's
  7. (Bi)simplicial log convergent topoi
  8. Weight-filtered convergent complexes and $p$-adic purity
  9. The functoriality of the weight-filtered convergent and isozariskian complexes
  10. Boundary morphisms
  11. Comparison theorem
  12. Log convergent cohomologies with compact supports

Key Result

Theorem 1.1

There exists a canonical isomorphism where $\varpi^{(k)}_{\rm conv}(D/S)$ is the convergent orientation sheaf which will be defined in § sec:lcs below.

Theorems & Definitions (185)

  • Theorem 1.1: $p$-adic purity
  • Theorem 1.2: Comparison theorem between $(E_{\rm conv},P)$ and $(C_{\rm conv},P)$
  • Theorem 1.3: Comparison theorem between $(C_{\rm isozar},P)$ and $(C_{\rm zar},P)$
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • ...and 175 more