Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yukiyoshi Nakkajima

TL;DR

This work develops a derived PD-Hirsch extension framework for the log crystalline complex of proper SNCL schemes over a p-adic family of log points in characteristic $p>0$. It constructs a fundamental filtered log crystalline complex $H_{ m zar}(X/S)$ and a filtered log crystalline dga $H_{ m zar,TW}(X/S)$, equipped with a weight filtration $P$ designed to harmonize the cup product with the weight structure. A key result is the canonical comparison when the base is the log point of a perfect field, yielding isomorphisms with Kim and Hain's filtered complex and dga, respectively. The paper further develops pre-weight filtrations and the PD-Hirsch extensions of log crystalline complexes, culminating in a robust, functorial, and choice-independent theory that underpins $p$-adic weight filtrations, spectral sequences, and duality results in log crystalline cohomology for SNCL schemes.

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.

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 work develops a derived PD-Hirsch extension framework for the log crystalline complex of proper SNCL schemes over a p-adic family of log points in characteristic . It constructs a fundamental filtered log crystalline complex and a filtered log crystalline dga , equipped with a weight filtration designed to harmonize the cup product with the weight structure. A key result is the canonical comparison when the base is the log point of a perfect field, yielding isomorphisms with Kim and Hain's filtered complex and dga, respectively. The paper further develops pre-weight filtrations and the PD-Hirsch extensions of log crystalline complexes, culminating in a robust, functorial, and choice-independent theory that underpins -adic weight filtrations, spectral sequences, and duality results in log crystalline cohomology for SNCL schemes.

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 and a fundamental filtered complex for a simple normal crossing log scheme 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 . When the base log scheme is the log point of a perfect field of characteristic , we prove that and is canonically isomorphic to Kim and Hain's filtered dga and their filtered complex in [KH], respectively.

Paper Structure

This paper contains 21 sections, 177 theorems, 817 equations.

Table of Contents

  1. Introduction
  2. SNCL schemes
  3. PD-Hirsch extensions of dga's
  4. PD-Hirsch extensions of log crystalline complexes
  5. Preweight filtrations on PD-Hirsch extensions of log crystalline complexes
  6. PD-Hirsch pre-weight-filtered log crystalline complexes
  7. Hirsch weight-filtered log crystalline dga's
  8. Contravariant functorialities of PD-Hirsch pre-weight-filtered log crystalline complexes and Hirsch weight-filtered log crystalline dga's
  9. Filtered base change theorem of PD-Hirsch pre-weight-filtered complexes
  10. Monodromy operators of PD-Hirsch pre-weight-filtered log crystalline complexes
  11. Cup products of PD-Hirsch pre-weight-filtered log crystalline complexes
  12. Infinitesimal deformation invariance of Hirsch weight-filtered log crystalline complexes and Hirsch weight-filtered log crystalline dga's
  13. The $E_2$-degeneration of the $p$-adic weight spectral sequence
  14. Convergence of the weight filtration
  15. Strict compatibility
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\varpi_{\rm crys}^{(m)}(\overset{\circ}{X}/\overset{\circ}{S})$$(m\in {\mathbb N})$ be the crystalline orientation sheaf associated to the set $\{\overset{\circ}{X}_{\lambda}\}_{\lambda \in \Lambda}$. That is, $\varpi_{\rm crys}^{(m)}(\overset{\circ}{X}/\overset{\circ}{S})$ is the extension to with a canonical isomorphism in $D^+(f^{-1}({\cal O}_S))$ such that in $D^+(f^{-1}({\cal O}_S))$.

Theorems & Definitions (368)

  • Theorem 1.1: nb
  • Theorem 1.2
  • Theorem 1.3: Existence of the PD-Hirsch weight-filtered complex
  • Corollary 1.4
  • Theorem 1.5: cf. fup
  • Theorem 1.6: Comparison theorem of weight filtrations
  • Theorem 1.7: Comparison theorem of the filtered complexes
  • Conjecture 1.8: $p$-adic variational filtered log hard Lefschetz conjecture
  • Theorem 1.9
  • Theorem 1.10
  • ...and 358 more