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Logarithmic A$_{\rm inf}$-cohomology

Hansheng Diao, Zijian Yao

Abstract

We extend the construction of A$_{\rm inf}$-cohomology by Bhatt-Morrow-Scholze to the context of log $p$-adic formal schemes over a log perfectoid base. In particular, using coordinates, we prove comparison theorems between log A$_{\rm inf}$-cohomology with other $p$-adic cohomology theories, including log de Rham, log (q-)crystalline, log prismatic, and Kummer étale cohomology, as well as the derived A$_{\rm inf}$-cohomology of certain infinite root stacks. Along the way, we define and give a combinatorial characterization of a new class of maps between saturated log schemes, called pseudo-saturated maps, which is of independent interest. They are related to (and slightly weaker than) the notion of quasi-saturated maps and maps of Cartier type studied by Tsuji.

Logarithmic A$_{\rm inf}$-cohomology

Abstract

We extend the construction of A-cohomology by Bhatt-Morrow-Scholze to the context of log -adic formal schemes over a log perfectoid base. In particular, using coordinates, we prove comparison theorems between log A-cohomology with other -adic cohomology theories, including log de Rham, log (q-)crystalline, log prismatic, and Kummer étale cohomology, as well as the derived A-cohomology of certain infinite root stacks. Along the way, we define and give a combinatorial characterization of a new class of maps between saturated log schemes, called pseudo-saturated maps, which is of independent interest. They are related to (and slightly weaker than) the notion of quasi-saturated maps and maps of Cartier type studied by Tsuji.
Paper Structure (61 sections, 134 theorems, 660 equations, 4 figures)

This paper contains 61 sections, 134 theorems, 660 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fontaine's period ring $\mathop{\mathrm{\textup{A}_{\textup{inf}}}}\nolimits$
  4. The décalage operator $L\eta$
  5. Conventions on monoids
  6. Pushouts of saturated monoids
  7. Pseudo-saturated maps
  8. (Quasi-)saturated and pseudo-saturated homomorphisms
  9. A combinatorial criterion for pseudo-saturated homomorphisms
  10. Log adic spaces
  11. Log cotangent complexes
  12. Pre-log rings
  13. Log cotangent complexes
  14. Homologically log flat descent
  15. The log quasisyntomic site
  16. ...and 46 more sections

Key Result

Proposition 1.2

Let $u: P \rightarrow Q$ be a pseudo-saturated homomorphism between saturated monoids. Then for any map $P \rightarrow P'$ where $P'$ is a saturated and divisible monoid (which means that it is $n$-divisible for every positive integer $n$), the natural map from the naive pushout to the saturated pushout is of finite type.

Figures (4)

  • Figure 1: minimal decomposition of $(3,1)$
  • Figure 2: two minimal decompositions of $(3,1)$
  • Figure 3: minimal decomposition of $(2,2,1)$
  • Figure 4: two minimal decompositions of $(1, 1, 1)$

Theorems & Definitions (357)

  • Definition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 1.9: The primitive Hodge--Tate comparison
  • Theorem 1.10: The Hodge--Tate comparison
  • ...and 347 more