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Analytic Geometry and Hodge-Frobenius Structure

Xin Tong

TL;DR

The work develops a multidimensional $p$-adic analytic framework for Frobenius structures, introducing relative multidimensional Robba rings, Fréchet–Stein algebras, and commuting multi-Frobenius actions. It proves finiteness and coherence results for the cohomology of multidimensional $(oldsymbol i,oldsymbol migtriangleup)$-modules via Cartan–Serre methods and $p$-adic functional analysis, and extends these constructions to relative settings and to noncommutative coefficients. The paper also studies triangulations and residue pairings in the multidimensional Robba context, and outlines conjectural noncommutative analogs of Tamagawa-type statements and equivariant Iwasawa theory. Collectively, these results provide foundational tools for higher-dimensional relative $p$-adic Hodge theory, with potential applications to generalized Tamagawa-number conjectures and noncommutative Iwasawa theory.

Abstract

In this paper, we study Frobenius structures in higher dimensional $p$-adic analytic geometry and the corresponding $p$-adic functional analysis. This will build up foundations for further study on some generalized cohomology of Frobenius modules and the corresponding generalized Iwasawa theory and generalized noncommutative Tamagawa number conjectures in the spirit of Burns-Flach-Fukaya-Kato and Nakamura (as well as certainly the original noncommutative Tamagawa number conjectures as observed by Pal-Zábrádi). We will work in the program proposed by Carter-Kedlaya-Zábrádi and after Pal-Zábrádi, and we will follow closely the approach from Kedlaya-Pottharst-Xiao to investigate the corresponding deformation of the generalized $p$-adic Hodge structures.

Analytic Geometry and Hodge-Frobenius Structure

TL;DR

The work develops a multidimensional -adic analytic framework for Frobenius structures, introducing relative multidimensional Robba rings, Fréchet–Stein algebras, and commuting multi-Frobenius actions. It proves finiteness and coherence results for the cohomology of multidimensional -modules via Cartan–Serre methods and -adic functional analysis, and extends these constructions to relative settings and to noncommutative coefficients. The paper also studies triangulations and residue pairings in the multidimensional Robba context, and outlines conjectural noncommutative analogs of Tamagawa-type statements and equivariant Iwasawa theory. Collectively, these results provide foundational tools for higher-dimensional relative -adic Hodge theory, with potential applications to generalized Tamagawa-number conjectures and noncommutative Iwasawa theory.

Abstract

In this paper, we study Frobenius structures in higher dimensional -adic analytic geometry and the corresponding -adic functional analysis. This will build up foundations for further study on some generalized cohomology of Frobenius modules and the corresponding generalized Iwasawa theory and generalized noncommutative Tamagawa number conjectures in the spirit of Burns-Flach-Fukaya-Kato and Nakamura (as well as certainly the original noncommutative Tamagawa number conjectures as observed by Pal-Zábrádi). We will work in the program proposed by Carter-Kedlaya-Zábrádi and after Pal-Zábrádi, and we will follow closely the approach from Kedlaya-Pottharst-Xiao to investigate the corresponding deformation of the generalized -adic Hodge structures.

Paper Structure

This paper contains 13 sections, 33 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Higher Dimensional Analytic Geometry
  3. Results and Applications
  4. What's Next
  5. Fréchet Rings and Sheaves in the Commutative Case
  6. Fréchet Objects
  7. Further Discussion and Residue Pairings
  8. Relative Frobenius Sheaves over Affinoids and Fréchet Rings
  9. Der Endlichkeitssatz
  10. Finiteness through $p$-adic Functional Analaysis
  11. Application to Triangulation over Rigid Analytic Spaces
  12. Noncommutative Coefficients
  13. Frobenius Modules over Noncommutative Rings

Key Result

Theorem 1.5

With the same assumptions as in assumption1 as below, we set $M$ to be the global section of a locally free coherent sheaf of module over the sheaf of the Robba ring $\Pi_{\mathrm{an},\mathrm{con},I,X}(\pi_{K_I})$ over an affinoid domain $X$ attached to an affinoid algebra $A$ in the rigid analytic

Theorems & Definitions (104)

  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Conjecture 1.8
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • ...and 94 more