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The tempered disk and the tempered cohomology

Federico Bambozzi, Bruno Chiarellotto, Pietro Vanni

TL;DR

The paper develops a tempered, derived-analytic framework for non-archimedean geometry by associating a spectrum $\mathfrak{S}(A)$ to ind-Banach rings and introducing tempered tubes and tempered power series. It defines tempered convergent cohomology $H^{\bullet}_{\mathrm{temp}}(X)$ via the Hodge-completed derived de Rham complex on tempered tubes and proves independence from choices, with a crystalline-cohomology comparison for smooth proper $X$. A key innovation is the reinterpretation of log-growth transfer theorems for $p$-adic differential equations as continuity statements within the derived analytic spectrum, using tempered open subsets of the affine line. The work also develops a robust infrastructure—quasi-abelian categories, Ind-Banach modules, bornological models, derived analytic stacks, and an analytic cotangent complex—to underpin tempered cohomology and its geometric and cohomological applications in non-archimedean settings.

Abstract

Consider a non-archimedean valuation ring V (K its fraction field, in mixed characteristic): inspired by some views presented by Scholze, we introduce a new point of view on the non-archimedean analytic setting in terms of derived analytic geometry (then associating a "spectrum" to each ind-Banach algebra). We want to look at the behaviour of this spectrum from a differential point of view. In such a spectrum, for example, there exist open subsets having functions with log-growth as sections for the structural sheaf. In this framework, a transfer theorem for the log-growth of solutions of p-adic differential equations can be interpreted as a continuity theorem (analogue to the transfer theorem for their radii of convergence in the Berkovich spaces). As a dividend of such a theory, we define a new cohomology theory in terms of the Hodge-completed derived de Rham cohomology of the ind-Banach derived analytic space associated to a smooth k-scheme, X_k (k residual field of V), via the use of "tempered tubes". We finally compare our tempered de Rham cohomology with crystalline cohomology.

The tempered disk and the tempered cohomology

TL;DR

The paper develops a tempered, derived-analytic framework for non-archimedean geometry by associating a spectrum to ind-Banach rings and introducing tempered tubes and tempered power series. It defines tempered convergent cohomology via the Hodge-completed derived de Rham complex on tempered tubes and proves independence from choices, with a crystalline-cohomology comparison for smooth proper . A key innovation is the reinterpretation of log-growth transfer theorems for -adic differential equations as continuity statements within the derived analytic spectrum, using tempered open subsets of the affine line. The work also develops a robust infrastructure—quasi-abelian categories, Ind-Banach modules, bornological models, derived analytic stacks, and an analytic cotangent complex—to underpin tempered cohomology and its geometric and cohomological applications in non-archimedean settings.

Abstract

Consider a non-archimedean valuation ring V (K its fraction field, in mixed characteristic): inspired by some views presented by Scholze, we introduce a new point of view on the non-archimedean analytic setting in terms of derived analytic geometry (then associating a "spectrum" to each ind-Banach algebra). We want to look at the behaviour of this spectrum from a differential point of view. In such a spectrum, for example, there exist open subsets having functions with log-growth as sections for the structural sheaf. In this framework, a transfer theorem for the log-growth of solutions of p-adic differential equations can be interpreted as a continuity theorem (analogue to the transfer theorem for their radii of convergence in the Berkovich spaces). As a dividend of such a theory, we define a new cohomology theory in terms of the Hodge-completed derived de Rham cohomology of the ind-Banach derived analytic space associated to a smooth k-scheme, X_k (k residual field of V), via the use of "tempered tubes". We finally compare our tempered de Rham cohomology with crystalline cohomology.

Paper Structure

This paper contains 24 sections, 68 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Quasi-abelian categories
  4. Quasi-abelian categories and derived categories
  5. Closed symmetric monoidal quasi-abelian categories
  6. Elementary quasi-abelian categories
  7. Model structures on cochain complexes
  8. Ind-completion of quasi-abelian categories
  9. Ind-Banach modules
  10. The quasi-abelian structure
  11. Bornological vector spaces
  12. Tempered power series
  13. The derived analytic spectrum of an Ind-Banach ring
  14. Preliminaries of lattice theory
  15. The spectrum of a closed symmetric stable $\infty$-category
  16. ...and 9 more sections

Key Result

Proposition 3.4

Let $\sf{C}$ be a quasi-abelian category. The canonical embedding induces an equivalence of triangulated categories The left $t$-structure on $\mathcal{D}(\sf{C})$ corresponds to the canonical $t$-structure of $\mathcal{D}(\mathcal{LH}(\sf{C}))$ via this equivalence.

Theorems & Definitions (188)

  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Definition 3.5
  • Remark 3.6
  • Proposition 3.7
  • Remark 3.8
  • Definition 3.9
  • ...and 178 more