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Characteristic cycles for coadmissible D-modules on smooth rigid analytic curves

Raoul Hallopeau

TL;DR

The paper develops a rigid-analytic analog of holonomicity for coadmissible $\wideparen{\mathcal{D}}$-modules on smooth rigid analytic curves by introducing characteristic varieties and finite-length-ensuring cycles in the one-dimensional setting. It builds a robust microlocalization framework on formal curves and their admissible blow-ups, extends it to the Zariski–Riemann space, and leverages the specialization map to transfer sub-holonomic structure between the formal model and the rigid-analytic side, including the $\mathcal{D}_{\langle\mathfrak{X}\rangle}$-theory of Ardakov–Bode–Wadsley. A core outcome is that sub-holonomic modules are generically integrable connections and have finite length, with an Artinian subcategory structure in the quasi-compact case, albeit not being stable under all six cohomological operations. The results unify microlocal techniques with blow-up invariance and provide a finite, multiplicity-driven description of characteristic cycles for both $\mathcal{D}_{\mathfrak{X},\infty}$ and $\wideparen{\mathcal{D}}_{\mathfrak{X}_K}$-modules, enabling precise invariants and finiteness statements in the $p$-adic setting.

Abstract

Let $\mathfrak{X}$ be a formal smooth curve over a complete discrete valuation ring of mixed characteristic and let $\mathfrak{X}\_K$ be its generic fiber. We consider respectively over $\mathfrak{X}$ and $\X\_K$ the sheaves of differential operators $\mathcal{D}\_{\mathfrak{X}, \infty}$ and $\wideparen{\D}\_{\mathfrak{X}\_K}$ with a rapid convergence condition. In this article, we define a characteristic variety as a subset of the cotangent space $T^*\mathfrak{X}\_K$ together with a characteristic cycle for coadmissible $\wideparen{\D}\_{\mathfrak{X}\_K}$-modules. We deduce a notion of ''sub-holonomicity'' for coadmissible $\wideparen{\D}\_{\mathfrak{X}\_K}$-modules which is equivalent to being generically an integrable connection. When $\mathfrak{X}$ is quasi-compact, we get an Artinian category of sub-holonomic $\wideparen{\D}\_{\mathfrak{X}\_K}$ which are weakly-holonomic. Moreover, we prove that a coadmissible $\wideparen{\D}\_{\mathfrak{X}\_K}$-modules is sub-holonomic if and only if the corresponding coadmissible $\Di$-module is.

Characteristic cycles for coadmissible D-modules on smooth rigid analytic curves

TL;DR

The paper develops a rigid-analytic analog of holonomicity for coadmissible -modules on smooth rigid analytic curves by introducing characteristic varieties and finite-length-ensuring cycles in the one-dimensional setting. It builds a robust microlocalization framework on formal curves and their admissible blow-ups, extends it to the Zariski–Riemann space, and leverages the specialization map to transfer sub-holonomic structure between the formal model and the rigid-analytic side, including the -theory of Ardakov–Bode–Wadsley. A core outcome is that sub-holonomic modules are generically integrable connections and have finite length, with an Artinian subcategory structure in the quasi-compact case, albeit not being stable under all six cohomological operations. The results unify microlocal techniques with blow-up invariance and provide a finite, multiplicity-driven description of characteristic cycles for both and -modules, enabling precise invariants and finiteness statements in the -adic setting.

Abstract

Let be a formal smooth curve over a complete discrete valuation ring of mixed characteristic and let be its generic fiber. We consider respectively over and the sheaves of differential operators and with a rapid convergence condition. In this article, we define a characteristic variety as a subset of the cotangent space together with a characteristic cycle for coadmissible -modules. We deduce a notion of ''sub-holonomicity'' for coadmissible -modules which is equivalent to being generically an integrable connection. When is quasi-compact, we get an Artinian category of sub-holonomic which are weakly-holonomic. Moreover, we prove that a coadmissible -modules is sub-holonomic if and only if the corresponding coadmissible -module is.

Paper Structure

This paper contains 22 sections, 42 theorems, 66 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Notations
  4. Sub-holonomic $\mathcal{D}_{\mathfrak{X}, \infty}$-modules
  5. Microlocalization of $\mathcal{D}_{\mathfrak{X}, \infty}$
  6. Sub-holonomic $\mathcal{D}_{\mathfrak{X}, \infty}$-modules
  7. Microlocalization for an admissible blow-up $\mathfrak{X}'$ of $\mathfrak{X}$
  8. Microlocalization of $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}', k, \mathbb{Q}}$
  9. Microlocalization of $\mathcal{D}_{\mathfrak{X}', \infty}$
  10. Sub-holonomic $\mathcal{D}_{\mathfrak{X}', \infty}$-modules
  11. Microlocalization of coadmissible $\mathcal{D}_{\mathfrak{X}', \infty}$-modules
  12. Weakly holonomic $\mathcal{D}_{\mathfrak{X}', \infty}$-modules
  13. Sub-holonomic $\mathcal{D}_{\mathfrak{X}', \infty}$-modules
  14. Equivalence of sub-holonomicity over $\mathfrak{X}'$ and $\mathfrak{X}$
  15. Sub-holonomic $\mathcal{D}_{\langle \mathfrak{X} \rangle}$-modules
  16. ...and 7 more sections

Key Result

Theorem 1.1

We can associate to any coadmissible $\wideparen{\mathcal{D}}_{\mathfrak{X}_K}$-module $\mathcal{M}$ a characteristic variety $\mathop{\mathrm{Char}}\nolimits(\mathcal{M})$ which is a closed subset of $T^*\mathfrak{X}_K$ and which satisfies Bernstein's inequality: if $\mathcal{M} \neq 0$, then $\mat

Figures (1)

  • Figure 1: $\mathop{\mathrm{Char}}\nolimits(\mathcal{D}_{\mathfrak{X}, \infty} /P)$ for $P = \sum_{n = 0}^d a_n \cdot \partial^n$

Theorems & Definitions (88)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Example 2.7
  • ...and 78 more