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Weakly holonomic equivariant $\mathcal{D}$-modules on rigid analytic spaces

Tobias Schmidt, Thi Minh Phuong Vu

Abstract

Let G be a $p$-adic Lie group. We develop a dimension theory for coadmissible G-equivariant $\mathcal{D}$-modules on smooth rigid analytic spaces. We introduce the category of weakly holonomic G-equivariant $\mathcal{D}$-modules, study its duality and its preservation under various operations.

Weakly holonomic equivariant $\mathcal{D}$-modules on rigid analytic spaces

Abstract

Let G be a -adic Lie group. We develop a dimension theory for coadmissible G-equivariant -modules on smooth rigid analytic spaces. We introduce the category of weakly holonomic G-equivariant -modules, study its duality and its preservation under various operations.

Paper Structure

This paper contains 20 sections, 64 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Reminder on coadmissible equivariant $\mathop{\mathrm{\mathcal{D}}}\nolimits$-modules
  3. Some algebraic background
  4. Lie-Rinehart algebras
  5. Crossed products
  6. Coadmissible equivariant $\mathop{\mathrm{\mathcal{D}}}\nolimits$-modules on rigid analytic spaces
  7. Ext groups and dimension for coadmissible $\wideparen{\mathcal{D}}(\mathop{\mathrm{\mathbf{X}}}\nolimits,G)$-modules
  8. Ext functors for coadmissible equivariant $\mathcal{D}$-modules
  9. Modules over the sheaf of rings $\mathcal{Q}$
  10. Globalization of Ext functors
  11. The Ext functors on the category $\mathcal{C}_{\mathop{\mathrm{\mathbf{X}}}\nolimits/G}$
  12. Dimension and weakly holonomic equivariant $\mathcal{D}$-modules
  13. Dimension theory for coadmissible equivariant $\mathcal{D}$-modules
  14. Bernstein's inequality
  15. Weakly holonomic equivariant $\mathcal{D}$-modules and duality
  16. ...and 5 more sections

Key Result

Lemma \oldthetheorem

Let $\varphi: R \rightarrow A$ be a morphism of rings such that $\varphi$ is left (resp. right) flat and that it factors through Then the morphism $R\ast G \rightarrow A$ is left (resp. right) flat.

Theorems & Definitions (151)

  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • ...and 141 more