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Six operations for D-cap-modules on rigid analytic spaces

Andreas Bode

TL;DR

This work develops a six-functor formalism for complete bornological $\wideparen{\mathcal{D}}_X$-modules on smooth rigid analytic spaces, by working in the derived category of complete bornological $K$-vector spaces and isolating a triangulated subcategory of '$\mathcal{C}$-complexes' whose cohomology is coadmissible. The authors establish a sturdy algebraic and analytic foundation: completed enveloping algebras $\wideparen{U_A(L)}$ of smooth Lie–Rinehart algebras are Fréchet–Stein with a faithfully flat map from $U_A(L)$, enabling a derived theory compatible with Kashiwara’s equivalence and Beilinson–Bernstein localization analogues. On the analytic side, complete bornological spaces and their left-heart $LH(\widehat{\mathcal{B}}c_K)$ provide a flexible setting for derived functors and six operations, with a precise comparison to Ind-Banach categories to ensure exactness and monoidal compatibilities. The paper also proves that coadmissible modules over Fréchet–Stein algebras are nuclear and of countable type, guaranteeing good behavior of completed tensor products and Hom functors, which underpins the main results including Kashiwara equivalence and stability under inverse and direct image functors in the rigid-analytic context. Overall, the framework consolidates D-cap-module theory on rigid spaces into a robust derived category approach, enabling de Rham cohomology computations and localization techniques in non-archimedean geometry.

Abstract

We introduce all six operations for D-cap-modules on smooth rigid analytic spaces by considering the derived category of complete bornological D-cap-modules. We then focus on a full subcategory which should be thought of as consisting of complexes with coadmissible cohomology, and establish analogues of some classical results: Kashiwara's equivalence, stability of coadmissibility under extraordinary inverse image for smooth morphisms and direct image for projective morphisms, as well as the computation of relative de Rham cohomology.

Six operations for D-cap-modules on rigid analytic spaces

TL;DR

This work develops a six-functor formalism for complete bornological -modules on smooth rigid analytic spaces, by working in the derived category of complete bornological -vector spaces and isolating a triangulated subcategory of '-complexes' whose cohomology is coadmissible. The authors establish a sturdy algebraic and analytic foundation: completed enveloping algebras of smooth Lie–Rinehart algebras are Fréchet–Stein with a faithfully flat map from , enabling a derived theory compatible with Kashiwara’s equivalence and Beilinson–Bernstein localization analogues. On the analytic side, complete bornological spaces and their left-heart provide a flexible setting for derived functors and six operations, with a precise comparison to Ind-Banach categories to ensure exactness and monoidal compatibilities. The paper also proves that coadmissible modules over Fréchet–Stein algebras are nuclear and of countable type, guaranteeing good behavior of completed tensor products and Hom functors, which underpins the main results including Kashiwara equivalence and stability under inverse and direct image functors in the rigid-analytic context. Overall, the framework consolidates D-cap-module theory on rigid spaces into a robust derived category approach, enabling de Rham cohomology computations and localization techniques in non-archimedean geometry.

Abstract

We introduce all six operations for D-cap-modules on smooth rigid analytic spaces by considering the derived category of complete bornological D-cap-modules. We then focus on a full subcategory which should be thought of as consisting of complexes with coadmissible cohomology, and establish analogues of some classical results: Kashiwara's equivalence, stability of coadmissibility under extraordinary inverse image for smooth morphisms and direct image for projective morphisms, as well as the computation of relative de Rham cohomology.

Paper Structure

This paper contains 42 sections, 172 theorems, 504 equations.

Table of Contents

  1. Introduction
  2. Algebraic background: Completed enveloping algebras
  3. Fréchet--Stein algebras and coadmissible modules
  4. Faithful flatness
  5. Categorical background: Sheaves valued in quasi-abelian categories
  6. Quasi-abelian categories
  7. Homotopy limits
  8. Modules over monoids
  9. Categories of sheaves
  10. Sheaves on bases
  11. Derived tensors and Homs
  12. Derived direct and inverse image
  13. Analytic background: Complete bornological vector spaces
  14. Bornological vector spaces
  15. Comparison with $\mathrm{Ind}(\mathrm{Ban}_K)$
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $X$ be a smooth (equidimensional) rigid analytic $K$-variety. The category $\mathrm{Mod}_{\mathrm{Shv}(X, LH(\widehat{\mathcal{B}}c_K))}(\wideparen{\mathcal{D}}_X)$ is a Grothendieck abelian category admitting flat resolutions. In particular, there are functors For any morphism $f: X\to Y$ of smooth (equidimensional) rigid analytic $K$-varieties, there are functors

Theorems & Definitions (354)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • proof
  • Theorem 1.3
  • proof
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • ...and 344 more