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D-cap modules are quasi-coherent sheaves on an analytic stack

Arun Soor

TL;DR

The paper addresses a long-standing link between analytic $D$-modules and quasi-coherent sheaves on an analytic stack arising from rigid geometry. It constructs a stratifying stack $X_{ ext{str}}$ and a monad $oxed{egin{aligned} \\ \\ \, \mathcal{D}^ty_X\end{aligned}}$ on QCoh$(X)$, yielding a fully faithful functor from complexes of $D$-cap modules with Fréchet cohomology to QCoh$(X_{ ext{str}})$. In the étale-over-polydisk case, it proves an equivalence QCoh$(X_{ ext{str}}) \simeq \text{Mod}_{\mathcal{D}^ty_X}\text{QCoh}(X)$ and develops descent for wideparen{D}-modules in the analytic topology, along with compatibility with restrictions and a global Kan-extension construction. It also analyzes the essential image of the embedding, showing it does not lie inside dualizable or $f$-prim objects in general and proposing an $f$-suave criterion as a natural target for a precise intrinsic description.

Abstract

We construct a fully-faithful functor of $\infty$-categories from complexes of D-cap modules with Fréchet cohomology to quasi-coherent sheaves on an analytic stack. We prove various descent results for $\infty$-categories of D-cap modules in the analytic topology.

D-cap modules are quasi-coherent sheaves on an analytic stack

TL;DR

The paper addresses a long-standing link between analytic -modules and quasi-coherent sheaves on an analytic stack arising from rigid geometry. It constructs a stratifying stack and a monad on QCoh, yielding a fully faithful functor from complexes of -cap modules with Fréchet cohomology to QCoh. In the étale-over-polydisk case, it proves an equivalence QCoh and develops descent for wideparen{D}-modules in the analytic topology, along with compatibility with restrictions and a global Kan-extension construction. It also analyzes the essential image of the embedding, showing it does not lie inside dualizable or -prim objects in general and proposing an -suave criterion as a natural target for a precise intrinsic description.

Abstract

We construct a fully-faithful functor of -categories from complexes of D-cap modules with Fréchet cohomology to quasi-coherent sheaves on an analytic stack. We prove various descent results for -categories of D-cap modules in the analytic topology.

Paper Structure

This paper contains 14 sections, 63 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Reminders on quasi-abelian categories
  4. Properties of the derived category
  5. Monoidal structure
  6. Reminders on complete-bornological spaces
  7. Proof of the main result
  8. Descent for $\wideparen{\mathcal{D}}$-modules
  9. Compatibility with restrictions
  10. Globalising the embedding of $\mathcal{C}$-complexes
  11. On the essential image of $\mathcal{C}$-complexes
  12. A way to produce monoids from monads
  13. Noncommutative notion of descendability
  14. Barr--Beck--Lurie in families

Key Result

Theorem 1.1

soor_six-functor_2024 Suppose that $X = \operatorname{dSp}(A)$ is a smooth classical affinoid rigid space which is étale over a polydisk. Then the adjunction $p_! \dashv p^!$ is monadic, so that there is an equivalence of $\infty$-categories

Theorems & Definitions (132)

  • Theorem 1.1
  • Theorem 1.2: = Lemma \ref{['lem:naturaltransformlema']}
  • Theorem 1.3: = Theorem \ref{['thm:PNFtensor']}, Proposition \ref{['prop:algebraiso']}
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6: c.f. § \ref{['sec:DescentforDCap']}
  • Theorem 1.7: c.f. §\ref{['sec:global']}
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • ...and 122 more