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Quasicoherent sheaves for dagger analytic geometry

Arun Soor

TL;DR

The paper develops a robust framework for quasicoherent sheaves on dagger analytic varieties by leveraging the derived category of IndBan spaces to form a stable, monoidal $\infty$-category of modules. It builds a $CAlg(\mathsf{Pr}^L_{st})$-valued QCoh on the dagger affinoid site, proves descent in the analytic topology, and introduces pushforwards with compact supports $f_!$ alongside a right adjoint $f^!$, together with an excision sequence for suitable open decompositions. It establishes base-change and projection formulas for transversal morphisms, proves coherence descends, and develops Grothendieck duality in this setting, including explicit results for open polydisk projections, étale and smooth morphisms, and open immersions. Collectively, these results provide a six-functor-like toolkit for dagger analytic geometry, bridging non-archimedean analytic spaces with a derived, descent-friendly formalism. The work lays groundwork toward a coherent duality theory in p-adic analytic geometry and connects to classical rigid-analytic duality results while remaining within a dagger-analytic framework.

Abstract

We develop a theory of quasicoherent sheaves on dagger analytic varieties based on Ind-Banach spaces. We show that they satisfy descent in the analytic topology. We define compactly supported pushforwards and produce an adjunction $f_! \dashv f^!$, and produce an excision sequence for certain inclusions of admissible open subsets. Finally, we prove a Grothendieck duality theorem.

Quasicoherent sheaves for dagger analytic geometry

TL;DR

The paper develops a robust framework for quasicoherent sheaves on dagger analytic varieties by leveraging the derived category of IndBan spaces to form a stable, monoidal -category of modules. It builds a -valued QCoh on the dagger affinoid site, proves descent in the analytic topology, and introduces pushforwards with compact supports alongside a right adjoint , together with an excision sequence for suitable open decompositions. It establishes base-change and projection formulas for transversal morphisms, proves coherence descends, and develops Grothendieck duality in this setting, including explicit results for open polydisk projections, étale and smooth morphisms, and open immersions. Collectively, these results provide a six-functor-like toolkit for dagger analytic geometry, bridging non-archimedean analytic spaces with a derived, descent-friendly formalism. The work lays groundwork toward a coherent duality theory in p-adic analytic geometry and connects to classical rigid-analytic duality results while remaining within a dagger-analytic framework.

Abstract

We develop a theory of quasicoherent sheaves on dagger analytic varieties based on Ind-Banach spaces. We show that they satisfy descent in the analytic topology. We define compactly supported pushforwards and produce an adjunction , and produce an excision sequence for certain inclusions of admissible open subsets. Finally, we prove a Grothendieck duality theorem.
Paper Structure (11 sections, 56 theorems, 106 equations)

This paper contains 11 sections, 56 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. The derived $\infty$-category of a quasi-abelian category
  4. Quasicoherent sheaves
  5. Transversal morphisms
  6. Coherent sheaves
  7. Pushforward with compact supports
  8. The functor $f_!$ and its properties
  9. The excision sequence
  10. Composition of the functors $f_!$.
  11. Grothendieck duality

Key Result

Theorem 1.1

(c.f. §sec:quasicoherent, §sec:coherentsheaves).

Theorems & Definitions (131)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • ...and 121 more