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The analytic de Rham stack in rigid geometry

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arxiv 2401.07738 v1 pith:GHEW25NS submitted 2024-01-15 math.AG math.RT

Juan Esteban Rodríguez Camargo This is my paper · ORCID
classification math.AGmath.RT
keywords analytictheorymodulesadicformalismgeometrypreviousrham
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abstract

Applying the new theory of analytic stacks of Clausen and Scholze we introduce a general notion of derived Tate adic spaces. We use this formalism to define the analytic de Rham stack in rigid geometry, extending the theory of $D$-cap-modules of Ardakov and Wadsley to the theory of analytic $D$-modules. We prove some foundational results such as the existence of a six functor formalism and Poincar\'e duality for analytic $D$-modules, generalizing previous work of Bode. Finally, we relate the theory of analytic $D$-modules to previous work of the author with Rodrigues Jacinto on solid locally analytic representations of $p$-adic Lie groups.

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Cited by 4 Pith papers

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  1. Cartier duality for gerbes of vector bundles

    math.AG 2025-12 conditional novelty 8.0 of 10

    The Hodge-Tate stack of a smooth rigid variety is Cartier dual to the Simpson gerbe, so its solid quasi-coherent sheaves equal the weight-1 sheaves on the gerbe.

  2. Bernstein-Zelevinsky duality for locally analytic principal series representations

    math.RT 2025-01 conditional novelty 7.0 of 10

    For locally algebraic integral weights, the Bernstein-Zelevinsky dual of a locally analytic principal series representation equals another locally analytic principal series built from the dual Verma module and the smo...

  3. A 6-functor formalism for solid quasi-coherent sheaves on the Fargues-Fontaine curve

    math.AG 2024-12 accept novelty 7.0 of 10

    A 6-functor formalism for Z_p-linear solid quasi-coherent sheaves on small v-stacks is constructed, yielding Poincare duality for pro-etale Q_p-cohomology.

  4. An axiomatic approach to analytic $1$-affineness

    math.AG 2025-09 conditional novelty 6.0 of 10

    An axiomatic framework proves 1-affineness for analytic Betti stacks, analytic de Rham stacks, and rigid analytic varieties, giving categorical Künneth formulas.

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