arXiv subjects

Fernando Peña Vázquez

Publications and source records attributed to Fernando Peña Vázquez.

4 recordsLinked to original sources

arXiv2026

p-adic Cherednik algebras on rigid analytic spaces

Let $X$ be a smooth rigid space with an action of a finite group $G$ satisfying that $X/G$ is represented by a rigid space. We construct sheaves of $p$-adic Cherednik algebras on the small étale site of the quotient $X/G$, and study some of their properties. The sheaves of $p$-adic Cherednik algebras are sheaves of Fréchet-Stein $K$-algebras on $X/G$, which can be regarded as $p$-adic analytic versions of the sheaves of Cherednik algebras associated to the action of a finite group on a smooth algebraic variety defined by P. Etingof.

math.NT↗

arXiv2026

Category O for p-adic rational Cherednik algebras

We introduce the concept of a triangular decomposition for Banach and Fréchet-Stein algebras over $p$-adic fields, which allows us to define a category $\mathcal{O}$ for a wide array of topological algebras. In particular, we apply this concept to $p$-adic rational Cherednik algebras, which allows us to obtain an analytic version of the category $\mathcal{O}$ developed by Ginzburg, Guay, Opdam and Rouquier. Along the way, we study the global sections of $p$-adic Cherednik algebras on smooth Stein spaces, and determine their behavior with respect to the rigid analytic GAGA functor.

math.NT↗

arXiv2026

Hochschild (Co)homology of D-modules on rigid analytic spaces I

We introduce a formalism of Hochschild (co)-homology for $\mathcal{D}$-cap modules on smooth rigid analytic spaces based on the homological tools of Ind-Banach $\mathcal{D}$-cap modules. We introduce several categories of $\mathcal{D}$-cap bimodules for which this theory is well-behaved. Among these, the most important example is the category of diagonal $\mathcal{C}$-complexes. We give an explicit calculation of the Hochschild complex for diagonal $\mathcal{C}$-complexes, and show that the Hochschild complex of $\mathcal{D}$-cap is canonically isomorphic to the de Rham complex of $X$. In particular, we obtain a Hodge-de Rham spectral sequence converging to the Hochschild cohomology groups of $\mathcal{D}$-cap. We obtain explicit formulas relating the Hochschild cohomology and homology of a given diagonal $\mathcal{C}$-complex.

math.NT↗

arXiv2025

Hochschild (Co)homology of D-modules on rigid analytic spaces II

Let $X$ be a smooth $p$-adic Stein space with free tangent sheaf. We use the notion of Hochschild cohomology for sheaves of Ind-Banach algebras developed in our previous work to study the Hochschild cohomology of the algebra of infinite order differential operators $\mathcal{D}_X$-cap. In particular, we show that the Hochschild cohomology complex of $\mathcal{D}_X$-cap is a strict complex of nuclear Fréchet spaces which is quasi-isomorphic to the de Rham complex of $X$. We then use this to compare the first Hochschild cohomology group of $\mathcal{D}_X$-cap with a wide array of Ext functors. Finally, we investigate the relation of the Hochschild cohomology of $\mathcal{D}_X$-cap with the deformation theory of $\mathcal{D}_X(X)$-cap. Assuming some finiteness conditions on the de Rham cohomology of $X$, we define explicit isomorphisms between the first Hochschild cohomology group of $\mathcal{D}_X$-cap and the space of bounded outer derivations of $\mathcal{D}_X(X)$-cap, and between the second Hochschild cohomology group of $\mathcal{D}_X$-cap and the space of infinitesimal deformations of $\mathcal{D}_X(X)$-cap.

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