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.

💡 Research Summary

The paper develops a comprehensive theory of Hochschild (co)homology for sheaves of completed p‑adic differential operators (\widehat{\mathcal D}_X) on a smooth, separated rigid analytic space (X) over a complete non‑archimedean field (K). The authors work within the framework of Ind‑Banach modules, which provides a quasi‑abelian category suitable for homological algebra in the analytic setting. After reviewing the necessary background on bornological spaces, Ind‑Banach spaces, and the tensor‑Hom adjunction, they construct derived tensor products (\widehat\otimes^{\mathbf L}) and derived Hom functors (\mathbf R!\operatorname{Hom}) in this context.

A central object is the “diagonal (C)-complex”, a sheaf that is simultaneously a left and right (\widehat{\mathcal D}_X)-module with identical structures. The authors introduce the bi‑enveloping algebra (\widehat{\mathcal D}_X^{e}) (the completed tensor product of (\widehat{\mathcal D}_X) with its opposite) and develop side‑switching operations that allow one to pass freely between left and right module viewpoints. Co‑admissible modules over (\widehat{\mathcal D}_X^{e}) are studied both locally and globally, using an immersion functor that embeds global modules into a local co‑admissible category.

Hochschild homology and cohomology are defined for any (\widehat{\mathcal D}_X^{e})-module (\mathcal M) by the standard derived formulas: \