Hochschild (Co)homology of D-modules on rigid analytic spaces I
Fernando Peña Vázquez
TL;DR
The paper develops a p-adic analytic theory of Hochschild (co)homology for sheaves of completed differential operators $\wideparen{\mathcal{D}}_X$ on smooth rigid analytic spaces, using Ind-Banach techniques and quasi-abelian categories. It introduces the bi-enveloping algebra $\wideparen{E}_X$ and a Kashiwara-type immersion framework for diagonal $\mathcal{C}$-complexes, culminating in an explicit computation: the inner Hochschild cohomology $\mathcal{HH}^{\bullet}(\wideparen{\mathcal{D}}_X,\widehat{\mathcal{D}}_X)$ matches the de Rham complex $\Omega^{\bullet}_{X/K}$, hence $HH^{\bullet}(\wideparen{\mathcal{D}}_X)=R\Gamma(X,\Omega^{\bullet}_{X/K})$ and a Hodge–de Rham spectral sequence appears. The work develops the requisite homological machinery in the setting of Ind-Banach spaces, including derived tensor–hom functors, base-change, and closed immersions, and extends these to Lie algebroids via Fréchet-Stein enveloping algebras and co-admissible modules. It then treats products of Lie algebroids, defining the product $\mathscr{L}_X\times\mathscr{L}_Y$ and its enveloping algebras, and introduces the auxiliary algebras $\wideparen{E}(\mathscr{L})$ and $\wideparen{U}(\mathscr{L}^2)$ to facilitate Hochschild calculations on $X^2$ with respect to the diagonal.
Abstract
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.