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 Fr\'echet 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.