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 DX-cap. In particular, we show that the Hochschild cohomology complex of DX-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 DX-cap with a wide array of Ext functors. Finally, we investigate the relation of the Hochschild cohomology of DX-cap with the deformation theory of DX(X)-cap. Assuming some finiteness conditions on the de Rham cohomology of X, we define explicit isomorphisms between the first Hochschild cohomology group of DX-cap and the space of bounded outer derivations of DX(X)-cap, and between the second Hochschild cohomology group of DX-cap and the space of infinitesimal deformations of DX(X)-cap.