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Ardakov-Wadsley defined the sheaf
X of p-adic analytic differential operators on a smooth rigid analytic variety by restricting to the case where X is affinoid and the tangent sheaf admits a smooth Lie lattice. We generalise their results by dropping the assumption of a smooth Lie lattice throughout, which allows us to describe the sections of
for arbitrary affinoid subdomains and not just on a suitable base of the topology. The structural results concerning
-modules can then be generalised in a natural way.
The main ingredient for our proofs is a study of completed tensor products over normed K-algebras, for K a discretely valued field of mixed characteristic. Given a normed right module U over a normed K-algebra A, we provide several exactness criteria for the functor
- applied to complexes of strict morphisms, including a necessary and sufficient condition in the case of short exact sequences.
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