Completed tensor products and a global approach to p-adic analytic differential operators

Copyright © Cambridge Philosophical Society 2018 Ardakov-Wadsley defined the sheaf (Formula presented.)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 (Formula presented.) for arbitrary affinoid subdomains and not just on a suitable base of the topology. The structural results concerning (Formula presented.) and coadmissible (Formula presented.)-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 (Formula presented.) - applied to complexes of strict morphisms, including a necessary and sufficient condition in the case of short exact sequences.