Cartan Theorems for Stein Manifolds Over a Discrete Valuation Base

Let X be a complex manifold, let A be a topological discrete valuation ring, and write for the sheaf of functions on X with values in A . We prove Cartan theorems A and B for coherent -modules, when X is a Stein manifold and A satisfies some requirements like being a nuclear direct limit of Banach algebras. The result is motivated by questions in the work of the second author with Kashiwara in the proof of the codimension-three conjecture for holonomic microdifferential systems.