Descent theory and Morita theory for ultrametric Banach modules (Q1383228)

For a spherically complete non-Archimedean valued field \(K\) and a non-Archimedean unitary Banach algebra \(R\) over \(K\) the category \(\text{Mod}_R\) of Banach \(R\)-modules and \(R\)-linear contractions is studied. In particular, it is shown that, for a Banach algebra morphism \(f: R\to S\), \(f\) is a weak retract in \(\text{Mod}_R\) iff \(f\) is a descent morphism. If in addition \(R\), \(S\) are commutative, if \(f\) is a weak retract and \(S\) is a retract of some \(\mathbb{R}^n\) then \(R\) and \(\text{Lin}_R(S, S)\) are Morita equivalent. Results and proofs have a strong category theory flavour and do not contain much ingredients from ultrametric analysis. It seems that only the Hahn-Banach theorem (guaranteed by the spherical completeness of \(K\)) and the additivity of the category \(\text{Mod}_R\) (caused by the strong triangle inequality) are essential.