The direct image theorem in formal and rigid geometry (Q1345934)

It is proved the following direct image theorem for formal schemes over an arbitrary valuation ring \(R\) of height 1: Let \(f : X \to Y\) be a proper morphism of formal \(R\)-schemes which are locally of topologically finite presentation and \({\mathcal M}\) be a coherent \({\mathcal O}_ X\)-module. Then \(R^ q f_ *\) is a coherent \({\mathcal O}_ Y\)-module for each integer \(q\). The main interest of this theorem is that the formal schemes are not supposed to be noetherian.