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The direct image theorem in formal and rigid geometry. (English) Zbl 0821.32029
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.

32P05 Non-Archimedean analysis
32H35 Proper holomorphic mappings, finiteness theorems
32C38 Sheaves of differential operators and their modules, \(D\)-modules
11G99 Arithmetic algebraic geometry (Diophantine geometry)
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