Valuative criteria for adic spaces

Abstract

We establish valuative criteria for separatedness and (partial) properness for morphisms of adic spaces, which account for both vertical and horizontal specializations. The lifting test diagrams used in Huber's analytic valuative criteria are naturally adapted to vertical specializations. In the non-analytic setting, one must additionally include lifting conditions for horizontal specializations. Combining these horizontal criteria with Huber's vertical criteria yields unified valuative characterizations of separatedness, universal specialization, and (partial) properness for arbitrary adic spaces.