Specialization Morphisms

Abstract

Abstract We define the notion of a specialization morphism from a locally noetherian analytic adic space to a scheme. This captures the (classical) specialization morphism associated with a formal scheme. There is a well-behaved theory of compactifications and it turns out that the classical specialization morphism is proper in this setup. As an application, we show that the nearby cycles functor commutes with lower shriek in great generality.