Specialization morphisms
Ildar Gaisin, John Welliaveetil
TL;DR
This paper introduces specialization morphisms from locally noetherian analytic adic spaces to schemes, establishing their properties and applications, including properness and compatibility of nearby cycles functor.
Contribution
It defines a new notion of specialization morphisms in the adic space context, extending classical concepts and proving their properness and compatibility with nearby cycles.
Findings
Specialization morphisms are well-behaved and proper.
The classical specialization morphism is a special case.
Nearby cycles functor commutes with lower shriek in general.
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 to a formal scheme. There is a well behaved theory of compactifications and it turns out that the classical specialization morphism is \emph{proper} in this setup. As an application, we show that the nearby cycles functor commutes with lower shriek in great generality.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Polynomial and algebraic computation