On relative vertical compactification of weakly square complete adic spaces

TL;DR

This paper generalizes the universal vertical compactification of morphisms between adic spaces, removing the need for local Noetherian assumptions, and provides an explicit construction of the compactification.

Contribution

It introduces a generalized universal vertical compactification for certain adic space morphisms without requiring local Noetherian conditions.

Findings

Existence of a universal vertical compactification for specified morphisms.

Construction of the compactification without local Noetherian assumptions.

Explicit and simple method for constructing the compactification.

Abstract

We prove for a morphism $f \colon X \rightarrow S$ locally of $^+$weakly finite type, separated and taut, where $X$ is a weakly square complete adic space and $S$ a square complete and stable adic space, there exists a universal vertical compactification $f' \colon X' \rightarrow S$. This provides a generalized version of Huber's proof of universal compactification of morphisms between analytic adic spaces. Notably, we will see that for the compactification, it is not necessary to assume that the adic spaces are locally Noetherian. In the fourth section we give an explicit and simple construction of $X'$.