Beauville-Laszlo gluing of algebraic spaces

TL;DR

This paper extends the Beauville-Laszlo gluing technique to algebraic spaces over discrete valuation rings, establishing an equivalence between gluing data and algebraic spaces, with applications to non-scheme algebraic spaces.

Contribution

It generalizes the Beauville-Laszlo theorem from schemes to algebraic spaces, providing a new gluing framework over discrete valuation rings and excellent bases.

Findings

Gluing algebraic spaces from formal and rigid data is always possible.

An equivalence between gluing triples and algebraic spaces is established.

Examples show the glued space can be a non-scheme algebraic space, even projective.

Abstract

For a complete discrete valuation field $K$, we show that one may always glue a separated formal algebraic space $\mathfrak{X}$ over $\mathcal{O}_K$ to a separated algebraic space $U$ over $K$ along an open immersion of rigid spaces $j\colon \mathfrak{X}^{\rm rig}\to U^{\rm an}$, producing a separated algebraic space $X$ over $\mathcal{O}_K$. This process gives rise to an equivalence between such `gluing triples' $(U,\mathfrak{X},j)$ and separated algebraic spaces $X$ over $\mathcal{O}_K$, which one might interpret as a version of the Beauville--Laszlo theorem for algebraic spaces rather than coherent sheaves. Moreover, an analogous equivalence exists over any excellent base. Examples due to Matsumoto imply that the result of such a gluing might be a genuine algebraic space (not a scheme) even if $U$ and the special fiber of $\mathfrak{X}$ are projective. The proof is a combination of Nagata compactification theorem for algebraic spaces and of Artin's contraction theorem. We give multiple examples and applications of this idea.