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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.20500 |
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Table of Contents:
- 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.