Beauville-Laszlo gluing of algebraic spaces

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 $j:{\mathfrak{X}}^{\mathrm{rig}}\to U^{\mathrm{an}}$, 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.

30 pages