2024/10/27 by Piotr Achinger, Achinger, Piotr, Alex Youcis +1 · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.2410.20500
For a complete discrete valuation field K, we show that one may always glue a separated formal algebraic space \mathfrakX over OK to a separated algebraic space U over K along an open immersion of rigid spaces j\colon \mathfrakX\rm rig→ U\rm an, producing a separated algebraic space X over OK. This process gives rise to an equivalence between such `gluing triples' (U,\mathfrakX,j) and separated algebraic spaces X over OK, 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 \mathfrakX 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.