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 $j\colon \mathfrak {X}^{\mathrm {rig}}\to U^{\mathrm {an}}$ , 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 fibre of $\mathfrak {X}$ are projective. The proof is a combination of the Nagata compactification theorem for algebraic spaces and of Artin’s contraction theorem. We give multiple examples and applications of this idea.