## 78.1 Introduction

The goal of this chapter is to discuss pushouts in the category of algebraic spaces. This can be done with varying assumptions. A fairly general pushout construction is given in : one of the morphisms is affine and the other is a closed immersion. We discuss a particular case of this in Section 78.2 where we assume one of the morphisms is affine and the other is a thickening, a situation that often comes up in deformation theory.

In Sections 78.5 and 78.6 we discuss diagrams

$\xymatrix{ f^{-1}(X \setminus Z) \ar[r] \ar[d] & Y \ar[d]^ f \\ X \setminus Z \ar[r] & X }$

where $f$ is a quasi-compact and quasi-separated morphism of algebraic spaces, $Z \to X$ is a closed immersion of finite presentation, the map $f^{-1}(Z) \to Z$ is an isomorphism, and $f$ is flat along $f^{-1}(Z)$. In this situation we glue quasi-coherent modules on $X \setminus Z$ and $Y$ (in Section 78.5) to quasi-coherent modules on $X$ and we glue algebraic spaces over $X \setminus Z$ and $Y$ (in Section 78.6) to algebraic spaces over $X$.

In Section 78.8 we discuss how proper birational morphisms of Noetherian algebraic spaces give rise to coequalizer diagrams in algebraic spaces in some sense.

In Section 78.9 we use the construction of elementary distinguished squares in Section 78.4 to prove Nagata's theorem on compactifications in the setting of algebraic spaces.

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