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I need to know when a glueing prcedure makes sense in the scheme world. Let $X$ be a smooth, projective scheme over $S=Spec(A)$, $A$ a complete ring. Let $x,y\in X$ closed points such that $\phi:\hat{\mathcal{O}}_{X,x}\cong \hat{\mathcal{O}}_{X,y}$ (completion w.r.t. maximal ideals). I want to produce a scheme where I identify the points $x$ and $y$ via this isomorphism. I guess that in general this is not possible. When does this operation make sense? When does it do at least as formal/rigid/analytic space?Do I get a stack?

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