Sorry if this is a stupid question, but what are the uses of a completion of a scheme along a closed subscheme? Are there any nice universal properties it satisfies, or do certain morphisms factor through completions? I understand that they are just colimits of closed subschemes, but is it just interesting that they exist or is there something useful one can do with them? What was the motivation for defining them?
In Hartshorne they are defined but he doesn't really seem to do anything with them. A reference where they are used would be great, thanks!