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Let $X$ be a scheme with closed subscheme $Z$.

There is a natural way to think of $X$ as a functor from schemes to sets, $$X : S \mapsto X(S) = \mathrm{Mor}(S,X).$$

It seems there will be a similar way to understand the completion $\hat X$ of $X$ at $Z$ as a functor, but I am not sure how we do this. Please tell me if you know.

user82625
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1 Answers1

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We can think of a formal scheme as a particular kind of Ind-scheme, namely let $\mathcal I_Z$ be the ideal sheaf cutting out $Z$, and let $X_n$ denote the closed subscheme of $X$ cut out by $\mathcal I_Z^n$. Then each $X_n$ defines a contravariant functor on the category of rings (equivalently, on the category of affine schemes), and we can consider $\widehat{X}$ as the direct limit functor: $\widehat{X}(A) = \varinjlim_n X_n(A).$

(There are many variations on this; e.g. we can extend $\widehat{X}$ to a functor on adic rings: if $A$ is a ring complete with respect to an ideal $A$, then we can define $\widehat{X}(A) = \varprojlim_n X(A/I^n).$)

Matt E
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  • Great, I wasn't sure my question was clear but this is just what I wanted! – user82625 Jun 15 '13 at 20:37
  • @user82625: Dear user, I'm glad this helped. I have been thinking about these kinds of questions recently, because I am trying to apply Artin's results about representing functors in some work I am doing. I highly recommend his papers if you want to see the functorial view-point at work. (Start with The implicit function theorem in algebraic geometry, and then look at Algebraization of formal moduli I and II.) Cheers, – Matt E Jun 16 '13 at 00:50