# Two definitions of formal schemes

I have read two seemingly completely different definitions of formal schemes.

Hartshorne defined a formal scheme as the formal completion of a noetherian scheme $$X$$ along a closed subscheme $$Y$$ of $$X$$.

In Demazure's Lectures on p-Divisible Groups, a $$k$$-formal scheme is defined to be a left exact covariant functor from the category of finite dimensional $$k$$-algebras to the category of sets, where $$k$$ is a field.

Are these two definitions of formal schemes the same (or related to each other)?

(update) Now I have read about the definition in EGA I, which defines an affine formal scheme as the formal spectrum $$\mathop{\mathrm{Spf}}(A)$$ of an admissible ring $$A$$. As Murray answered, Demazure defines an affine formal scheme over $$k$$ as $$\mathop{\mathrm{Spf}}(A)$$ for some profinite $$k$$-algebra. It a profinite $$k$$-algebra always admissible? (It seems to me that this is not true, for example consider the the product of infinitely many $$k$$. So the definition in EGA is not the most general case?)

• What you quote from Hartshorne is an example, not the definition, of a formal scheme. Demazure's definition isn't very general either. You can go to the original source EGA I section 10 for the actual definition. Probably the stacks project also treats this, but I haven't checked. May 9, 2020 at 15:38
• @DonuArapura: see Tag 0AHY. May 9, 2020 at 16:00

I was reading those lectures only recently. It is very important to acknowledge that Demazure uses the functor-of-points perspective for discussing schemes (see Chapter I.4 of those notes to see a discussion of the equivalence of this approach with the more standard one). Demazure gives 4 different (but equivalent) definitions of a formal scheme, one of those being the definition you stated. Another definition is stated in terms of profinite $$k$$-algebras. These are topological $$k$$-algebras $$A$$ such that (for some inverse system of open ideals $$\{I_n\}$$)
1. $$A\cong \varprojlim_n A/I_n$$
2. $$A/I_n$$ is finitely generated as a $$k$$-algebra.
Then the formal spectrum $$\mathrm{Spf}(A)$$ of $$A$$ is defined for a $$k$$-algebra $$R$$ as the set of continuous maps $$\mathrm{Spf}(A)(R)$$ (where $$R$$ is equipped with the discrete topology). Even better it is $$\mathrm{Spf}(A)(R)\cong \varinjlim_n \mathrm{Spf}(A/I_n)(R)$$ Back to considering Hartshorne: A closed subscheme $$Y\subset X$$ gives rise to an ideal sheaf $$\mathcal{I}\subset \mathcal{O}_X$$. If we consider the case that $$X=\mathrm{Spec}A$$ is affine then $$\mathcal{I}$$ is simply an ideal of $$A$$ and $$Y\cong \mathrm{Spec}{(A/\mathcal{I})}$$. We then place the $$\mathcal{I}$$-adic topology on $$A$$, check that it satisfies 1 and 2 above, and then the formal completion as given in Hartshorne exactly becomes $$\mathrm{Spf}(A)(R)\cong \varinjlim_n \mathrm{Spf}(A/\mathcal{I}^n)(R)$$ As @DonuArapura commented the case presented in Hartshorne is an example of a more general definition where rather than considering adic topological rings, we consider admissible topological rings.