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?)

  • 7
    $\begingroup$ 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. $\endgroup$ – Donu Arapura May 9 '20 at 15:38
  • 2
    $\begingroup$ @DonuArapura: see Tag 0AHY. $\endgroup$ – R. van Dobben de Bruyn May 9 '20 at 16:00

Disclaimer: I am far from an expert. Feel free to ask for more details!

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.