Formal scheme

In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. For this reason, formal schemes frequently appear in topics such as deformation theory.

Definition

Formal schemes are usually defined only in the Noetherian case. While there have been several definitions of non-Noetherian formal schemes, these encounter technical problems. Consequently we will make the assumption that all rings are Noetherian.

All rings will be assumed to be commutative and with unit. Let "A" be a (Noetherian) topological ring, that is, a ring "A" which is a topological space such that the operations of addition and multiplication are continuous. "A" is linearly topologized if zero has a base consisting of ideals. An ideal of definition $mathcal\{J\}$ for a linearly topologized ring is an open ideal such that for every open neighborhood "V" of 0, there exists a positive integer "n" such that $mathcal\{J\}^n\; subseteq\; V$. A linearly topologized ring is preadmissible if it admits an ideal of definition, and it is admissible if it is also complete. (In the terminology of Bourbaki, this is "complete and separated".)

Assume that "A" is an admissible, and let $mathcal\{J\}$ be an ideal of definition. A prime ideal is open if and only if it contains $mathcal\{J\}$. The set of open prime ideals of "A", or equivalently the set of prime ideals of $A/mathcal\{J\}$, is the underlying topological space of the formal spectrum of "A", denoted Spf "A". Spf "A" has a structure sheaf which is defined using the structure sheaf of the spectrum of a ring. Let $mathcal\{J\}\_lambda$ be a neighborhood basis for zero consisting of ideals of definition. All the spectra of $A/mathcal\{J\}\_lambda$ have the same underlying topological space but a different structure sheaf. The structure sheaf of Spf "A" is the projective limit $varprojlim\_lambda\; mathcal\{O\}\_\{\; ext\{Spec\}\; A/mathcal\{J\}\_lambda\}$.

It can be shown that if "f" ∈ "A" and "D"_"f" is the set of all open prime ideals of "A" not containing "f", then $mathcal\{O\}\_\{\; ext\{Spf\}\; A\}(D\_f)\; =\; widehat\{A\_f\}$, where $widehat\{A\_f\}$ is the completion of the localization "A"_"f".

Finally, a Noetherian formal scheme is a topologically ringed space $(mathfrak\{X\},\; mathcal\{O\}\_\{mathfrak\{X)$ (that is, a ringed space whose sheaf of rings is a sheaf of topological rings) such that each point of $mathfrak\{X\}$ admits an open neighborhood isomorphic (as topologically ringed spaces) to the formal spectrum of a noetherian ring.

References

*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1960
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : I. Le langage des schémas | journal = Publications Mathématiques de l'IHÉS
volume = 4
pages = 5–228
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1960__4_

See also

*Deformation theory
*Scheme (mathematics)
*Spectrum of a ring