- Formal scheme
In

mathematics , specifically inalgebraic 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 asdeformation 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 atopological 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 ofBourbaki , 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 thespectrum 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, aringed 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

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