Editing Galois connection (section)

====Algebraic geometry====
In [[algebraic geometry]], the relation between sets of [[polynomial]]s and their zero sets is an antitone Galois connection.

Fix a [[natural number]] {{mvar|n}} and a [[field (mathematics)|field]] {{mvar|K}} and let {{mvar|A}} be the set of all subsets of the [[polynomial ring]] {{math|''K''[''X''<sub>1</sub>, ..., ''X<sub>n</sub>'']}} ordered by inclusion ⊆, and let {{mvar|B}} be the set of all subsets of {{math|''K''<sup>&hairsp;''n''</sup>}} ordered by inclusion ⊆. If {{mvar|S}} is a set of polynomials, define the [[Algebraic geometry#Affine varieties|variety]] of zeros as

:<math>V(S) = \{x \in K^n : f(x) = 0 \mbox{ for all } f \in S\},</math>

the set of common [[root of a polynomial|zeros]] of the polynomials in {{mvar|S}}. If {{mvar|U}} is a subset of {{math|''K''<sup>&hairsp;''n''</sup>}}, define {{math|''I''(''U''&hairsp;)}} as the [[ideal (ring theory)|ideal]] of polynomials vanishing on {{mvar|U}}, that is

:<math>I(U) = \{f \in K[X_1,\dots,X_n] : f(x) = 0 \mbox{ for all } x \in U\}.</math>

Then {{mvar|V}} and ''I'' form an antitone Galois connection.

The closure on {{math|''K''<sup>&hairsp;''n''</sup>}} is the closure in the [[Zariski topology]], and if the field {{mvar|K}} is [[Algebraically closed field|algebraically closed]], then the closure on the polynomial ring is the [[Radical of an ideal|radical]] of ideal generated by {{mvar|S}}.

More generally, given a [[commutative ring]] {{mvar|R}} (not necessarily a polynomial ring), there is an antitone Galois connection between radical ideals in the ring and Zariski closed subsets of the [[Algebraic geometry#Affine varieties|affine variety]] {{math|[[Spectrum of a ring|Spec]](''R'')}}.

More generally, there is an antitone Galois connection between ideals in the ring and [[subscheme]]s of the corresponding [[Algebraic geometry#Affine varieties|affine variety]].