Noetherian room

The Noetherian topological space , named after Emmy Noether , is a mathematical term from the branch of topology . It is motivated by the algebraic concept of Noether's ring and is mainly used in algebraic geometry .

definition

If one considers open sets of a topological space in analogy to the ideals of a ring , the following definition is obvious with a view to the concept of Noether's ring:

A topological space is called Noetherian if every ascending chain of open sets becomes stationary, that is: If there is a family of open sets, there is one with for all . ${\ displaystyle U_ {1} \ subset U_ {2} \ subset \ ldots}$ ${\ displaystyle n_ {0} \ in \ mathbb {N}}$ ${\ displaystyle U_ {n} = U_ {n_ {0}}}$ ${\ displaystyle n \ geq n_ {0}}$

As in algebra, a simple argument shows:

A topological space is Noetherian if and only if there is a maximum condition for open sets , that is: Every non-empty family of open sets contains a maximal element .

Since the closed sets are exactly the complements of open sets, one has:

A topological space is Noetherian if and only if every descending chain of closed sets becomes stationary, that is: If there is a family of closed sets, there is one with for all . ${\ displaystyle A_ {1} \ supset A_ {2} \ supset \ ldots}$ ${\ displaystyle n_ {0} \ in \ mathbb {N}}$ ${\ displaystyle A_ {n} = A_ {n_ {0}}}$ ${\ displaystyle n \ geq n_ {0}}$

A topological space is Noetherian if and only if a minimal condition for closed sets holds, that is: Every non-empty family of closed sets contains a minimal element .

Examples

Spaces with finite topologies, especially topological spaces with a finite basic set, are Noetherian.
The affine space over a body with the Zariski topology a Noetherian space. ${\ displaystyle k ^ {n}}$ ${\ displaystyle k}$

{\ displaystyle \ mathbb {R}}

with the Euclidean topology is not Noetherian, because the open intervals form an ascending sequence of open sets that does not become stationary.

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meaning

The Zariski topology is usually viewed on the spectrum of a ring . It is easy to show that the spectrum of a Noetherian commutative ring is a Noetherian topological space. Since affine varieties correspond to the radical ideals in the ring of polynomials in finitely many variables over the coordinate field ( Hilbert's zero theorem ), and this ring is Noetherian ( Hilbert's basis theorem ), one obtains that affine varieties with the Zariski topology are Noetherian. Hence, this term plays a role in algebraic geometry in which such varieties are studied.

application

A Noetherian topological space has only finitely many irreducible components .

In particular, an affine variety consists of finitely many irreducible components.

Since the simple proof clarifies the typical Noetherian inference, it should be reproduced here briefly: Let be the set of all closed subsets that are not a finite union of irreducible sets. If it is assumed that this set is not empty, it contains a minimal element because of the minimum condition for closed sets . As an element of, this can not be irreducible, so it is a union of two true closed sets and . As is minimal, are and not and therefore finite union of irreducible amounts. But then there is also a finite union of irreducible sets, which is a contradiction in terms. Therefore it is empty, in particular space itself, is a finite union of irreducible sets, what was to be shown. ${\ displaystyle {\ mathcal {A}}}$ ${\ displaystyle A_ {0}}$ ${\ displaystyle {\ mathcal {A}}}$ ${\ displaystyle A_ {1}}$ ${\ displaystyle A_ {2}}$ ${\ displaystyle A_ {0}}$ ${\ displaystyle A_ {1}}$ ${\ displaystyle A_ {2}}$ ${\ displaystyle {\ mathcal {A}}}$ ${\ displaystyle A_ {0} = A_ {1} \ cup A_ {2}}$ ${\ displaystyle A_ {0} \ in {\ mathcal {A}}}$ ${\ displaystyle {\ mathcal {A}}}$

compactness

If compactness is defined by the covering property and if the Hausdorff property is dispensed with , some authors then also speak of quasi-compact spaces, the following applies:

Every Noetherian space is quasi-compact.
A topological space is Noetherian if and only if every subset is quasi-compact with the relative topology .

Other properties

Every subspace of a Noetherian space is again Noetherian.
If the topological space is a union of the subspaces and if each is Noetherian, then it is also Noetherian. ${\ displaystyle X}$ ${\ displaystyle X_ {1}, \ ldots, X_ {n}}$ ${\ displaystyle X_ {i}}$ ${\ displaystyle X}$

Individual evidence

^ Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6 , definition I.2.13
^ Ernst Kunz: Introduction to commutative algebra and algebraic geometry , Vieweg (1980), ISBN 3-528-07246-6 , sentence I.2.14
^ IG MacDonald: Algebraic Geometry, Introduction to Schemes , WA Benjamin Inc. (1968), Chapter 2: Noetherian Spaces
↑ IG MacDonald: Algebraic Geometry, Introduction to Schemes , WA Benjamin Inc. (1968), Theorem (2.2) (ii)
^ IG MacDonald: Algebraic Geometry, Introduction to Schemes , WA Benjamin Inc. (1968), Theorem (2.2) (iii)

[1] Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6 , definition I.2.13

[2] Ernst Kunz: Introduction to commutative algebra and algebraic geometry , Vieweg (1980), ISBN 3-528-07246-6 , sentence I.2.14

[3] IG MacDonald: Algebraic Geometry, Introduction to Schemes , WA Benjamin Inc. (1968), Chapter 2: Noetherian Spaces

[4] IG MacDonald: Algebraic Geometry, Introduction to Schemes , WA Benjamin Inc. (1968), Theorem (2.2) (ii)

[5] IG MacDonald: Algebraic Geometry, Introduction to Schemes , WA Benjamin Inc. (1968), Theorem (2.2) (iii)