Hilbert's basic set

The Hilbert's base rate (after David Hilbert ) is a basic set of algebraic geometry , he combines various finiteness conditions.

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. For more details, see Commutative Algebra .

formulation

Hilbert's basic theorem states in its general form:

• If a ring is Noetherian , then every polynomial ring with coefficients is Noetherian. ${\displaystyle A}$ ${\displaystyle A[X_{1},\ldots ,X_{n}]}$ ${\displaystyle A}$

Since the algebras of finite type are exactly the quotient rings of polynomial rings, this statement is equivalent to:

• If a Noetherian ring and an algebra are of finite type, then also Noetherian. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$

The version proved by Hilbert (except for the language usage) in 1888 deals with the special case of the body :

• The polynomial ring over a body is Noetherian. ${\displaystyle k[X_{1},\ldots ,X_{n}]}$ ${\displaystyle k}$

Inference

An important application is the following statement: If a subset of one for a body is described by an infinite number of polynomial equations, then a finite number of them are sufficient. ${\displaystyle k^{n}}$ ${\displaystyle k}$

More formal: Let be an arbitrary set of polynomials with the set of common zeros (also called a vanishing set of ): ${\displaystyle {\mathcal {F}}\subset k[X_{1},\ldots ,X_{n}]}$ ${\displaystyle {\mathcal {F}}}$

${\displaystyle V=V({\mathcal {F}})=\{(x_{1},\ldots ,x_{n})\in k^{n};\,f(x_{1},\ldots ,x_{n})=0{\mbox{ für alle }}f\in {\mathcal {F}}\}}$

Then there are finally many , so that is true ${\displaystyle f_{1},\ldots ,f_{m}\in {\mathcal {F}}}$

${\displaystyle V=\{(x_{1},\ldots ,x_{n})\in k^{n};\,f_{i}(x_{1},\ldots ,x_{n})=0{\mbox{ für alle }}i=1,\ldots ,m\}}$.

This is the hardest part of proving the statement that the Zariski topology is a topology.

See also

• algebraic variety
• Groebner base

Web links

• VI Danilov, Hilbert's basis theorem, Encyclopedia of Mathematics, Springer

Individual evidence

1. ↑ Hilbert, Ueber die theory der Algebraischenformen, Mathematische Annalen, Volume 36, 1890, pp. 473-534
2. ↑ Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6 , Chapter I, §2, Theorem 2.3 (very brief proof)
3. ^ BL van der Waerden : Algebra II, Springer-Verlag (1967), ISBN 3-540-03869-8 , §115