Noether normalization theorem: Difference between revisions

Revision as of 00:05, 8 January 2008

Statement

Let $k$ be a field and $R$ a finitely generated $k$ -algebra. Let $P_{r} \geq P_{r - 1} \geq \dots \geq P_{0}$ be a chain of descending ]]prime ideal]]s, maximal in the sense that no further prime ideals can be inserted in the chain. Then there exists a subring $S$ of $R$ with $S ≅ k [x_{1}, x_{2}, \dots, x_{r}]$ sich that $R$ is a finitely generated $S$ -module and $P_{i} \cap S = (x_{1}, x_{2}, \dots, x_{i})$ .

Applications

Hilbert's nullstellensatz

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 ==Statement==
-Let <math>k</math> be a [[field]] and <math>R</math> a finitely generaetd <math>k</math>-algebra. Let <math>P_r \ge P_{r-1} \ge \ldots \ge P_0</math> be a chain of descending ]]prime ideal]]s, maximal in the sense that no further prime ideals can be inserted in the chain. Then there exists a subring <math>S</math> of <math>R</math> with <math>S \cong k[x_1, x_2, \ldots, x_r]</math> sich that <math>R</math> is a finitely generated <math>S</math>-module and <math>P_i \cap S = (x_1, x_2, \ldots, x_i)</math>.
+Let <math>k</math> be a [[field]] and <math>R</math> a finitely generated <math>k</math>-algebra. Let <math>P_r \ge P_{r-1} \ge \ldots \ge P_0</math> be a chain of descending ]]prime ideal]]s, maximal in the sense that no further prime ideals can be inserted in the chain. Then there exists a subring <math>S</math> of <math>R</math> with <math>S \cong k[x_1, x_2, \ldots, x_r]</math> sich that <math>R</math> is a finitely generated <math>S</math>-module and <math>P_i \cap S = (x_1, x_2, \ldots, x_i)</math>.
 ==Applications==
 * [[Hilbert's nullstellensatz]]