Noether normalization theorem: Difference between revisions

Revision as of 21:13, 2 February 2008

Let $k$ be a field and $R$ a finitely generated $k$ -algebra. Then:

Weak version: There exists a subring $S$ of $R$ such that $S ≅ k [x_{1}, x_{2}, \dots, x_{r}]$ and $R$ is a finitely generated $S$ -module
Strong version: Suppose $P_{0} \leq P_{1} \leq \dots P_{r}$ is an ascending chain of prime ideals, such that no further prime ideals can be and $P_{i} \cap S = (x_{1}, x_{2}, \dots, x_{i})$ Then we can choose $S$ as above, with the further constraint that $P_{i} \cap S = (x_{1}, \dots, x_{i})$ .

weak nullstellensatz for arbitrary fields follows very directly from the weak version of the Noether normalization theorem.

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 ==Statement==
-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>.
+Let <math>k</math> be a [[field]] and <math>R</math> a finitely generated <math>k</math>-algebra. Then:
+* ''Weak version'': There exists a subring <math>S</math> of <math>R</math> such that <math>S \cong k[x_1, x_2, \ldots, x_r]</math> and <math>R</math> is a finitely generated <math>S</math>-module
+* ''Strong version'': Suppose <math>P_0 \le P_1 \le \ldots P_r</math> is an ascending chain of [[prime ideal]]s, such that no further prime ideals can be  and <math>P_i \cap S = (x_1, x_2, \ldots, x_i)</math> Then we can choose <math>S</math> as above, with the further constraint that <math>P_i \cap S = (x_1,\ldots,x_i)</math>.
 ==Applications==
-* [[Hilbert's nullstellensatz]]
+* [[weak nullstellensatz for arbitrary fields]] follows very directly from the weak version of the Noether normalization theorem.