Krull's height theorem: Difference between revisions

Latest revision as of 16:26, 12 May 2008

Statement

Let $R$ be a Noetherian commutative unital ring and $x_{1}, x_{2}, \dots, x_{c}$ be elements in $R$ . Let $P$ be minimal among primes containing all the $x_{i}$ s. Then, the codimension of $P$ is at most $c$ .

There is also a converse of this statement viz converse of Krull's height theorem.

Proof

Starting assumptions

Replacing $R$ by $R_{P}$ if necessary, we may assume that $R$ is a local ring with unique maximal ideal $P$ .

In particular, we see that the ring $R / (x_{1}, x_{2}, \dots, x_{c})$ is a local Artinian ring with unique maximal ideal $P / (x_{1}, x_{2}, \dots, x_{c})$ , hence $P$ is nilpotent modulo $(x_{1}, x_{2}, \dots, x_{c})$ .

Main proof

Supose $P_{1}$ is a prime contained in $P$ , with no primes between. Then, it suffices to show, inductively, that $P_{1}$ is minimal over an ideal generated by $c - 1$ elements.

@@ Line 3: / Line 3: @@
 Let <math>R</math> be a [[Noetherian ring|Noetherian]] [[commutative unital ring]] and <math>x_1, x_2, \ldots, x_c</math> be elements in <math>R</math>. Let <math>P</math> be minimal among primes containing all the <math>x_i</math>s. Then, the codimension of <math>P</math> is at most <math>c</math>.
-There is also a converseof this statement viz [[converse of Krull's height theorem]].
+There is also a converse of this statement viz [[converse of Krull's height theorem]].
+==Proof==
+===Starting assumptions===
+Replacing <math>R</math> by <math>R_P</math> if necessary, we may assume that <math>R</math> is a [[local ring]] with unique [[maximal ideal]] <math>P</math>.
+In particular, we see that the ring <math>R/(x_1,x_2,\ldots,x_c)</math> is a [[local Artinian ring]] with unique maximal ideal <math>P/(x_1,x_2,\ldots,x_c)</math>, hence <math>P</math> is nilpotent modulo <math>(x_1,x_2,\ldots,x_c)</math>.
+===Main proof===
+Supose <math>P_1</math> is a prime contained in <math>P</math>, with no primes between. Then, it suffices to show, inductively, that <math>P_1</math> is minimal over an ideal generated by <math>c-1</math> elements.