Determinantal ideal theorem - Revision history

Vipul: 2 revisions

2008-05-12T16:19:39Z

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Vipul: /* Statement */

2007-08-10T08:37:43Z

Statement

← Older revision		Revision as of 08:37, 10 August 2007
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	==Statement==		==Statement==

	Let <math>M</math> be a <math>p \times q</math> matrix with entries over a [[Noetherian ring]] <math>R</math>. Denote by <math>I_k(M)</math> the ideal generated by the <math>k \times k</math> minors of <math>M</math>. Then, the [[codimension of a prime ideal\|codimension of any prime ideal]] minimal over <math>~~I_p~~(M)</math> is at most <math>q - p + 1</math>.		Let <math>M</math> be a <math>p \times q</math> matrix with entries over a [[Noetherian ring]] <math>R</math>. Denote by <math>I_k(M)</math> the ideal generated by the <math>k \times k</math> minors of <math>M</math>. Then, the [[codimension of a prime ideal\|codimension of any prime ideal]] minimal over <math>I_k(M)</math> is at most <math>(p - k + 1)(q - k + 1)</math>.

	The case <math>p = 1</math> yields [[Krull's height theorem]] and the case <math>p = q = 1</math> yields [[Krull's principal ideal theorem]].		The case <math>k = p = 1</math> yields [[Krull's height theorem]] and the case <math>p = q = k = 1</math> yields [[Krull's principal ideal theorem]].

Vipul at 08:36, 10 August 2007

2007-08-10T08:36:45Z

New page

==History==

The result was proved for polynomial rings by [[Macaulay]] and for arbitrary Noetherian rings by [[Eagon]].

==Statement==

Let <math>M</math> be a <math>p \times q</math> matrix with entries over a [[Noetherian ring]] <math>R</math>. Denote by <math>I_k(M)</math> the ideal generated by the <math>k \times k</math> minors of <math>M</math>. Then, the [[codimension of a prime ideal|codimension of any prime ideal]] minimal over <math>I_p(M)</math> is at most <math>q - p + 1</math>.

The case <math>p = 1</math> yields [[Krull's height theorem]] and the case <math>p = q = 1</math> yields [[Krull's principal ideal theorem]].