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2008-05-12T16:19:16Z

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Vipul at 20:46, 9 March 2008

2008-03-09T20:46:26Z

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	==Definitions used==		==Definitions used==

			===Cohen-Macaulay ring===
			{{further\|[[Cohen-Macaulay ring]]}}

			A [[Noetherian ring]] is termed Cohen-Macaulay if for every ideal, the [[depth of an ideal\|depth]] equals the [[codimension of an ideal\|codimension]].

	==Facts used==		==Facts used==

			* [[Cohen-Macaulay is strongly local]]: A ring is Cohen-Macaulay iff its localization at every maximal ideal is Cohen-Macaulay.
			* [[Krull's principal ideal theorem]]
			==Proof==

			''Given'': A [[Cohen-Macaulay ring]] <math>R</math>, and the polynomial ring <math>R[x]</math>

			''To prove'': <math>R[x]</math> is a Cohen-Macaulay ring

			===Reduction to a local case===

			By the strongly local nature of the Cohen-Macaulay property, it suffices to show that for every maximal ideal <math>P</math> of <math>R[x]</math>, the localization <math>R[x]_P</math> is a Cohen-Macaulay ring. Let <math>Q</math> be the [[contraction]] of <math>P</math> to <math>R</math> (in other words, <math>Q = P \cap R</math>). Clearly <math>Q</math> is a prime ideal in <math>R</math> (because [[primeness is contraction-closed]]).

			A little thought reveals that:

			<math>R[x]_P = R_Q[x]_{P_Q}</math>

			where <math>R_Q</math> denotes localization at the prime ideal <math>Q</math>, and <math>P_Q</math> denotes the ideal generated by <math>P</math> in <math>R_Q[x]</math>. Thus, by passing to <math>R_P</math>, we may without loss of generality assume that <math>R</math> is a local ring with maximal ideal <math>Q</math>. In particular, we may asume that <math>R/Q</math> is a [[field]].

			Note that for this step of the reduction, we're using the fact that since <math>R</math> is Cohen-Macaulay, so is <math>R_Q</math>.

			===The depth increases by at least one===

			Notation as before: <math>R</math> is a local Cohen-Macaulay ring, <math>Q</math> is its unique maximal ideal, and <math>P</math> is an ideal of <math>R[x]</math> that contracts to <math>Q</math>.

			Now, consider <math>R[x]/QR[x]</math>. This is the same as <math>(R/Q)[x]</math>, which is a [[polynomial ring over a field]], hence a [[principal ideal]]. The image <math>P/QR[x]</math> is thus a principal ideal in this field, and since it is also prime, it must be generated by a monic polynomial, say <math>f</math>. Pulling back, we see that <math>P</math> is generated by <math>Q</math> and a monic polynomial (any pullback of <math>f</math>).

			Now, any regular sequence in <math>Q</math> in <math>R</math> continues to remain a regular sequence inside the ring <math>R[x]</math>. Augmenting with <math>f</math> at the end, we get a regular sequence for <math>P</math> in <math>R</math> (we use the fact that by our construction <math>f</math> is not a zero divisor in the quotient). Thus, the depth of <math>P</math> in <math>R[x]</math> is at least 1 more than the depth of <math>Q</math> in <math>R</math>.

			===The codimension increases by at most one===

			This follows from [[Krull's principal ideal theorem]].

			===Putting the pieces together===

			* Since <math>R</math> is Cohen-Macaulay, the depth and codimension of <math>Q</math> are equal
			* The codimension of <math>P</math> is at most one more than this, and the depth of <math>P</math> is at least one more than this
			* Hence the depth of <math>P</math> is at least equal to the codimension of <math>P</math>
			* On the other hand, we know that the depth of <math>P</math> is at most equal to the codimension of <math>P</math>, for any <math>P</math>
			* Hence, the depth and codimension of <math>P</math> are equal

			==References==

			===Textbook references===
			* {{booklink\|Eisenbud}}, Page 456-457

Vipul: New page: {{curing metaproperty satisfaction}} ==Statement== ===Property-theoretic statement=== The property of commutative unital rings of being Cohen-Macaulay satisfi...

2008-03-09T20:02:10Z

New page: {{curing metaproperty satisfaction}} ==Statement== ===Property-theoretic statement=== The property of commutative unital rings of being Cohen-Macaulay satisfi...

New page

{{curing metaproperty satisfaction}}
==Statement==

===Property-theoretic statement===

The [[property of commutative unital rings]] of being [[Cohen-Macaulay ring|Cohen-Macaulay]] satisfies the [[metaproperty of commutative unital rings]] of being [[polynomial-closed property of commutative unital rings|polynomial-closed]].

===Verbal statement===

The polynomial ring in one variable over a Cohen-Macaulay ring, is again Cohen-Macaulay.

==Definitions used==

==Facts used==

Cohen-Macaulay is polynomial-closed - Revision history

Vipul: 2 revisions

Vipul at 20:46, 9 March 2008

Vipul: New page: {{curing metaproperty satisfaction}} ==Statement== ===Property-theoretic statement=== The property of commutative unital rings of being Cohen-Macaulay satisfi...