Vipul: Vipul moved page Equivalence of definitions of Cohen-Macaulay to Equivalence of definitions of Cohen-Macaulay ring

2013-07-18T04:34:37Z

Vipul moved page Equivalence of definitions of Cohen-Macaulay to Equivalence of definitions of Cohen-Macaulay ring

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2008-05-12T16:19:55Z

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Vipul: New page: {{definition equivalence|Cohen-Macaulay ring}} ==The definitions that we have to prove as equivalent== The ring is assumed to be Noetherian. # For every [[maximal id...

2008-03-09T19:47:37Z

New page: {{definition equivalence|Cohen-Macaulay ring}} ==The definitions that we have to prove as equivalent== The ring is assumed to be Noetherian. # For every [[maximal id...

New page

{{definition equivalence|Cohen-Macaulay ring}}

==The definitions that we have to prove as equivalent==

The ring is assumed to be [[Noetherian ring|Noetherian]].

# For every [[maximal ideal]], the [[depth of an ideal|depth]] equals the [[codimension of an ideal|codimension]]
# For every [[prime ideal]], the [[depth of an ideal|depth]] equals the [[codimension of an ideal|codimension]]
# For every [[ideal]], the [[depth of an ideal|depth]] equals the [[codimension of an ideal|codimension]]

Note that (3) implies (2) implies 1, so we need to show that (1) implies (3). In other words, we need to show that assuming depth = codimension for ''maximal'' ideals is enough to show that depth = codimension for ''all'' ideals.

Equivalence of definitions of Cohen-Macaulay ring - Revision history

Vipul: Vipul moved page Equivalence of definitions of Cohen-Macaulay to Equivalence of definitions of Cohen-Macaulay ring

Vipul: 1 revision

Vipul: New page: {{definition equivalence|Cohen-Macaulay ring}} ==The definitions that we have to prove as equivalent== The ring is assumed to be Noetherian. # For every [[maximal id...