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| | In [[commutative algebra]], a field of mathematics, the '''monomial conjecture''' of [[Melvin Hochster]] says the following:<ref>http://www5a.biglobe.ne.jp/~tomari/hamana/roberts.pdf</ref> |
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| | Let ''A'' be a [[Noetherian ring|Noetherian]] [[local ring]] of [[Krull dimension]] ''d'' and let ''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub> be a system of parameters for ''A'' (so that ''A''/(''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>) is an [[Artinian ring]]. Then for all positive integers ''t'', we have |
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| | : <math> x_1^t \cdots x_d^t \not\in (x_1^{t+1},\dots,x_d^{t+1}). \, </math> |
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| | The statement can relatively easily be shown in [[characteristic zero]]. |
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| | == References == |
| | <!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically --> |
| | {{Reflist}} |
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| | {{DEFAULTSORT:Monomial Conjecture}} |
| | [[Category:Commutative algebra]] |
| | [[Category:Conjectures]] |
Revision as of 14:05, 27 January 2014
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In commutative algebra, a field of mathematics, the monomial conjecture of Melvin Hochster says the following:[1]
Let A be a Noetherian local ring of Krull dimension d and let x1, ..., xd be a system of parameters for A (so that A/(x1, ..., xd) is an Artinian ring. Then for all positive integers t, we have
The statement can relatively easily be shown in characteristic zero.
References
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