Perfect ideal
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In commutative algebra, a perfect ideal is a proper ideal [math]\displaystyle{ I }[/math] in a Noetherian ring [math]\displaystyle{ R }[/math] such that its grade equals the projective dimension of the associated quotient ring.[1]
[math]\displaystyle{ \textrm{grade}(I)=\textrm{proj}\dim(R/I). }[/math]
A perfect ideal is unmixed.
For a regular local ring [math]\displaystyle{ R }[/math] a prime ideal [math]\displaystyle{ I }[/math] is perfect if and only if [math]\displaystyle{ R/I }[/math] is Cohen-Macaulay.
The notion of perfect ideal was introduced in 1913 by Francis Sowerby Macaulay[2] in connection to what nowadays is called a Cohen-Macaulay ring, but for which Macaulay did not have a name for yet. As Eisenbud and Gray[3] point out, Macaulay's original definition of perfect ideal [math]\displaystyle{ I }[/math] coincides with the modern definition when [math]\displaystyle{ I }[/math] is a homogeneous ideal in polynomial ring, but may differ otherwise. Macaulay used Hilbert functions to define his version of perfect ideals.
References
- ↑ Matsumura, Hideyuki (1987). Commutative Ring Theory. Cambridge: Cambridge University Press. p. 132. ISBN 9781139171762. https://www.cambridge.org/core/books/commutative-ring-theory/02819830750568B06C16E6199F3562C1.
- ↑ Macaulay, F. S. (1913). "On the resolution of a given modular system into primary systems including some properties of Hilbert numbers". Math. Ann. 74 (1): 66–121. doi:10.1007/BF01455345. https://link.springer.com/article/10.1007/BF01455345. Retrieved 2023-08-06.
- ↑ Eisenbud, David; Gray, Jeremy (2023). "F. S. Macaulay: From plane curves to Gorenstein rings". Bull. Amer. Math. Soc. 60 (3): 371–406. doi:10.1090/bull/1787. https://www.ams.org/journals/bull/2023-60-03/S0273-0979-2023-01787-4/. Retrieved 2023-08-06.
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