Perfect ideal

In commutative algebra, a perfect ideal is a proper ideal $I$ in a Noetherian ring $R$ such that its grade equals the projective dimension of the associated quotient ring.^[1]

${\textrm {grade}}(I)={\textrm {proj}}\dim(R/I).$

A perfect ideal is unmixed.

For a regular local ring $R$ a prime ideal $I$ is perfect if and only if $R/I$ 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 $I$ coincides with the modern definition when $I$ 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.
^ 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. S2CID 123229901. 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. Retrieved 2023-08-06.

[Matsumura1-1] Matsumura, Hideyuki (1987). Commutative Ring Theory. Cambridge: Cambridge University Press. p. 132. ISBN 9781139171762.

[Macauley-2] 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. S2CID 123229901. Retrieved 2023-08-06.

[EisenbudGray-3] 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. Retrieved 2023-08-06.

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