Perfect ideal explained

I in a Noetherian ring R such that its grade equals the projective dimension of the associated quotient ring.^[1] rm{grade}(I)= rm{proj}\dim(R/I).

A perfect ideal is unmixed.

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.

Notes and References

1. Book: Matsumura, Hideyuki . Hideyuki Matsumura . 1987 . Commutative Ring Theory . Cambridge . Cambridge University Press . 132 . 9781139171762.
2. Macaulay . F. S. . Francis Sowerby Macaulay . 1913 . On the resolution of a given modular system into primary systems including some properties of Hilbert numbers . Math. Ann. . 74 . 1 . 66–121 . 10.1007/BF01455345 . 123229901 . 2023-08-06. subscription .
3. Eisenbud . David . David Eisenbud . Gray . Jeremy . 2023 . F. S. Macaulay: From plane curves to Gorenstein rings . Bull. Amer. Math. Soc. . 60 . 3 . 371–406 . 10.1090/bull/1787 . 2023-08-06. free .