Perfect ideal

In commutative algebra, a perfect ideal is a proper ideal ${\displaystyle I}$in a Noetherian ring ${\displaystyle R}$such that its grade equals the projective dimension of the associated quotient ring.

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

A perfect ideal is unmixed.

For a regular local ring ${\displaystyle R}$a prime ideal ${\displaystyle I}$is perfect if and only if ${\displaystyle R/I}$is Cohen-Macaulay.

The notion of perfect ideal was introduced in 1913 by Francis Sowerby Macaulay 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 point out, Macaulay's original definition of perfect ideal ${\displaystyle I}$coincides with the modern definition when ${\displaystyle I}$is a homogeneous ideal in a polynomial ring, but may differ otherwise. Macaulay used Hilbert functions to define his version of perfect ideals.