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Perfect ideal

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

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

A perfect ideal is unmixed.

For a regular local ring R {\displaystyle R} a prime ideal I {\displaystyle I} is perfect if and only if R / I {\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 I {\displaystyle I} coincides with the modern definition when I {\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.

References

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