Example 7

For a noetherian local ring $A$ to be a Macaulay ring, it is necessary and sufficient that it possess an ideal of definition generated by an $A$-regular sequence: this follows from Prop. 3 of No. 3, and from the fact that an $A$-regular sequence of elements of $m_A$ generates an ideal of definition if and only if it is of length $\dim(A)$ (VIII, § 3, No. 2, Th. 1 and cor. to Prop. 3). In particular, every regular noetherian local ring is a Macaulay ring (VIII, § 5, No. 2, Th. 1). More generally, the quotient of a regular noetherian local ring $A$ by an ideal generated by an $A$-regular sequence is a Macaulay ring (Example 6).