Remark 2

One says that a noetherian ring $A$ is a complete intersection ring if, for every maximal ideal $m$ of $A$, the complete local ring $\widehat{A_m}$ is isomorphic to the quotient of a complete regular local noetherian ring by a completely secant ideal. It results from Cor. 3 above and from Cor. 2 of Prop. 12 of § 3, No. 8 that such a ring is a Gorenstein ring.