Exercise 24, § 7

\* Let A be a noetherian local ring, Ā its completion. In this exercise we propose to prove the equivalence of the properties: (i) A is regular ; (ii) Ā is equidimensional and without immersed prime ideals, and e(Ā) = 1 ; (iii) Ā is integral and e(Ā) = 1. Prove the implications (i) ⇒ (ii) ⇒ (iii). Notice that (iii) entails the equality e(Ā) = 1 and that, in order to prove (iii) ⇒ (i), it suffices to prove that Ā is regular. We may therefore, in order to prove (iii) ⇒ (i), assume A complete, which we shall do henceforth. We shall first treat the case where the residue field κ_A possesses an infinite number of elements. We proceed by induction on the dimension of A. Examine the case dim(A) = 0, and suppose henceforth dim(A) > 0. There then exists a superficial element x ∈ A (§ 7, no 5, remark 4). By virtue of the catenary property (exercise 22) and § 3, no 1, prop. 1, for every minimal ideal p among those containing xA, one has dim(A/p) = dim(A/xA). Since e(A/xA) = 1 (§ 7, no 5, th. 1), there exists a single minimal prime ideal p among those containing xA, and one has xA_p = pA_p, e(A/p) = 1 (exercise 23). By the induction hypothesis, A/p is regular, hence integrally closed (§ 5, no 2, cor. 1 to th. 1). By Hironaka's lemma (exercise 22), one deduces that p = xA. Therefore A/xA is regular and since A is integral, A is regular (§ 5, no 3, prop. 2).

Suppose now that κ_A is a finite field. Prove that it suffices to prove that the extension A]X[ (IX, App., no 2) is regular. Notice that A and consequently A]X[ is isomorphic to a quotient of a regular ring (IX, § 2, no 5, th. 3) and e(A]X[) = 1. According to the theory of Macaulay rings (chapter X) the completion Ā]X[ is equidimensional and without immersed prime ideals. Deduce that Ā]X[ is integral (exercise 23) and that e(Ā]X[) = 1, hence that Ā]X[ is regular. Conclude. \*