Conference on Commutative Algebra by James W. Brewer, Edgar A. Rutter

By James W. Brewer, Edgar A. Rutter

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X n]. r(Ql[X2, .... Xn] ). assumption implies that r(Q) = r(Ql[X2, X n]) ~< (n-l) + r(Ql[X 2 . . . X n]) ~< r(P[X I . . . Since i + r(Q/QI[X2, r(P[X I . . . as Moreover, x n]). Combining these Xn] ) + x n]) + n. x n]) ~< r(Q/p[x I . . . x n]) + r(Q/P[X I . . . the induction .... Xn] ) + results we have that r(Q) = i + r(P[X I . . . r(Q/QI[X 2 . . . Xn ] If P[X I] c QI' then Lemma i implies that .... Xn] ) = i + r(P[Xl, r(Q/QI[X 2 . . . 9 x n]). x n] ), r(Q) ~< But the reverse of this last inequality is clear, so r(Q) = r(P[X I .

That this is not the case. We shall show, In fact, we exhibit in Example 3 of Section 2 a strong S-ring R such that dim (R[XI, X 2] ) > 2 + dim (R). As Robert Gilmer pointed out to us, the first assertion of Theorem 1 can be deduced from the "Special Chain Theorem" of Jaffard 15, p. 35]. In fact, the two results are equivalent. A chain C = {Q0 c QI c ... , X n] belongs to C. With this notation we prove 32 COROLLARY of finite rank, 3. (Jaffard) If Q is a prime ideal of R[ X 1 , .. , X n] with terminal lar, if R is finite dimensional, element Q.

2. Examples We present here the examples promised in the preceding section. , Xn] would profit from studying them. Despite the fact that our first construction examples of least importance to this paper, generality and for its possible usefulness the construction is used to give the it does possess a certain in other contexts, we give in that generality. Let R be a Prufer domain w i t h quotient field K. to find a domain D whose local behavior is identical We would like to that of R except at a single prime ideal P of D and at P we w i s h certain pathology--namely, we want Dp to be a one-dimensional, quasi-local, integrally closed domain which is not a valuation domain.

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