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Valuations d’un anneau noethérien et théorie de la dimension. (French) Zbl 0111.04103

Algèbre Théorie Nombres, Sém. P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot 13 (1959/60), Exp. No. 12, 21 p. (1961).
A valuation of an integral domain \(A\) is a valuation of the quotient field \(K\) of \(A\) whose valuation ring contains \(A\). The valuation dimension of \(A\) \((\dim_v A)\) is the least upper bound of the ranks of the valuations of \(A\). More generally, if \(A\) is a commutative ring with unity \(\dim_v A\) is the least upper bound of \(\dim_v A/\mathfrak p\) taken over all primes \(\mathfrak p\) of \(A\). If \(v\) is a valuation of an integral domain \(A\) with center \(\mathfrak p\) on \(A\), \(\dim_A v\) denotes the transcendence degree of the residue field of \(v\) over the field of fractions of \(A/\mathfrak p\). The author gives a proof of the following theorem of S. Abhyankar [Am. J. Math. 78, 321–348 (1956; Zbl 0074.26301)]. If \(v\) is a valuation of a (Noetherian) local domain \(A\) whose center is the maximal ideal of \(A\), then \(\dim_A v +\mathrm{rk}\,v\leq \dim A\), where \(\mathrm{rk}\,v\) denotes the rank of \(v\) and \(\dim A\) is the usual dimension of \(A\). The proof is somewhat more simple than Abhyankar’s proof.
He shows also that this theorem is equivalent to the following corollary to a theorem of Krull. If \(A\) is a noetherian ring, then \(\dim_v A = \dim A\). Let \(\mathfrak p\supset \mathfrak q\) be two prime ideals of a ring \(A\). \(\delta(\mathfrak p,\mathfrak q)\) denotes the least upper bound of the numbers \(d\) such that the homomorphism \(A/\mathfrak q\to A/\mathfrak p\) can be extended to a specialization of the field of fractions of \(A/\mathfrak q\) onto an extension of transcendence degree \(d\) of the field of fractions of \(A/\mathfrak p\). Properties of \(\delta(\mathfrak p,\mathfrak q)\) are given for certain classes of rings, for example when \(A\) is a Noetherian local domain with maximal ideal \(\mathfrak m\), \(\delta(0,\mathfrak m)= \sup (0, \dim A-1)\).

MSC:

13A18 Valuations and their generalizations for commutative rings
13H99 Local rings and semilocal rings

Citations:

Zbl 0074.26301
Full Text: Numdam EuDML