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

Zbl 0074.26301
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