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Dimension de Krull des anneaux de séries formelles sur un produit fibré. (Krull dimension of formal power series rings on a fibre product). (French) Zbl 0748.13005
Let \(T\) be a (commutative but not necessarily Noetherian) local integral domain, with maximal ideal \({\mathcal M}\) and residue field \(K\). Let \(\varphi:T\to K\) be the canonical homomorphism and let \(D\) be a proper subring of \(K\). The author considers the Krull dimension of \(R[[X]]\), where \(R=\varphi^{-1}(D)\). J. T. Arnold [Trans. Am. Math. Soc. 177, 299-304 (1973; Zbl 0262.13007)] has introduced the concept of SFT- ring, and proved that if a commutative ring \(A\) is not an SFT-ring, then \(\dim A[[X]]\) is infinite. Some sample results from the present paper are the following:
(a) \(R\) is an SFT-ring if and only if \(T\) and \(D\) are both SFT-rings; (b) if \(T\) is Noetherian, or if \(T\) is a discrete valuation ring (with value group possibly of rank greater than one), or if \(D\) is a field, then \(\dim R[[X]]=\dim D[[X]]+\dim T[[X]]-1\).
Using this result the author produces an example of a domain \(R\) of Krull dimension \(n\) such that: (1) \(\dim R[[X]]=\dim R+1\); (2) \(R[[X]]\) is catenary; (3) \(R\) is neither Noetherian nor a discrete valuation ring. Thus \(R\) is a new type of ring with properties (1) and (2).

MSC:
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13F25 Formal power series rings
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[1] Anderson D.F., Bouvier A., Dobbs D.E., Fontana M., Kabbaj S.,On Jaffard Domains, Expositiones Mathematicae6 (1988), 145–175.
[2] Arnold J.,Krull Dimension in Power Series Rings, Trans. Amer. Math. Soc.177 (1973), 299–304. · Zbl 0262.13007 · doi:10.1090/S0002-9947-1973-0316451-8
[3] Arnold J.,Power Series Rings over Prüfer Domains, Pacific J. Math.44 (1973), 1–11. · Zbl 0223.13016
[4] Arnold J.,Power Series Rings with finite Krull Dimension, Ind. Univ. Math. J.31 n. 6, (1982), 897–911. · Zbl 0498.13008 · doi:10.1512/iumj.1982.31.31061
[5] Arnold J.,The catenarian Property of Power Series Rings over Prüfer Domains, Proc. Amer. Math. Soc.94 (1985). · Zbl 0577.13010
[6] Brewer J.W.,Power Series Rings over Commutative Rings, Marcel Dekker, INC., (1983). · Zbl 0512.13013
[7] Fields D.E.,Dimension Theory in Power Series Rings, Pacific J. Math.35 (1970), 601–611. · Zbl 0192.38701
[8] Girolami F.,Power Series Rings over Globalized Pseudo-Valuation Domaines, J. of Pure and App. Algebra50 (1988), 259–269. · Zbl 0664.13007 · doi:10.1016/0022-4049(88)90104-1
[9] Nachar G.,Sur la caténarité des anneaux de séries formelles sur un anneau de Prüfer, Communications in Agebra15 (3), (1987), 479–489. · Zbl 0632.13006 · doi:10.1080/00927878708823429
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