an:00010755
Zbl 0748.13005
Khalis, M.
Dimension de Krull des anneaux de s??ries formelles sur un produit fibr??. (Krull dimension of formal power series rings on a fibre product)
FR
Rend. Circ. Mat. Palermo, II. Ser. 39, No. 3, 395-411 (1990).
00157077
1990
j
13C15 13F25
formal power series rings; Krull dimension; SFT-ring
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 \(R=\varphi^{-1}(D)\). \textit{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).
L.G.Roberts (Kingston/Ontario)
Zbl 0262.13007