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Universally going-down homomorphisms of commutative rings. (English) Zbl 0544.13004

The paper begins with a unified treatment of universally i-, universally unibranched, and universally mated homomorphisms (unital, of commutative rings). Theorems 2.1 and 2.5 show in each of the three cases that the ”universal” property is equivalent to the conjunction of the ”ordinary” property and the condition that the induced inclusions of residue fields be purely inseparable extensions. The ”universally mated” property plays a key role in the analysis of ”universally going-down” in section 3. Section 2 also contains some useful technical results (lemma 2.4 (a)) concerning stability of various properties under direct limit. Section 3 obtains some examples of universally going-down homomorphisms, including all \(R\to T\) for which \(\dim(R)=0\) (cf. proposition 3.3). Then, in an attempt to obtain an ”internal” characterization of ”universally going- down,” we modify some constructions of A. Andreotti and E. Bombieri [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 23, 431- 450 (1969; Zbl 0184.245)] and C. Traverso [ibid. 24 (1970), 585- 595 (1971; Zbl 0205.505)], and thus introduce the notion of a UGD homomorphism. In case \(f:\quad R\to T\) is injective and integral, we find that f is UGD if and only if f satisfies GD and T is the weak normalization of R with respect to f [in the sense of Andreotti-Bombieri (op. cit.)]. For arbitrary (not necessarily integral), f, UGD is shown to have several useful consequences, notably ”radiciel” and ”universally mated” (cf. corollary 3.12). After observing that UGD implies a weak variant of going-up, we infer our main result, theorem 3.15, concerning universality of UGD. Its specializations include theorem 3.17, a characterization of ”universally going-down” for (the inclusion map of) an arbitrary overring of an integral domain. This entails consequences for certain nonintegral maps, which cannot be handled by the riding hypotheses of S. McAdam [Can. J. Math. 23, 704-711 (1971; Zbl 0223.13006); Duke Math. J. 39, 633-636 (1972; Zbl 0252.13001)]. The upshot for an integral overring of an integral domain is that ”universally going-down” and UGD are equivalent (cf. corollary 3.20). In particular, there is but one type of integral overring extension \(R\to T\) of the type studied by McAdam (op. cit.), viz. for which the induced homomorphisms \(R[X_ 1,...,X_ n]\to T[X_ 1,...,X_ n]\) satisfy GD or are unibranched: T must be the weak normalization of R inside T.

MSC:

13B02 Extension theory of commutative rings
14A05 Relevant commutative algebra
13A15 Ideals and multiplicative ideal theory in commutative rings
13G05 Integral domains
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