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A spectral construction of a treed domain that is not going-down. (English) Zbl 1089.13502

From the paper: A ring homomorphism of commutative rings \(f:A\to B\) is said to satisfy going-down if, whenever \(P_2\subseteq P_1\) are prime ideals of \(A\) and \(Q_1\) is a prime ideal of \(B\) such that \(f^{-1}(Q_1)= P_1\), there exists a prime ideal \(Q_2\) of \(B\) such that \(Q_2\subseteq Q_1\) and \(f^{-1}(Q_2)=P_2\). A ring extension \(A\subseteq B\) is said to satisfy going-down if the inclusion map \(i:A\hookrightarrow B\) satisfies going-down.
In the paper under review, it is proved that if \(2\leq d\leq \infty\), then there exist a treed domain \(R\) of Krull dimension \(d\) and an integral domain \(T\) containing \(R\) as a subring such that the extension \(R\subset T\) does not satisfy the going-down property. Rather than proceeding ring-theoretically, we construct a suitable spectral map \(\varphi\) connecting spectral (po)sets, then use a realization theorem of M. Hochster [Trans. Am. Math. Soc. 142, 43–60 (1969; Zbl 0184.29401)] to infer that \(\varphi\) is essentially Spec\((f)\) for a suitable ring homomorphism \(f\), and finally replace \(f\) with an inclusion map \(R\hookrightarrow T\) having the asserted properties.

MSC:

13B24 Going up; going down; going between (MSC2000)
13A15 Ideals and multiplicative ideal theory in commutative rings
13G05 Integral domains
06A11 Algebraic aspects of posets

Citations:

Zbl 0184.29401
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References:

[1] Bourbaki, N.. Topologie Générale, Chapitres 1-4. Hermann, Paris, 1971. · Zbl 0249.54001
[2] Dobbs, D.E.. On going-down for simple overrings, II. Comm. Algebra, 1:439-458, 1974. · Zbl 0285.13001
[3] Dobbs, D.E.. Posets admitting a unique order-compatible topology. Discrete Math., 41:235-240, 1982. · Zbl 0498.06004
[4] Dobbs, D.E.. On treed overrings and going-down domains. Rend. Mat., 7:317-322, 1987. · Zbl 0683.13003
[5] Dobbs, D.E. and Fontana, M.. Universally going-down homomorphisms of commutative rings. J. Algebra, 90:410-429, 1984. · Zbl 0544.13004
[6] Dobbs, D.E., Fontana, M., and Papick, I.J.. On certain distinguished spectral sets. Ann. Mat. Pura Appl., 128:227-240, 1980. · Zbl 0472.54021
[7] Dobbs, D.E. and Papick, I.J.. On going-down for simple overrings, III. Proc. Amer. Math. Soc., 54:35-38, 1976. · Zbl 0285.13002
[8] Dobbs, D.E. and Papick, I.J.. Going down: a survey. Nieuw Arch. v. Wisk., 26:255-291, 1978. · Zbl 0383.13005
[9] Fontana, M.. Topologically defined classes of commutative rings. Ann. Mat. Pura Appl., 123:331-355, 1980. · Zbl 0443.13001
[10] Gilmer, R.. Multiplicative Ideal Theory. Dekker, New York, 1972. · Zbl 0248.13001
[11] Hochster, M.. Prime ideal structure in commutative rings. Trans. Amer. Math. Soc., 142:43-60, 1969. · Zbl 0184.29401
[12] Lewis, W.J.. The spectrum of a ring as a partially ordered set. J. Algebra, 25:419-434, 1973. · Zbl 0266.13010
[13] Nagata, M.. Local rings. Wiley/Interscience, New York, 1962. · Zbl 0123.03402
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