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It is proved that if 2 < d < ∞, 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 C T does not satisfy the going-down property. Rather than proceeding ring-theoretically, we construct a suitable spectral map φ connecting spectral (po)sets, then use a realization theorem of Hochster to infer that φ is essentially Spec(f) for a suitable ring homomorphism f, and finally replace f with an inclusion map R→ T having the asserted properties.

A spectral construction of a treed domain that is not going-down