Irredundant intersections of valuation overrings of two-dimensional noetherian domains
Throughout the paper \(D\) is a noetherian domain of Krull dimension 2. Any ring \(R\) included between \(D\) and its quotient field is called an overring. Denote two sets of valuation overrings of \(D\) by \(\Sigma\) and \(\Gamma\). The author is interested in investigation when (with various irredundancy conditions) the condition \((\bigcap_{V\in\Sigma}V)\cap R=(\bigcap_{W\in\Gamma}W)\cap R\) implies \(\Sigma=\Gamma\). A number of technical conditions are introduced to this end. Thus it is shown for instance, that if \(R\) is integrally closed, the above intersections are strongly irredundant (none of the \(V\) or \(W\) can be replaced by their proper overrings), and every member of \(\Sigma\cup\Gamma\) is of Krull dimension 2, then \(\Sigma=\Gamma\). The latter condition may be curtailed, if instead, another condition is imposed: that \(\Sigma\) and \(\Gamma\) are noetherian subspaces of the Zariski-Riemann space of the quotient field of \(D\). A number of examples are given to illustrate various points. Among them an example where it is shown that these results cannot be extended to overrings of noetherian domains of dimension 3.
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