On the Krull dimension of \(\text{Int}(D)\) when \(D\) is a pullback (Q2785952)

Let \(D\) be a finite dimensional domain with quotient field \(K\). One is interested in the Krull dimension of the ring \(\text{Int} (D):= \{f\in K[X]: f(D) \subset D\}\) of integer valued polynomials. If \(D\) has a nonzero ideal in common with an overring \(B\) relations between \(\dim \text{Int} (D)\) and \(\dim B\) are established. In particular an explicit formula for \(\dim \text{Int} (D)\) is obtained for a locally finite domain \(D\) (i.e. every nonzero element of \(D\) is contained only in finitely many maximal ideals) which is locally a pullback of a Jaffard domain [cf. \textit{D. F. Anderson}, \textit{A. Bouvier}, \textit{D. E. Dobbs}, \textit{M. Fontana} and \textit{S. Kabbaj}, Expo. Math. 6, No. 2, 145-175 (1988; Zbl 0657.13011)]. Seidenberg's construction [\textit{A. Seidenberg}, Pac. J. Math. 4, 603-614 (1954; Zbl 0057.26802)] of domains \(D\) with different dimensions of \(D[X]\) is transferred to the rings \(\text{Int} (D)\). In case of pseudo-valuation domains of type \(n\) [cf. \textit{D. E. Dobbs} and \textit{M. Fontana}, C. R. Acad. Sci., Paris, Sér. I 306, No. 1, 11-16 (1988; Zbl 0642.13010)] a precise formula for \(\dim \text{Int} (D)\) is given.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].