Catenarity of Nagata Rings in Pullback Construction: A Characterization Approach

In this paper, we investigate the catenarity of Nagata rings arising from pullback constructions of type (T,I,D), where I is the intersection of finitely many maximal ideals of T. Assuming that D is a Prüfer domain, T is universally catenarian, and the residue field extensions satisfy suitable transcendence degree conditions, we establish necessary and sufficient conditions for the Nagata rings R(n) to be catenarian for all positive integers n. Our characterization extends previous results obtained for pullbacks defined by a single maximal ideal and provides a unified treatment of a broader class of pullback constructions. As applications, we exhibit examples highlighting the distinction between the catenarity of Nagata rings and that of the corresponding Serre conjecture rings, and we provide an affirmative answer to an open question from the literature.