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Prime ideals in birational extensions
Abstract
Let B be a finitely generated birational extension of R[x], a ring of polynomials in one variable over a one-dimensional Noetherian domain R. We study Spec( B), the set of prime ideals of B partially ordered under inclusion. We separate into two cases: (1) R is a local domain. (Chapter II) (2) R is an order in an algebraic number field. (Chapter III) Among other results in Chapters II and III, we show (1) If R is a one-dimensional local non-Henselian Hilbertian domain, [special characters omitted] where [special characters omitted] and [special characters omitted] is an R[x]-sequence, and T is a finite set of height-two maximal ideals of B, then there are infinitely many height-one primes of B which are contained in every element of T but contained in no other maximal ideals. (2) If D is an order in an algebraic number field, then a birational extension of D[x] of the form [special characters omitted] has prime spectrum order-isomorphic to Spec([special characters omitted]). Chapter IV deals with sets of prime ideals of finitely generated birational extensions of Noetherian unique factorization domains. Also we investigate the “going down” property for some finitely generated birational extensions. In Chapter V, we study relationships among properties of partially ordered sets.
Subject Area
Mathematics
Recommended Citation
Saydam, Azime Serpil, "Prime ideals in birational extensions" (1998). ETD collection for University of Nebraska-Lincoln. AAI9917858.
https://digitalcommons.unl.edu/dissertations/AAI9917858