Jaffard families and localizations of star operations

Abstract

We generalize the concept of localization of a star operation to flat overrings; subsequently, we investigate the possibility of representing the set Star(R) of star operations on R as the product of Star(T ), as T ranges in a family of overrings of R with special properties. We then apply this method to study the set of star operations on a Prüfer domain R, in particular the set of stable star operations and the star-class groups of R.Date: November 7, 2018. 2010 Mathematics Subject Classification. 13A15, 13C20, 13F05, 13F30, 13G05.

Towards a classification of stable semistar operations on a Prüfer domain

“…is an order isomorphism [29,Theorem 5.4]. An inspection of the proof of this result shows that the same reasoning also gives a bijection from FStar(D) to {FStar(T ) | T ∈ Θ}.…”

“…In this paper, we deepen this study, linking it to the concept of Jaffard family (whose tie with star operations was established in [29]) and extending it to semistar operations. In particular, we focus on which information about a Prüfer semilocal domain D is sufficient to determine the sets SStar(D) and Star(D) of, respectively, semistar and star operations.…”

The derived sequence of a pre-Jaffard family

“…(1) Definition 3.7 is not the original one of a Jaffard family, but it is the one most useful for our purpose; see [6, Section 6.3] and [15,Proposition 4.3]. (2) By Proposition 3.6 (see also [6,Theorem 6.3.1(4)]) the two conditions "Θ is independent" and "Θ is locally finite" can be unified into the single one "Θ is strongly independent".…”

“…A stable semistar operation is uniquely determined by its action on proper ideals of D. Hence, if ⋆ is a stable semistar operation fixing D, then the notion of extension of a star operation studied in [15] can be used to show that if Θ is a Jaffard family then I ⋆ = T ∈Θ (IT ) ⋆ (see, in particular, [15, Theorems 5.4 and 5.6]); a similar result, without the hypothesis D = D ⋆ , can be shown joining the results in Sections 3 and 6 of [17] (passing through length functions), so that any Jaffard family is stable-preserving. We want to generalize this case, but we first point out why stable-preserving properties are useful.…”

“…This notion allows to extend several results of h-local domains to more general rings; in particular, it was used to extend factorization properties from domains of Dedekind type to a wider class of domains [6,Chapter 6]. Jaffard families were subsequently used to factorize the set of star operations [15,18] and the set of length functions [17] on an integral domain as the product of the analogous sets on the members of a Jaffard family.…”