Topological Aspects of Irredundant Intersections of Ideals and Valuation Rings

Abstract

An intersection of sets A = i∈I B i is irredundant if no B i can be omitted from this intersection. We develop a topological approach to irredundance by introducing a notion of a spectral representation, a spectral space whose members are sets that intersect to a given set A and whose topology encodes set membership. We define a notion of a minimal representation and show that for such representations, irredundance is a topological property. We apply this approach to intersections of valuation rings and ideals. In the former case we focus on Krull-like domains and Prüfer v-multiplication domains, and in the latter on irreducible ideals in arithmetical rings. Some of our main applications are to those rings or ideals that can be represented with a Noetherian subspace of a spectral representation.

Non-compact subsets of the Zariski space of an integral domain

“…In particular, Zariski showed that Zar(K|D), endowed with a natural topology, is always a compact space [34, Chapter VI, Theorem 40]; this result has been subsequently improved by showing that Zar(K|D) is a spectral space (in the sense of Hochster [18]), first in the case where K is the quotient field of D [4,5], and then in the general case [8,Corollary 3.6(3)]. The topological aspects of the Zariski space has subsequently been used, for example, in real and rigid algebraic geometry [19,31] and in the study of representation of integral domains as intersections of valuation overrings [26,27,28]. In the latter context, i.e., when K is the quotient field of D, two important properties for subspaces of Zar(K|D) to investigate are the properties of compactness and of Noetherianess.…”

The upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applications

“…An overring of D is an integral domain contained between D and its quotient field K; the collection of all overrings of D is denoted by Overr(D). Under the Zariski topology, this space is closed in the constructible topology of SMod D (K) (this essentially follows from [36]); in particular, Overr(D) is a spectral space and a subbase for the open sets of Overr(D) is formed by the sets in the form O F := {B ∈ Overr(D) | B ⊇ F }, where F runs among the finite subsets of K.…”

Directed unions of local quadratic transforms of a regular local ring

“…As follows in Remark 4.4, properties of the Zariski topology, which are the focus of the next section, can be derived from the patch topology, so our approach in this section is to focus on the patch limit points of subsets of Q * (D) and use this description in the next section to describe properties of the Zariski topology of Q * (D). The patch topology is a common tool for studying the Zariski-Riemann space of valuation rings of a field; see for example [3,4,17,25,26,27,28].…”