Towards a classification of stable semistar operations on a Prüfer domain

Abstract

We study stable semistar operations defined over a Prüfer domain, showing that, if every ideal of a Prüfer domain R has only finitely many minimal primes, every such closure can be described through semistar operations defined on valuation overrings of R.2010 Mathematics Subject Classification. 13A15, 13A18, 13F05, 13F30, 13G05.

The Zariski topology on sets of semistar operations without finite-type assumptions

“…We shall consider one case where we can prove that the space SStar st (D) is spectral: namely, the case when D is a Prüfer domain where every ideal has only finitely many minimal primes. To this aim, we shall use the following characterization, proved in [19]: under this hypothesis, if is a stable semistar operation, then…”

“…By [19,Proposition 4.10], π is surjective. Moreover, we claim that, for every Λ ⊆ SStar sv (D), we have inf Λ = inf Λ, where Λ is the closure in the inverse topology of Λ.…”

“…Keep the notation of the previous proof. By [19,Proposition 4.10], π is actually bijective; since ι is injective and π = π • ι is bijective, it follows that also ι is bijective. But this implies that every subset of SStar sv (D) closed by generizations is compact; however, this can happen only if SStar sv (D) is a Noetherian space.…”

Jaffard families and localizations of star operations

“…The following proposition may also be proved, in a slightly more generalized setting, using a different, more direct, approach; see [31]. Then, every stable star operation * on R is in the form…”

Decomposition and classification of length functions

“…The normalized stable version of ⋆ is [17, Section 4]⋆ : I → P ∈Σ 1 (⋆) ID P ∩ P ∈Σ 2 (⋆) (ID P ) v D P ,where v D P is the v-operation on D P (i.e., if J is an ideal of D P then J v D P = {yD P | J ⊆ yD P }), andΣ 1 (⋆) :={P ∈ Spec(D) | 1 / ∈ P ⋆ }, Σ 2 (⋆) :={P ∈ Spec(D) | 1 ∈ P ⋆ , 1 / ∈ Q ⋆ for some P -primary ideal Q}. (In the terminology of[17], Σ 1 (⋆) =: QSpec ⋆ (D) is the quasi-spectrum of ⋆, while Σ 2 (⋆) =: PsSpec ⋆ (D) is the pseudo-spectrum.) By [17, Proposition 3.4], and in the terminology introduced in Definition 5.9, furthermore, {Σ 1 (⋆), Σ 2 (⋆)} is a layered family with core Σ 1 (⋆).…”