2024/11/03 by Thomas Polstra, Polstra, Thomas
Computer Science · Mathematics · #13A15 #13A35 #Advanced Topology and Set Theory #Commutative Algebra (math.AC) #Computability, Logic, AI Algorithms #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2411.01480
openalex publication_date 2024/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the containment problem of symbolic and ordinary powers of ideals in a commutative Noetherian domain R. Let R be a normal domain of prime characteristic p>0 that is F-finite or essentially of finite type over an excellent local ring. Assume there exists a finite extension R→ S so that the non-strongly F-regular locus of Spec(S) consists only of isolated points, then there exists a constant C such that for all ideals I ⊆ R and n ∈ ℕ, the symbolic power I(Cn) is contained in the ordinary power In. In other words, R enjoys the Uniform Symbolic Topology Property. Moreover, if R is F-finite and strongly F-regular, then R enjoys a property that is proven to be stronger: there exists a constant e0 ∈ ℕ such that for any ideal I ⊆ R and all e ∈ ℕ, if x ∈ R ∖ I[pe], then there exists an R-linear map φ: Fe+e0_*R → R such that φ(Fe+e0_*x) ∉ I.