Arc closures and the local isomorphism problem

2018/11/30 by Devlin Mallory, Mallory, Devlin
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1811.12577

openalex publication_date 2018/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give an answer in the "geometric" setting to a question of de Fernex, Ein, and Ishii, asking when local isomorphisms of k-schemes can be detected on the associated maps of local arc or jet schemes. In particular, we show that their ideal-closure operation \mathfrak a↦ \mathfrak a^ac (the arc-closure) on a local k-algebra (R,\mathfrak m,L) is trivial when R is Noetherian and k\hookrightarrow L is separable, and thus that such a germ Spec R has the (embedded) local isomorphism property.

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