The defect formula

2022/07/22 by Josnei Novacoski, Novacoski, Josnei, Enric Nart +1 · 1 citation
Mathematics · #Advanced Topics in Algebra #Advanced Topology and Set Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2207.11119

openalex publication_date 2022/07/22 · openalex created_date 2022/07/27 · openalex updated_date 2026/07/28

Abstract

In this paper we present a characterization for the defect of a simple algebraic extensions of valued fields. This characterization generalizes the known result for the henselian case, namely that the defect is the product of the relative degrees of limit augmentations. The main tool used here is the graded algebra associated to a valuation on a polynomial ring. Let \kh be a henselization of a valued field K. Another relevant result proved in this paper is that for every valuation \muh on \khx, with restriction μ on \kx, the corresponding map \mathcal G_μ\hk\mathcal G_\muh of graded algebras is an isomorphism.

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