A generalization of Nakai's theorem on locally finite iterative higher derivations

2014/12/04 by Shigeru Kuroda, Kuroda, Shigeru · 1 citation
Mathematics · #14R10 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Functional Equations Stability Results #Nonlinear Differential Equations Analysis #Primary 13N15 #Secondary 13A50 #advanced mathematical theories #math.AC #math.AG #msc:13A50 #msc:13N15 #msc:14R10

paper · pdf · doi:10.48550/arxiv.1412.1598

arxiv created 2014/12/04 · openalex publication_date 2014/12/04 · arxiv updated 2014/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k be a field of arbitrary characteristic. Nakai (1978) proved a structure theorem for k-domains admitting a nontrivial locally finite iterative higher derivation when k is algebraically closed. In this paper, we generalize Nakai's theorem to cover the case where k is not algebraically closed. As a consequence, we obtain a cancellation theorem of the following form: Let A and A' be finitely generated k-domains with A[x]≃ _kA'[x]. If A and k⊗ _kA are UFDs and \mathop\rm trans.deg\nolimits_kA=2, then we have A≃ _kA'. This generalizes the cancellation theorem of Crachiola (2009).

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