Locally nilpotent derivations of graded integral domains and cylindricity

2021/05/04 by Michael Chitayat, Daniel Daigle, Chitayat, Michael +1
Mathematics · #14R05 #14R20. Secondary: 14C20 #14R25 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #Commutative Algebra (math.AC) #FOS: Mathematics #Holomorphic and Operator Theory #Primary: 13N15

paper · pdf · doi:10.48550/arxiv.2105.01729

openalex publication_date 2021/05/04 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

Let B be a commutative ℤ-graded domain of characteristic zero. An element f of B is said to be cylindrical if it is nonzero, homogeneous of nonzero degree, and such that B_(f) is a polynomial ring in one variable over a subring. We study the relation between the existence of a cylindrical element of B and the existence of a nonzero locally nilpotent derivation of B. Also, given d > 0, we give sufficient conditions that guarantee that every derivation of B^(d) = ⊕_i B_di can be extended to a derivation of B. We generalize some results of Kishimoto, Prokhorov and Zaidenberg that relate the cylindricity of a polarized projective variety (Y,H) to the existence of a nontrivial G_a-action on the affine cone over (Y,H).

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