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The automorphism theorem and additive group actions on the affine plane. (English) Zbl 1402.14079

Let \(k\) be a field, \(A\) a \(k\)-domain, \(A[T]\) the polynomial ring in one variable over \(A\), and \(\sigma :A\to A[T]\) a homomorphism of \(k\)-algebras. Then \(\sigma \) defines an action of the additive group \({\mathbf G}_a =\mathbf{Spec}\;k[T]\) on \(\mathbf{Spec} A\)
Due to R. Rentschler [C. R. Acad. Sci., Paris, Sér. A 267, 384–387 (1968; Zbl 0165.05402)], M. Miyanishi [Nagoya Math. J. 41, 97–100 (1971; Zbl 0191.19201)] and H. Kojima [Colloq. Math. 137, No. 2, 215–220 (2014; Zbl 1315.14079)], the invariant ring for a \({\mathbf G}_a\)-action on the affine plane over an arbitrary field is generated by one coordinate.
This nice paper provide a new short and very elegant proof for this result using the automorphism theorem of H. W. E. Jung [J. Reine Angew. Math. 184, 161–174 (1942; Zbl 0027.08503)] and W. van der Kulk [Nieuw Arch. Wiskd., III. Ser. 1, 33–41 (1953; Zbl 0050.26002)].

MSC:

14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
13A50 Actions of groups on commutative rings; invariant theory
14R20 Group actions on affine varieties
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Full Text: arXiv Euclid

References:

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