A universal deformation ring of higher conductor. (Un anneau de déformation universel en conducteur supérieur.) (French. English summary) Zbl 1298.14007

Byszewski, Jakub; Cornelissen, Gunther; Kato, Fumiharu

Proc. Japan Acad., Ser. A 88, No. 2, 25-27 (2012).

Let \(k\) be a perfect field of characteristic 5, and let \(\sigma: k[[t]]\to k[[t]]\) be given by \(t \mapsto \frac{t}{\sqrt{t^2+1}}\). \(\sigma\) is an automorphism of order 5 and Hasse conductor 2. The authors prove that the functor of formal deformations of \(\sigma\) is pro-representable, by explicitly demonstrating that the corresponding versal ring, as constructed by J. Bertin and A. Mézard, [“Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques”, Invent. Math. 141, No. 1, 195–238 (2000; Zbl 0993.14014)], is in fact universal. (It should be noted that since every automorphism of \(k[[t]]\) of order 5 and conductor 2 is conjugate to \(\sigma\), this theorem is valid for all such automorphisms.) Universality for weakly ramified actions of finite groups on \(k[[t]]\) was shown in [J. Byszewski and G. Cornelissen, “Which weakly ramified group actions admit a universal formal deformation?”, Ann. Inst. Fourier 59, No. 3, 877–902 (2009; Zbl 1226.14007)]. The example in this paper is the first known example of pro-representability for a non-weakly ramified action.

Reviewer: Emre Coskun (Ankara)

MSC:

14B12	Local deformation theory, Artin approximation, etc.
14D15	Formal methods and deformations in algebraic geometry
14H37	Automorphisms of curves
13D10	Deformations and infinitesimal methods in commutative ring theory
13F25	Formal power series rings

Keywords:

deformation; automorphism; formal power series; universality

Citations:

Zbl 0993.14014; Zbl 1226.14007

References:

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