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The Drinfeld-Grinberg-Kazhdan Theorem for formal schemes and singularity theory
David Bourqui1; Julien Sebag1
1 Institut de recherche mathématique de Rennes, CNRS UMR 6625, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France
Confluentes Mathematici, Volume 9 (2017) no. 1, pp. 29-64.

Let k be a field. In this article, we provide an extended version of the Drinfeld-Grinberg-Kazhdan Theorem in the context of formal geometry. We prove that, for every formal scheme V topologically of finite type over Spf(k[[T]]), for every non-singular arc γ∈ℒ ∞ (V)(k), there exists an affine noetherian adic formal k-scheme 𝒮 and an isomorphism of formal k-schemes

ℒ∞(V)γ≅𝒮×kSpf(k[[(Ti)i∈N]]).

We emphasize the fact that the proof is constructive and, when V is the completion of an affine algebraic k-variety, effectively implementable. Besides, we derive some properties of such an isomorphism in the direction of singularity theory.

Received:
Accepted:
Revised after acceptance:
Published online:
DOI: 10.5802/cml.35
Classification: 14E18, 14B05
Keywords: Arc scheme, formal neighborhood

David Bourqui 1; Julien Sebag 1

1 Institut de recherche mathématique de Rennes, CNRS UMR 6625, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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David Bourqui; Julien Sebag. The Drinfeld-Grinberg-Kazhdan Theorem for formal schemes and singularity theory. Confluentes Mathematici, Volume 9 (2017) no. 1, pp. 29-64. doi : 10.5802/cml.35. https://cml.centre-mersenne.org/articles/10.5802/cml.35/

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