The Drinfeld-Grinberg-Kazhdan Theorem for formal schemes and singularity theory

David Bourqui; Julien Sebag

doi:10.5802/cml.35

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

Confluentes Mathematici, Tome 9 (2017) no. 1, pp. 29-64.

Résumé

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 $γ \in ℒ_{\infty} (V) (k)$ , there exists an affine noetherian adic formal $k$ -scheme $𝒮$ and an isomorphism of formal $k$ -schemes

ℒ_{\infty} {(V)}_{γ} ≅ 𝒮 \times_{k} Spf (k [[{(T_{i})}_{i \in 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.

Reçu le : 2015-12-16
Accepté le : 2016-10-12
Accepté après révision le : 2016-12-09
Publié le : 2017-09-13

DOI : 10.5802/cml.35

Classification : 14E18, 14B05
Mots clés : Arc scheme, formal neighborhood

Affiliations des auteurs :

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

Licence :

CC-BY-NC-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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David Bourqui; Julien Sebag. The Drinfeld-Grinberg-Kazhdan Theorem for formal schemes and singularity theory. Confluentes Mathematici, Tome 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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