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David Bourqui; Julien Sebag
The Drinfeld-Grinberg-Kazhdan Theorem for formal schemes and singularity theory
Confluentes Mathematici, 9 no. 1 (2017), p. 29-64, doi: 10.5802/cml.35
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Class. Math.: 14E18, 14B05
Keywords: Arc scheme, formal neighborhood

Résumé - Abstract

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 $\mathrm{Spf}(k[[T]])$, for every non-singular arc $\gamma \in \mathscr{L}_{\infty }(V)(k)$, there exists an affine noetherian adic formal $k$-scheme $\mathscr{S}$ and an isomorphism of formal $k$-schemes

$$ \mathscr{L}_{\infty }(V)_\gamma \cong \mathscr{S}\times _k \mathrm{Spf}(k[[(T_i)_{i\in \mathbf{N}}]]). $$

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.


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