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Smooth affine group schemes over the dual numbers

Abstract : We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 \to \text{Lie}(G, I) \to E \to G \to 1$ where G is an affine, smooth group scheme over k. Here k is an arbitrary commutative ring and $k[I] = k \oplus I$ with $I^2 = 0$. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. It is compatible with the exact structures and the $\mathbb{O}_k$-module stack structures on both categories. Our constructions rely on the use of the group algebra scheme of an affine group scheme; we introduce this object and establish its main properties. As an application, we establish a Dieudonné classification for smooth, commutative, unipotent group schemes over $k[I]$.
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https://hal.archives-ouvertes.fr/hal-01712886
Contributor : Matthieu Romagny <>
Submitted on : Tuesday, June 25, 2019 - 11:48:36 AM
Last modification on : Monday, November 18, 2019 - 2:23:56 PM

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  • HAL Id : hal-01712886, version 4
  • ARXIV : 1802.06989

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Matthieu Romagny, Dajano Tossici. Smooth affine group schemes over the dual numbers. Épijournal de Géométrie Algébrique, EPIGA, 2019, 3, pp.article n°31. ⟨hal-01712886v4⟩

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