Smooth affine group schemes over the dual numbers
Épijournal de Géométrie Algébrique, Tome 3 (2019)
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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 $k[I]$.
@article{10_46298_epiga_2019_volume3_4792,
author = {Romagny, Matthieu and Tossici, Dajano},
title = {Smooth affine group schemes over the dual numbers},
journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
publisher = {mathdoc},
volume = {3},
year = {2019},
doi = {10.46298/epiga.2019.volume3.4792},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.46298/epiga.2019.volume3.4792/}
}
TY - JOUR AU - Romagny, Matthieu AU - Tossici, Dajano TI - Smooth affine group schemes over the dual numbers JO - Épijournal de Géométrie Algébrique PY - 2019 VL - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.46298/epiga.2019.volume3.4792/ DO - 10.46298/epiga.2019.volume3.4792 LA - en ID - 10_46298_epiga_2019_volume3_4792 ER -
%0 Journal Article %A Romagny, Matthieu %A Tossici, Dajano %T Smooth affine group schemes over the dual numbers %J Épijournal de Géométrie Algébrique %D 2019 %V 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.46298/epiga.2019.volume3.4792/ %R 10.46298/epiga.2019.volume3.4792 %G en %F 10_46298_epiga_2019_volume3_4792
Romagny, Matthieu; Tossici, Dajano. Smooth affine group schemes over the dual numbers. Épijournal de Géométrie Algébrique, Tome 3 (2019). doi: 10.46298/epiga.2019.volume3.4792
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