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 $k[I]$.

Source : oai:HAL:hal-01712886v4

Volume: Volume 3

Published on: July 1, 2019

Submitted on: August 30, 2018

Keywords: [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG],[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT],[MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT]

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