Smooth affine group schemes over the dual numbers

TL;DR

This paper establishes an equivalence between affine smooth group schemes over dual numbers and certain extensions, using Weil restriction, and applies this to classify unipotent group schemes via Dieudonné theory.

Contribution

It introduces a new equivalence between categories of group schemes over dual numbers and extensions, utilizing Weil restriction and group algebra schemes.

Findings

Established an equivalence between categories of group schemes over dual numbers and extensions.

Introduced the concept of group algebra schemes and proved their main properties.

Applied the theory to classify unipotent group schemes over dual numbers using Dieudonné classification.

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 $1 \rightarrow \text{Lie}(G, I) \rightarrow E \rightarrow G \rightarrow 1$ 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 Dieudonn\'e classification for smooth, commutative, unipotent group schemes over $k[I]$.