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arXiv 1802.06989math.AGmath.NTmath.RT
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对偶数上的光滑仿射群格式

Smooth affine group schemes over the dual numbers

Matthieu Romagny, Dajano Tossici

备注：
38 pages. Accepted in Epiga, final version
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摘要：

给出了广义对偶数环$k[I]$上的仿射光滑群概型范畴与$1 \rightarrow \text{Lie}(G, I) \rightarrow E \rightarrow G \rightarrow 1$形式的扩张范畴之间的等价性，其中G是k上的仿射光滑群概型.这里k是任意交换环，且$k[I] = k \oplus I$与$I^2 = 0$。等价性由Weil限制给出，并给出了一个拟逆，我们称之为Weil扩张。它与两个类别上的精确结构和$\mathbb{O}_k$模块堆栈结构兼容。我们的建设依赖于使用的群代数计划的仿射群计划，我们介绍了这个对象，并建立其主要性质。作为应用，我们建立了$k[I]$上光滑交换幂幺群格式的Dieudonne分类.

英文摘要：

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 Dieudonné classification for smooth, commutative, unipotent group schemes over $k[I]$.

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