Title: On the Quadratic Structure of Torsors over Affine Group Schemes Show Chinese title

Title: 关于仿射群概形上扭子的二次结构

Abstract: Let $\mathcal{G}=\mathrm{Spec}(A)$ be a finite and flat group scheme over the ring of algebraic integers $R$ of a number field $K$ and suppose that the generic fiber of $\mathcal{G}$ is the constant group scheme over $K$ for a finite group $G$. Then the $R$-dual $A^D$of $A$ identifies as a Hopf $R$-order in the group algebra $K[G]$. If $B$ is a principal homogeneous space for $A$, then it is known that $B$ is a locally free $A^D$-module. By multiplying the trace form of $B_K/K$ by a certain scalar we obtain a $G$-invariant form $Tr'_B$ which provides a non-degenerate $R$-form on $B$. If $G$ has odd order, we show that the $G$-forms $(B, Tr'_B)$ and $(A, Tr'_A)$ are locally isomorphic and we study the question of when they are globally isomorphic. Suppose now that $K$ is a finite extension of $\mathbb Q_p$ with valuation ring $R$. In the course of our study we are led to consider the extension of scalars map $\varphi_K: G_0(A^D)\rightarrow G_0(A^D_K)=G_0(K[G])$. When $A^D$ is the group ring $R[G]$, Swan showed that $\varphi_K$ is an isomorphism. Jensen and Larson proved that $\varphi_K$ is also an isomorphism for any Hopf $R$-order $A^D$ of $K[G]$ when $G$ is abelian and $K$ is large enough. Here we prove that $\ker \varphi_K$ is at most a finite abelian $p$-group. However, numerous examples lead us to conjecture that Swan's result extends to all Hopf $R$-orders in $K[G]$, i.e. $\ker \varphi_K$ is always trivial. Show Chinese abstract

Abstract: 设 $\mathcal{G}=\mathrm{Spec}(A)$ 是数域 $K$ 的代数整数环 $R$ 上的有限且平坦的群概形，并且假设 $\mathcal{G}$ 的一般纤维是有限群 $G$ 在 $K$ 上的常数群概形。 然后，$R$-对偶$A^D$的$A$被识别为群代数$K[G]$中的一个霍普夫$R$-阶。 如果$B$是$A$的主齐次空间，那么已知$B$是一个局部自由的$A^D$模块。 通过将$B_K/K$的迹形式乘以某个标量，我们得到一个$G$-不变形式$Tr'_B$，它在$B$上提供了一个非退化的$R$-形式。 如果$G$的阶是奇数，我们证明了$G$-形式$(B, Tr'_B)$和$(A, Tr'_A)$是局部同构的，并我们研究它们何时全局同构的问题。 现在假设$K$是$\mathbb Q_p$的有限扩张，其赋值环为$R$。 在我们的研究过程中，我们考虑了标量扩展映射$\varphi_K: G_0(A^D)\rightarrow G_0(A^D_K)=G_0(K[G])$。当$A^D$是群环$R[G]$时，Swan 表明$\varphi_K$是一个同构。 Jensen 和 Larson 证明了当 $G$ 是阿贝尔的且 $K$ 足够大时，对于任何霍普夫 $R$-阶 $A^D$ 的 $K[G]$， $\varphi_K$ 也是同构。 这里我们证明$\ker \varphi_K$最多是一个有限交换$p$-群。 然而，许多例子使我们猜想斯旺的结果可以推广到$K[G]$中的所有霍普夫$R$-阶，即 $\ker \varphi_K$总是平凡的。