克莱恩与唐金主题变奏曲
Variations on a theme of Cline and Donkin
摘要：
设$N$是群$G$的正规子群。对于每个$g\in G$，只要$Q$等价于$g$对$Q$的扭曲$Q^g$，$N$模块$Q$就是$G$稳定的.如果$N$对$Q$的作用扩展为$G$对$Q$的作用，则$Q$显然是$G$稳定的，但反过来不一定成立。约化代数群的模表示理论中的一个著名猜想$G$断言，$G$的Frobenius核的（显然是$G$ -稳定的）投射不可分解模（PIM）$Q$具有$G$ -模结构。对于一般的模块$Q$，知道$Q$的有限直和$Q^{\oplus n}$具有兼容的$G$ -模块结构有时也是有用的。在本文中，这种性质被称为数值稳定性。在最近的工作（arXiv：0909.5207v2）中，作者建立了PIM特殊情况下的数值稳定性。在本文中，我们提供了一个更一般的背景下的结果，工作组计划和一个合适的版本的$G$稳定性，称为强$G$稳定性的背景下。在我们的研究结果中，这里是一个同调障碍的存在下，一个$G$模块结构，强$G$稳定的模块，张量积的方法来杀死的障碍。
英文摘要：
Let $N$ be a normal subgroup of a group $G$. An $N$-module $Q$ is $G$-stable provided that $Q$ is equivalent to the twist $Q^g$ of $Q$ by $g$, for every $g\in G$. If the action of $N$ on $Q$ extends to an action of $G$ on $Q$, $Q$ is obviously $G$-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups $G$ asserts that the (obviously $G$-stable) projective indecomposable modules (PIMs) $Q$ for the Frobenius kernels of $G$ have a $G$-module structure. It is sometimes just as useful (for a general module $Q$) to know that a finite direct sum $Q^{\oplus n}$ of $Q$ has a compatible $G$-module structure. In this paper, this property is called numerical stability. In recent work (arXiv:0909.5207v2), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of group schemes and a suitable version of $G$-stability, called strong $G$-stability. Among our results here is the presentation of a homological obstruction to the existence of a $G$-module structure, on strongly $G$-stable modules, and a tensor product approach to killing the obstruction.
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