模不变量理论中的协变模
Modules of covariants in modular invariant theory
摘要：
设有限群$G$线性作用在任意特征的域$k$上的向量空间$V$上.如果$H<G$是一个子群的不变环$k[V]^G\subset k[V]^H$的扩张研究使用模的协变。我们的结果的一个例子如下。令$W$是由$G$中的反射生成的$G$的子群。一个经典的定理由于塞尔说，如果$k[V]$是一个自由的$k[V]^G$ -模，那么$G=W$。我们把这个结果推广如下。如果$k[V]^H$是自由$k[V]^G$ -模，则$G$由$H$和$W$生成，不变环$k[V]^{H\cap W}$在$k[V]^W$上是自由的，并由$H$ -不变量和$W$ -不变量生成为代数。
英文摘要：
Let the finite group $G$ act linearly on the vector space $V$ over the field $k$ of arbitrary characteristic. If $H<G$ is a subgroup the extension of invariant rings $k[V]^G\subset k[V]^H$ is studied using modules of covariants. An example of our results is the following. Let $W$ be the subgroup of $G$ generated by the reflections in $G$. A classical theorem due to Serre says that if $k[V]$ is a free $k[V]^G$-module then $G=W$. We generalize this result as follows. If $k[V]^H$ is a free $k[V]^G$-module then $G$ is generated by $H$ and $W$, and the invariant ring $k[V]^{H\cap W}$ is free over $k[V]^W$ and generated as an algebra by $H$-invariants and $W$-invariants.
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