有限群概型的表示与projective varieties的态射
Representations of finite group schemes and morphisms of projective varieties
摘要：
给定一个特征为$\Char(k)=p>0$的代数闭域$k$上的有限群模式$\cG$，我们通过将某些态射$°^j_M : U_M \lra \Gr_d(M) \ \ (1\!\le\!j\!\le\! p\!-\!1)$与$M$相关联，引入了一个$\cG$ -模$M$的新的不变量，这些态射取$M$的Grassmannian中的值.这些映射研究了两类有限代数群，无穷小群计划和初等阿贝尔群计划。与所谓的恒定$j$ -秩的模块相关联的映射具有范围在$0$和$j\rk^j(M)$之间的明确定义的度，其中$\rk^j(M)$是$M$的通用$j$ -秩。当模块$M$具有等像性或等核性时，可得到极值。我们建立了联系$M$的$j$ -度及其对偶$M^\ast$的公式.对于常Jordan型的自对偶模$M$，这提供了关于$M$的拉回$α^\ast(M)$沿着$p$ -点$α: k[X]/(X^p) \lra k\cG$的不可分解成分的信息。
英文摘要：
Given a finite group scheme $\cG$ over an algebraically closed field $k$ of characteristic $\Char(k)=p>0$, we introduce new invariants for a $\cG$-module $M$ by associating certain morphisms $°^j_M : U_M \lra \Gr_d(M) \ \ (1\!\le\!j\!\le\! p\!-\!1)$ to $M$ that take values in Grassmannians of $M$. These maps are studied for two classes of finite algebraic groups, infinitesimal group schemes and elementary abelian group schemes. The maps associated to the so-called modules of constant $j$-rank have a well-defined degree ranging between $0$ and $j\rk^j(M)$, where $\rk^j(M)$ is the generic $j$-rank of $M$. The extreme values are attained when the module $M$ has the equal images property or the equal kernels property. We establish a formula linking the $j$-degrees of $M$ and its dual $M^\ast$. For a self-dual module $M$ of constant Jordan type this provides information concerning the indecomposable constituents of the pull-back $α^\ast(M)$ of $M$ along a $p$-point $α: k[X]/(X^p) \lra k\cG$.
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