Support varieties for restricted lie algebras
Support varieties for restricted lie algebras
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DOI：
10.1007/bf01389268
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发表时间：
1986-10
影响因子：
3.1
通讯作者：
E. Friedlander;B. Parshall
中科院分区：
文献类型：
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作者：
E. Friedlander;B. Parshall
本文考虑正特征p的固定代数闭域k，并且只考虑k上的代数群.设G是素域FP上具有Frobenius态射tr：G~G的仿射代数群，我们用g1表示tr的(群方案理论)核.如果V是有理G-模，我们写V1~对应的“扭”G-模，这是通过使G通过tr作用于V得到的。等价地，V1~具有与V相同的G结构，但有一种新的k结构，其中k通过tr-1作用于V。给定一个有理G-模V，使得G~在G对V的作用的核中，存在一个有理G-模W使得V-~W1~，我们经常记为V-~W。我们请读者参考I4；w来讨论这些问题。众所周知，有理G~-模范畴等价于g=Lie(G1)的受限g-模范畴。后者定义为限制李代数G的限制包络代数V(G)的模范畴，这是因为有理G-模定义为G~的坐标环k[G~]的余模，且V(G)自然同构于k[G~]的对偶代数。不作进一步的说明，我们通常认为一个受限g-模是有理Gl-模，其系数在这样一个模中的系数的代数V(G)的上同调是Gi的有理上同调。
Throughout this paper we consider a fixed algebraically closed field k of positive characteristic p and we consider only algebraic groups over k. Let G be an affine algebraic group defined over the prime field Fp with Frobenius morphism tr: G~ G. We denote by G1 the (group-scheme theoretic) kernel of tr. If V is a rational G-module, we write V 1~ for the corresponding" twisted" G-module, obtained by making G act on V though tr. Equivalently, V 1~ has the same G-structure as V, but a new k-structure in which k acts on V through tr-1. Given a rational G-module V such that G~ is in the kernel of the action of G on V, there exists a rational G-module W such that V-~ W 1~, and we often write V-~ for W. We refer the reader to I4; w for a discussion of these matters.As is well known, the category of rational G~-modules is equivalent to the category of restricted g-modules with g= Lie (G1). The latter is by definition the category of modules for the restricted enveloping algebra V (g) of the restricted Lie algebra g. This follows from the facts that a rational G~-module is by definition a comodule for the coordinate ring k [G~] of G~ and that V (g) is naturally isomorphic to the dual algebra of k [G~]. Without further comment, we frequently view a restricted g-module as a rational Gl-module and cohomology of the algebra V (g) with coefficients in such a module as rational cohomology of GI.