Weil restriction and support varieties

Abstract

Abstract. This paper extends the comparison results for cohomological support varieties for Chevalley groups of the form G( Consider a simple affine algebraic group G over k defined and split over F p with p-restricted Lie algebra Lie(G) = g. In [15], Z. Lin and D. Nakano showed for a rational G-module M that the rate of growth of a projective resolution for M as a kG(F p )-module is dominated by one-half the rate of growth of a projective resolution for M viewed as module for the restricted enveloping algebra for g. This result is illuminated and clarified by recent work of J. Carlson, Z. Lin, and D. Nakano who gave a direct relation between the cohomological varieties of G(F p ) and g [6]. For a restricted class of rational G-modules M , a comparison was also verified between the support variety of M for G and the support variety of M for g.In this paper, we extend the results of [15] and [6], so that an appropriate generalization of the formulation of their results now applies to G(F q ) where q is a prime power (with the same constraint on the prime, depending upon G, as required by [6]).

Restrictions to G(𝔽p) and G(r) of rational G-modules

“…In [Fri10], we showed how the Weil restriction functor enables one to extend techniques suitable for G(F p ) to G(F q ), q = p d , for algebraic groups G defined over F q . In this section, we indicate how this method applies to extend results of § 4 from G(F p ) to G(F q ).…”

On projective modules for Frobenius kernels and finite Chevalley groups

“…and by Friedlander [Fri1,Fri2] has sought to better understand the relationship between the varieties |G(F p )| M and |G 1 | M . In this note we provide an affirmative answer to Parshall's question for all r ≥ 1.…”

“…and by Friedlander [Fri1,Fri2] has sought to better understand the relationship between the varieties |G(F p )| M and |G 1 | M .…”

First cohomology for finite groups of Lie type: Simple modules with small dominant weights

“…[LN,Proposition 2.4] and [Fri,Theorem 4.3]). Put another way, u(u ⊕r ) acts on M via the surjection u(u ⊕r ) ։ u(u) discussed in Section 2.2 composed with the natural action of u(u) on M .…”