This paper extends the comparison results for cohomological support varieties for Chevalley groups of the form G(Fp) established by J. Carlson, Z. Lin, and D. Nakano to Chevalley groups of the form G(Fpd). Other topics considered are complexity and non-maximal support varieties for modules. Techniques developed by J. Pevtsova and the author, together with the use of Weil restriction functors, a...

This is the rst in a series of papers dedicated to the structure of Chevalley groups over commutative rings. The goal of this series is to systematically develop methods of calculations in Chevalley groups over rings, based on the use of their minimal modules. As an application we give new direct proofs for normality of the elementary subgroup, description of normal subgroups and similar result...

We prove that the trace of the nth power of the antipode of a Hopf algebra with the Chevalley property is a gauge invariant, for each integer n. As a consequence, the order of the antipode, and its square, are invariant under Drinfeld twists. The invariance of the order of the antipode is closely related to a question of Shimizu on the pivotal covers of finite tensor categories, which we affirm...

We refine the notion of variety over the “field with one element” developed by C. Soulé by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint developed by J. Tits. We then solve an open question of C. Soulé by proving, using results of J. Tits and C. Chevalley, that Chevalley group schemes are exampl...

In this handout we shall discuss a result that was conjectured by Emil Artin in 1935 and proved shortly thereafter by Claude Chevalley. A refinement was given by Artin’s graduate student Ewald Warning, who, as the story goes, was the one whom Artin had intended to prove the theorem before Chevalley came visiting Göttingen and got Artin to say a little too much about the mathematics his student ...

In the present paper we prove the main structure theorem for Chevalley groups G = G(Φ, R) of types Φ = E6,E7 over a commutative ring R. More precisely, we describe subgroups in G normalized by the elementary subgroup E(Φ, R). This result is not new, since structure theorems are known for all Chevalley groups [25, 27, 28, 30], [38]–[40], and [58, 61] (see [42, 65, 34, 56] for further references)...