Centrality of $\mathrm K_2$ for Chevalley groups: a pro-group approach

Abstract

We prove the centrality of K2(F4, R) for an arbitrary commutative ring R. This completes the proof of the centrality of K2(Φ, R) for any root system Φ of rank ≥ 3. Our proof uses only elementary localization techniques reformulated in terms of pro-groups. Another new result of the paper is the construction of a crossed module on the canonical homomorphism St(Φ, R) → Gsc(Φ, R), which has not been known previouly for exceptional Φ.

On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ modeled on linear and even orthogonal groups

“…For instance, St (∞) (Φ, R) is "generated" by x α in the sense that a pair of pro-group morphisms f, g : St (∞) (Φ, R) → G (∞) coincide if and only if f • x β = g • x β for all β ∈ Φ. It is enough to verify the latter equalities for β ∈ Φ \ {α} where α ∈ Φ is any fixed root (in fact, an even stronger result holds, see [11,Lemma 3.2]). Moreover, given a collection of progroup morphisms f α : R (∞) → G (∞) satisfying the "pro-analogues" of Chevalley commutator identities (see [11,Remark 2.15]), there exists a unique morphism f :…”

“…Recall also from the [26,Theorem 5.3] that under the assumptions on the rank Φ stated in Theorem 1.1 the first two homology groups of St(Φ, R) vanish. Consequently from the main result of [11] and § IV.1 of [36] it is easy to conclude that the group K 2 (Φ, R) coincides with the unstable K 2 -group defined by means of Quillen's +-construction:…”

“…These morphisms, which we call root subgroup morphisms, allow one to characterize the Steinberg pro-group St (∞) (Φ, R) by means of a universal property (cf. [11,Lemma 2.16]). For instance, St (∞) (Φ, R) is "generated" by x α in the sense that a pair of pro-group morphisms f, g : St (∞) (Φ, R) → G (∞) coincide if and only if f • x β = g • x β for all β ∈ Φ.…”

“…It is enough to verify the latter equalities for β ∈ Φ \ {α} where α ∈ Φ is any fixed root (in fact, an even stronger result holds, see [11,Lemma 3.2]). Moreover, given a collection of progroup morphisms f α : R (∞) → G (∞) satisfying the "pro-analogues" of Chevalley commutator identities (see [11,Remark 2.15]), there exists a unique morphism f :…”

“…The verification of this property for simply-laced Steinberg groups is another new result of the paper, see Theorem 4.1. Unlike M. Tulenbaev, who used the technique of van der Kallen's "another presentation" in the proof of the weaker Zariski excision property [31,Proposition 1.4], in our proof we use the new technique of pro-groups introduced by the third-named author in [35] (see also [11]).…”