Centrality of K2-functor revisited

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

“…This paper is the third installment in the series of articles by the thirdnamed author devoted to the proof of centrality of K 2 modeled on linear groups. In the previous articles of the series [30,31] the third-named author has already proven the centrality of K 2 for the so-called odd unitary groups (see [15]), while the present paper focuses on Chevalley groups.…”

Explicit presentation of relative Steinberg groups

“…For example, if I is an ideal of R, then the inclusion map d : I → R is a crossed module. Another examples are the ring homotopes used in [12]…”

“…In order to construct an action of St(R) on St(R, A) we use the root elimination technique from [12]. Let Ψ ⊆ Φ be a root subsystem (necessarily closed) with the span RΨ.…”

Another presentation of orthogonal Steinberg groups

“…The projective limits in Pro(Set) are denoted by lim ← − Pro and various pro-sets are labeled with an upper index (∞), such as X (∞) . We also use the following convention from [3,8,9]: if a morphism between pro-sets is given by a first order term (possibly many-sorted), then we add the upper index (∞) for the formal variables. For example, [g (∞) , h (∞) ] denotes the commutator morphism…”

“…In [8] we reproved that St(n, R) is a crossed module over GL(n, R) using progroups. This more powerful method allowed to generalize the result for isotropic linear groups over almost commutative rings and for matrix linear groups over non-commutative rings with a local stable rank condition.…”