Centrality of odd unitary $K_2$-functor

Abstract

Let (R, ∆) be an odd form algebra. We show that the unitary Steinberg group StU(R, ∆) is a crossed module over the odd unitary group U(R, ∆) in two major cases: if the odd form algebra has a free orthogonal hyperbolic family satisfying local stable rank condition and if the odd form algebra is sufficiently isotropic and quasi-finite. The proof uses only elementary localization techniques and stability results for KU1 and KU2.

Another presentation of orthogonal Steinberg groups

“…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. The same result for odd unitary groups (including the Chevalley groups of type A l , B l , C l , and D l ) was proved in [9]. Finally, together with Lavrenov and Sinchuk we generalized this result for all the remaining simple simply connected Chevalley groups of rank at least 3.…”

“…We would like to find analogs of ESD-transvections in the Steinberg group StO(2l, R). In order to do so, we use results from [3] or [9]. Recall that there is a "forgetful" functor from the category of pro-groups Pro(Grp) to the category of group objects in the category of pro-sets Pro(Set).…”

“…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…”

“…Finally, the constructions R (∞,S) and StO (∞,S) (2l, R) are contravariant on S, in the case S = {1} we get R and StO(2l, R) up to a canonical isomorphism. There are well-defined morphisms x α : R (∞,S) → StO (∞,S) (2l, R) of pro-groups, they generate the Steinberg progroup in the categorical sense by results from [3] or [9]. Lemma 2.…”

“…This endomorphism is denoted by the term a s n r (∞) with a free variable r (∞) . By results from [3] or [9], for any multiplicative subset S ⊆ R the local Steinberg group StO(2l, S −1 R) acts on the corresponding Steinberg pro-group StO (∞,S) (2l, R) by automorphisms. Note that this action is not given by a morphism StO(2l, S −1 R) × StO (∞,S) (2l, R) → StO (∞,S) (2l, R) of pro-sets, instead we have a homomorphism StO(2l, S −1 ) → Aut Pro(Grp) (StO (∞,S) (2l, R)) of abstract groups.…”