Twisted forms of classical groups

Centrality of odd unitary $K_2$-functor

“…It turns out that for any quadratic S-module there is a ring R with an involution and an odd form parameter ∆ ≤ Heis(B R ) (where B R (a, b) = a b) such that its unitary group is isomorphic to the unitary group of M , see [16,17]. Such a pair (R, ∆) is the same as a special unital odd form ring according to the definition below if we consider π and ρ are the first and the second projections from ∆ to R, and take φ(a) = (0, a − a).…”

“…The operations on the semi-direct product are uniquely determined by the operations on the factors and by the action. All axioms and all other claims follow from direct computation using the notion of quadratic maps from [5] and lemma 1 from [16].…”

“…Fix a commutative ring K. Odd form K-algebras from [16,17] are precisely odd form rings (R, ∆) with a unital action of the odd form ring (K, 0) (with the trivial involution on K) such that ak = ka for all a ∈ R, k ∈ K. Every odd form ring is an odd form Z-algebra with uniquely determined multiplication ∆ × Z → ∆.…”

“…Recall from [16] that a 2-step K-module is a pair (M, M 0 ), where M is a group with a right K • -action (the group operation and the action are denoted by m ∔ m ′ and m • k), M 0 is a subgroup of M and a left K-module, and there is a map τ : M → M 0 such that…”

“…Moreover, it is possible to define the unitary group of an odd form algebra and its elementary subgroup. In [16] we proved that all twisted forms of the most important classical reductive group schemes arise as parts of unitary and projective unitary groups of odd form algebras.…”