SK1 of Azumaya algebras over Hensel Pairs

Abstract

Let A be an Azumaya algebra of constant rank n 2 over a Hensel pair (R, I ) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK 1 (A) coincides with its reduction SK 1 (A/I A). This generalizes a result of Hazrat (J Algebra 305: [687][688][689][690][691][692][693][694][695][696][697][698][699][700][701][702][703] 2006) to non-local Henselian rings.Keywords Azumaya algebras · Reduced whitehead group · Reduced norm Let A be an Azumaya algebra over a ring R of constant rank n 2 . Then there is an étale faithfully flat commutative ring S over R which splits A, i.e., A ⊗ R S ∼ = M n (S). For a ∈ A, considering a ⊗ 1 as an element of M n (S), one then defines the reduced characteristic polynomial of a as char A (x, a) = det(x − a ⊗ 1) = x n − Trd(a)x n−1 + · · · + (−1) n Nrd(a).Using descent theory, one can show that char A (x, a) is independent of S and the isomorphism above and lies in R [x]. Furthermore, the element a is invertible in A if and only if Nrd A (a), the reduced norm of a, is invertible in R (see [10, III.

Transfers for non-stable K_1-functors of classical type

“…As a corollary, we obtain the following result. The corresponding statement was previously known for general isotropic simply connected groups G over Henselian discrete valuation rings [G1], [G2,Corollaire 7.3], and for groups G of type A l over arbitrary Henselian local rings [Haz10,Haz12].…”

Stability of unitary SK1 of Azumaya algebras

“…In [3], it was shown that if (C, I) is a Hensel pair and n is invertible in C, then the natural map SK 1 (A) → SK 1 (A ⊗ C C/I) is an isomorphism (see [2] for the notion of nonlocal Hensel rings and [9] for local Henselian rings). In this note we show that, if R is Henselian and 2 and n are invertible in R, then a similar statement is valid for the unitary SK 1 .…”

“…was established in [3] along with the following facts which we will use below; Nrd A (1 + mA) = 1 + mC, and 1 + mC is n-divisible. Consider the norm map N C/R : C * → R * defined by N C/R (a) = aτ (a).…”