Reduced Whitehead Group Research Articles

On triviality of the reduced Whitehead group over Henselian fields

Let $F$ be a Henselian field of $q$-cohomological dimension $3$, where $q$ is a prime. Let $\Gamma_F$ be the totally ordered abelian value group of $F$ and let $D$ be a central division algebra over $F$ of index a power of $q$ such that the characteristic of the residue field, $\overline{F}$ is coprime to $q$. We show that when $1\leq\dim_{\mathbb{F}_q}(\Gamma_F/q\Gamma_F)\leq 3$, the reduced Whitehead group of $D$ is trivial.

Suslin’s conjecture on the reduced Whitehead group of a simple algebra

Suslin’s conjecture on the reduced Whitehead group of a simple algebra

Reduced Whitehead Groups and the Conjugacy Problem for Special Unitary Groups of Anisotropic Hermitian Forms

Let K/k be a separable field extension of degree 2, D be a finite-dimensional central division algebra over K with a K/k-involution τ, h be an Hermitian anisotropic form on a right D-vector space with respect to τ, and let U(h) be the unitary group of h. Then the reduced Whitehead group of its special linear subgroup is defined as follows: $$ \mathrm{SUK}_1^{\mathrm{an}}(h)={{{\mathrm{SU}(h)}} \left/ {{\left[ {U(h),U(h)} \right]}} \right.} $$ , where [U(h), U(h)] is the commutator subgroup of U(h). The first main result establishes a link between the above group and its analog SUK1(h) for the case of isotropic h (with respect to the same τ). Theorem. There exists a surjective homomorphism from $$ \mathrm{SUK}_1^{\mathrm{an}}(h) $$ to SUK1(h). Furthermore, we also give a solution of the conjugacy problem for special unitary subgroups of anisotropic Hermitian forms over quaternion division algebras as subgroups of their multiplicative groups. Bibliography: 32 titles.

Comparing Invariants of SK1

In this text, we compare an invariant of the reduced Whitehead group SK 1 of a central simple algebra recently introduced by Kahn (2010) to other invariants of SK 1. Doing so, we prove the non-triviality of Kahn’s invariant using the non-triviality of an invariant introduced by Suslin (1991) which is non-trivial for Platonov’s examples of non-trivial SK 1 (Platonov, Math USSR Izv 10(2):211–243, 1976). We also give a formula for the value on the centre of the tensor product of two symbol algebras which generalises a formula of Merkurjev for biquaternion algebras (Merkurjev 1995).

When do reduced Whitehead groups of Iwasawa algebras vanish? A reduction step

Let l be an odd prime and K / k a Galois extension of totally real fields with Galois group G such that K / k ∞ and k / Q are finite. We reduce the conjectured triviality of the reduced Whitehead group S K 1 ( Q G ) of the algebra Q G = Quot ( Λ G ) with the Iwasawa algebra Λ G = Z l [ [ G ] ] to the case of pro- l Galois groups G and finite unramified coefficient extensions.

SK1 of graded division algebras

The reduced Whitehead group SK1 of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that SK1 of a tame valued division algebra over a henselian field coincides with SK1 of its associated graded division algebra. Furthermore, it is shown that SK1 of a graded division algebra is isomorphic to SK1 of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes SK1 for generic abelian crossed products.

SK1 of Azumaya algebras over Hensel Pairs

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 SK1(A) coincides with its reduction SK1(A/I A). This generalizes a result of Hazrat (J Algebra 305:687–703, 2006) to non-local Henselian rings.

Reduced K-theory of Azumaya algebras

In the theory of central simple algebras, often we are dealing with abelian groups which arise from the kernel or co-kernel of functors which respect transfer maps (for example K-functors). Since a central simple algebra splits and the functors above are “trivial” in the split case, one can prove certain calculus on these functors. The common examples are kernel or co-kernel of the maps K i ( F ) → K i ( D ) , where K i are Quillen K-groups, D is a division algebra and F its center, or the homotopy fiber arising from the long exact sequence of above map, or the reduced Whitehead group SK 1 . In this note we introduce an abstract functor over the category of Azumaya algebras which covers all the functors mentioned above and prove the usual calculus for it. This, for example, immediately shows that K-theory of an Azumaya algebra over a local ring is “almost” the same as K-theory of the base ring. The main result is to prove that reduced K-theory of an Azumaya algebra over a Henselian ring coincides with reduced K-theory of its residue central simple algebra. The note ends with some calculation trying to determine the homotopy fibers mentioned above.

On the Reduced Whitehead Group of an Affine Conic

In the framework of Milnor's K-theory, we compute SK 1 of the ring of regular functions on a conic. We also investigate unimodular rows of this ring from the point of view of the theory of quadratic forms. Bibliography: 8 titles.

SK1-like Functors for Division Algebras

We investigate the group valued functor G(D)=D*/F*D′ where D is a division algebra with center F and D′ the commutator subgroup of D*. We show that G has the most important functorial properties of the reduced Whitehead group SK1. We then establish a fundamental connection between this group, its residue version, and relative value group when D is a Henselian division algebra. The structure of G(D) turns out to carry significant information about the arithmetic of D. Along these lines, we employ G(D) to compute the group SK1(D). As an application, we obtain theorems of reduced K-theory which require heavy machinery, as simple examples of our method.

Cohomological Invariants of Simply Connected Groups of Rank 3

Let G be a linear algebraic group defined over a field F . One can define an equivalence relation (called R-equivalence) on the group GF‘ of points over F as follows (cf. [4, 9, 14]). Two points g0; g1 ∈ GF‘ are R-equivalent, if there is a rational morphism f x 1 F → G of algebraic varieties over F defined at points 0 and 1 such that f 0‘ = g0 and f 1‘ = g1. The group of R-equivalence classes is denoted by GF‘/R. For example, if G = SL1A‘ is the special linear group of a central simple F-algebra A, then the group of R-equivalence classes GF‘/R is equal to the reduced Whitehead group in algebraic K-theory (cf. [27])

Triviality of the reduced Whitehead group over certain fields

As is well known, the Tannaka--Artin problem on the triviality of SK~(A) was settled negatively by Platonov [1-3]. It follows from the results of [2, 3] that new fields K for which SKI(A ) = i must be sought among fields of sequential formal power series in several variables with large constant fields and among algebraic function fieldswith a small number ( ... , where k is algebraically closed or the field of real numbers, and we proved that SK, (A) can be an arbitrary cyclic group of finite order (when m > 4) in the first case and is trivial in the second. We remark that the latter result is also true over an arbitrary real closed field k.

THE TANNAKA-ARTIN PROBLEM AND REDUCEDK-THEORY

We solve the old Tannaka-Artin problem, which states the following in modern terminology: is the reduced Whitehead group SK1(A) of a finite-dimensional division algebra A trivial?We work out a method for computing the group SK1(A) based on a reduction to the computation of a group of special protective conorms—a new object in field theory—and we discover unexpected connections with number theory.Bibliography: 25 titles.

THE INFINITUDE OF THE REDUCED WHITEHEAD GROUP IN THE TANNAKA-ARTIN PROBLEM

Using the methods and results of preceding papers of the author (V. P. Platonov, The Tannaka-Artin problem and groups of projective conorms, Dokl. Akad. Nauk SSSR 222 (1975), 1229-1302="Soviet" Math. Dokl. 16 (1975), 782-786; The Tannaka-Artin problem and reduced -theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), 227-261 = Math. USSR Izv. 10 (1976), 211-243), in the first part of this paper we find conditions under which the reduced Whitehead group is infinite, and in the second, larger part we give the solution of the Tannaka-Artin problem for cyclic algebras. In particular, we completely calculate the reduced Whitehead group for cyclic algebras over formal power series fields and construct cyclic algebras of arbitrary degree with Whitehead group that is arbitrarily large but finite, and also with infinite Whitehead group. Bibliography: 15 titles.