Periodicity of hermitianK-groups

Abstract

Bott periodicity for the unitary and symplectic groups is fundamental to topologicalK-theory. Analogous to unitary topologicalK-theory, for algebraicK-groups with finite coefficients, similar results are consequences of the Milnor and Bloch-Kato conjectures, affirmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraicK-groups for any ring implies periodicity for the hermitianK-groups, analogous to orthogonal and symplectic topologicalK-theory.The proofs use in an essential way higherKSC-theories, extending those of Anderson and Green. They also provide an upper bound for the higher hermitianK-groups in terms of higher algebraicK-groups.We also relate periodicity to étale hermitianK-groups by proving a hermitian version of Thomason's étale descent theorem. The results are illustrated in detail for local fields, rings of integers in number fields, smooth complex algebraic varieties, rings of continuous functions on compact spaces, and group rings.

Hermitian periodicity and cohomology of infinite orthogonal groups

“…As an application of our papers [2], [4], in favourable cases we prove the periodicity of hermitian K-groups with a shorter period than previously obtained. We also compute the homology and cohomology with …eld coe¢ cients of in…nite orthogonal and symplectic groups of speci…c rings of integers in a number …eld.…”

“…Firstly, we prove a re…nement of our periodicity theorem proved in [4] which leads to a shorter period for hermitian K-groups of speci…c rings A. This is the case if A is an algebra over the real sub…eld R of a cyclotomic …eld.…”

“…In this section we establish a more re…ned periodicity theorem for a class of rings introduced in [4], and there called "hermitian regular". As seen in [3], this class includes many rings (and more generally schemes) of geometric nature.…”

“…In order to prove this theorem and more general ones below, we combine our general periodicity theorems [4,Theorem 0.7] with the more detailed information in [4, Theorem 2.6] for totally real 2-regular number …elds, so as to obtain sharper periodicity results for certain algebras A with involution.…”

“…We provide applications of our previous papers [2] and [4] in two directions. Firstly, we prove a re…nement of our periodicity theorem proved in [4] which leads to a shorter period for hermitian K-groups of speci…c rings A.…”