Periodicity of Hermitian K-groups

Bott periodicity for the unitary and symplectic groups is fundamental to topological K-theory. Analogous to unitary topological K-theory, for algebraic K-groups with …nite coe¢ cients, similar results are consequences of the Milnor and Bloch-Kato conjectures, a¢ rmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraic K-groups for any ring implies periodicity for the hermitian K-groups, analogous to orthogonal and symplectic topological K-theory. The proofs use in an essential way higher KSC-theories, extending those of Anderson and Green. They also provide an upper bound for the higher hermitian K-groups in terms of higher algebraic K-groups. We also relate periodicity to étale hermitian K-groups by proving a hermitian version of Thomason’s étale descent theorem. The results are illustrated in detail for local …elds, rings of integers in number …elds, smooth complex algebraic varieties, rings of continuous functions on compact spaces, and group rings. 0. Introduction and statements of main results By the fundamental work of Bott [11] it is known that the homotopy groups of classical Lie groups are periodic, of period 2 or 8. For instance, the general linear and symplectic groups satisfy the isomorphisms: n(GL(R)) = n+8(GL(R)) n(Sp(C)) = n+8(Sp(C)) n(GL(C)) = n+2(GL(C)) These periodicity statements were interpreted by Atiyah, Hirzebruch and others in the framework of topological K-theory of a Banach algebra A: recall that there are isomorphisms K n (A) = K n+p(A), where K n (A) = n 1(GL(A)) if n > 0 and K top 0 (A) = K(A) is the usual Grothendieck group. Here p is the period which is 2 or 8 according as A is complex or real. We refer to [37] and [50] for an overview of the subject, both algebraically and topologically. A few years later, after higher algebraic K-theory was introduced by Quillen, an analogous periodicity statement was sought, of the form