On the motivic commutative ring spectrum BO

TL;DR

This paper constructs a motivic commutative ring spectrum BO that aligns with hermitian K-theory, establishing a canonical isomorphism and a unique monoid structure compatible with Grothendieck-Witt groups.

Contribution

It introduces a new algebraic commutative ring spectrum BO in the motivic stable homotopy category with a canonical isomorphism to hermitian K-theory and a unique monoid structure.

Findings

BO spectrum is stably fibrant and (8,4)-periodic.

BO cohomology is canonically isomorphic to hermitian K-theory.

The monoid structure on BO is unique and compatible with Grothendieck-Witt groups.

Abstract

We construct an algebraic commutative ring T- spectrum BO which is stably fibrant and (8,4)- periodic and such that on SmOp/S the cohomology theory (X,U) -> BO^{p,q}(X_{+}/U_{+}) and Schlichting's hermitian K-theory functor (X,U) -> KO^{[q]}_{2q-p}(X,U) are canonically isomorphic. We use the motivic weak equivalence Z x HGr -> KSp relating the infinite quaternionic Grassmannian to symplectic $K$-theory to equip BO with the structure of a commutative monoid in the motivic stable homotopy category. When the base scheme is Spec Z[1/2], this monoid structure and the induced ring structure on the cohomology theory BO^{*,*} are the unique structures compatible with the products KO^{[2m]}_{0}(X) x KO^{[2n]}_{0}(Y) -> KO^{[2m+2n]}_{0}(X x Y). on Grothendieck-Witt groups induced by the tensor product of symmetric chain complexes. The cohomology theory is bigraded commutative with the switch map acting on BO^{*,*}(T^{2}) in the same way as multiplication by the Grothendieck-Witt class of the symmetric bilinear space <-1>.