On the motivic commutative ring spectrum BO

An algebraic commutative ring T-spectrum BO is constructed such that it is stably fibrant, $(8,4)$-periodic, and on Sm Op/S the cohomology theory (X,U) ↦ BOp,q(X₊/U₊) and Schlichting’s Hermitian K-theory functor (X,U) ↦ KO[q]2q-p(X,U) are canonically isomorphic. The motivic weak equivalence {equation*} Z × HGr {~ } KSp {equation*} relating the infinite quaternionic Grassmannian to symplectic K-theory is used to equip BO with the structure of a commutative monoid in the motivic stable homotopy category. When the base scheme is Spec Z[ 12], this monoid structure and the induced ring structure on the cohomology theory BO*,* are unique structures compatible with the products {equation*} KO[2m]_0(X) × KO[2n]_0(Y) → KO[2m+2n]_0(X × Y) {equation*} 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 T) in the same way as multiplication by the Grothendieck–Witt class of the symmetric bilinear space -1.