On the motivic commutative ring spectrum $\mathbf {BO}$

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) 7→ BO(X+/U+) and Schlichting’s hermitian K-theory functor (X, U) 7→ KO [q] 2q−p(X, U) are canonically isomorphic. We use the motivic weak equivalence Z×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 SpecZ[ 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)× KO [2n] 0 (Y ) → KO [2m+2n] 0 (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∧T ) in the same way as multiplication by the Grothendieck-Witt class of the symmetric bilinear space 〈−1〉.