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On the motivic commutative ring spectrum BO (1011.0650v2)

Published 2 Nov 2010 in math.AG and math.KT

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>.

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