# On the motivic commutative ring spectrum BO

* Corresponding author
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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Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-00531729
Contributor : Charles Walter <>
Submitted on : Wednesday, November 3, 2010 - 4:02:48 PM
Last modification on : Monday, October 12, 2020 - 10:27:28 AM

### Identifiers

• HAL Id : hal-00531729, version 1
• ARXIV : 1011.0650

### Citation

Ivan Panin, Charles Walter. On the motivic commutative ring spectrum BO. 2010. ⟨hal-00531729⟩

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