On the motivic commutative ring spectrum BO.

By Ivan Panin, C. Walter. 2010

51 citations recorded by the source corpus.

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 .

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ded48cad06018b5bbfc5b168672dafdd56c2892f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Fundamental classes in motivic homotopy theory (2018, background): “In particular, the ring spectrum HMW is symplectically oriented in the sense of Panin and Walter [PW19].”
Geometric models for higher Grothendieck-Witt groups in A1-homotopy theory (2013, background): “Also, it is known from [PW10] that BetO ∼= GrO• in H•(S); we give an alternative proof of a more precise version in Proposition 8.”
Slices of hermitian $K$–theory and Milnor's conjecture on quadratic forms (2013, methodology): “(The latter is deduced using the model for KQ from [39].”
Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem (2012, methodology): “Results of this article have already been used in [FS09], [Zib11], [PW10], in the solution of a conjecture of Suslin on the structure of stably free projective modules [FSR11] and in the solution of a conjecture of Williams and the analogue of the Quillen-Lichtenbaum conjecture for hermitian K-theor”
The generalized slices of Hermitian K‐theory (2016, background): “The geometric representability theorem of Panin-Walter [19] (for the case of symplecticK-theory, which is all we need here) and Schlichting-Tripathi [24] (for the general case) implies that GW [2] ≃ Z×HGr.”
A cohomological classification of vector bundles on smooth affine threefolds (2012, methodology): “ry 2.8. This result was stated by Wendt in a slightly different form in [Wen11, Theorem 6.8 and Remark 6.12]. The hypothesis on the characteristic of the base-ﬁeld is required to apply the results of [PW10]. Remark 2.11. In [MV99, §4], a different geometric model BgmSp 2n for the classifying space S”