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On the motivic commutative ring spectrum \(\mathbf{BO}\). (English) Zbl 1428.14011
St. Petersbg. Math. J. 30, No. 6, 933-972 (2019) and Algebra Anal. 30, No. 6, 43-96 (2018).
Let \({\mathrm{\mathcal{S}m}}/S\) be the category of smooth \(S\)-schemes of finite type. \(S\) is assumed to be regular, Noetherian separated of finite Krull dimension and such that \(\frac{1}{2}\in {\Gamma}(S,\mathcal{O}_S)\). Let \({\mathrm{\mathcal{S}m\mathcal{O}p}}/S\) be the category of pairs \((X,U)\) in \({\mathrm{\mathcal{S}m}}/S\) with the usual morphisms. The authors construct an algebraic commutative ring \(T\)-spectrum \(\mathbf{BO}\) with the following properties:
1)
\(\mathbf{BO}\) is stably fibrant,
2)
it is \((8,4)\)-periodic,
3)
on \({\mathrm{\mathcal{S}m\mathcal{O}p}}/S\) the resulting cohomology theory \((X,U)\rightarrow {\mathbf{BO}}^{p,q}(X_{+}/U_{+})\) is canonically isomorphic to the Schlichting’s Hermitian \(K\)-theory \((X,U)\rightarrow KO^{[q]}_{2q-p}(X,U)\)
The authors also equip \(\mathbf{BO}\) with the structure of a compatible monoid in motivic stable homotopy category. They show that if \(S={\mathrm{Spec}}{\mathbb Z}[\frac{1}{2}]\) this monoidal structure and the induced ring structure on the \(\mathbf{BO}\)-cohomology ring are compatible with the products: \[K_{0}^{[2m]}(X)\times K_{0}^{[2n]}(Y)\rightarrow K_{0}^{[2m+2n]}(X\times Y)\] induced on Grothendieck-Witt groups by the tensor product of symmetric chain complexes.

MSC:
14C15 (Equivariant) Chow groups and rings; motives
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