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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

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


Authors: I. Panin and C. Walter
Original publication: Algebra i Analiz, tom 30 (2018), nomer 6.
Journal: St. Petersburg Math. J. 30 (2019), 933-972
MSC (2010): Primary 14C15
DOI: https://doi.org/10.1090/spmj/1578
Published electronically: September 16, 2019
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Abstract: An algebraic commutative ring $ T$-spectrum $ \mathbf {BO}$ is constructed such that it is stably fibrant, $ (8,4)$-periodic, and on $ \mathcal {S}m\mathcal Op/S$ the cohomology theory $ (X,U) \mapsto \mathbf {BO}^{p,q}(X_+/U_+)$ and Schlichting's Hermitian $ K$-theory functor $ (X,U) \mapsto \mathrm {KO}^{[q]}_{2q-p}(X,U)$ are canonically isomorphic. The motivic weak equivalence

$\displaystyle \mathbb{Z} \times HGr \xrightarrow {\sim } \mathbf {KSp}$    

relating the infinite quaternionic Grassmannian to symplectic $ K$-theory is used to equip $ \mathbf {BO}$ with the structure of a commutative monoid in the motivic stable homotopy category. When the base scheme is $ \mathrm {Spec}\,\mathbb{Z}[\frac 12]$, this monoid structure and the induced ring structure on the cohomology theory $ \mathbf {BO}^{*,*}$ are unique structures compatible with the products

$\displaystyle \mathrm {KO}^{[2m]}_0(X) \times \mathrm {KO}^{[2n]}_0(Y) \to \mathrm {KO}^{[2m+2n]}_0(X \times Y)$    

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 $ \mathbf {BO}^{*,*}(T \wedge T)$ in the same way as multiplication by the Grothendieck-Witt class of the symmetric bilinear space $ \langle {-1}\rangle $.

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Additional Information

I. Panin
Affiliation: St. Petersburg Branch Steklov Mathematical Institute, 27 Fontanka, 191023 St. Petersburg, Russia
Email: paniniv@gmail.com

C. Walter
Affiliation: Laboratoire J.-A. Dieudonné, UMR 6621 du CNRS, Université de Nice – Sophia Antipolis, 28 Avenue Valrose 06108, Nice Cedex 02, France
Email: walter@unice.fr

DOI: https://doi.org/10.1090/spmj/1578
Keywords: Hermitian $K$-theory, Grothendieck--Witt groups, symplectic orientation
Received by editor(s): April 24, 2018
Published electronically: September 16, 2019
Additional Notes: The first author gratefully acknowledges excellent working conditions and support provided by Laboratoire J.-A. Dieudonné, UMR 6621 du CNRS, Université de Nice Sophia Antipolis, and by the RCN Frontier Research Group Project no. 250399 “Motivic Hopf equations” at University of Oslo, and by the RFBR-grant no. 16-01-00750
Article copyright: © Copyright 2019 American Mathematical Society