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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:
\(\mathbf{BO}\) is stably fibrant,
it is \((8,4)\)-periodic,
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.

14C15 (Equivariant) Chow groups and rings; motives
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[1] Adams:1974bk J. F. Adams, Stable homotopy and generalized homology, Univ. Chicago Lecture Notes, Univ. Chicago Press, Chicago, 1974. · Zbl 0309.55016
[2] Atiyah:1971zr M. F. Atiyah, Riemann surfaces and spin structures, Ann. Sci. \'Ecole Norm. Sup. (4)  4 (1971), 47-62. · Zbl 0212.56402
[3] Balmer:2000hb P. Balmer, Triangular Witt groups. I. The \(12\)-term localization exact sequence, \(K\)-Theory  19 (2000), no. 4, 311-363. · Zbl 0953.18003
[4] Balmer:2002rp P. Balmer and C. Walter, A Gersten-Witt spectral sequence for regular schemes, Ann. Sci. \'Ecole Norm. Sup. (4)  35 (2002), no. 1, 127-152. · Zbl 1012.19003
[5] Barge:2008it J. Barge and J. Lannes, Suites de Sturm, indice de Maslov et p\'eriodicit\'e de Bott, Progress Math., vol. 267, Birkh\"auser Verlag, Basel, 2008.
[6] CF:1966bk P. E. Conner and E. E. Floyd, The relation of cobordism to \(K\)-theory. Lecture Notes in Math., vol. 28, Springer-Verlag, Berlin, 1966. · Zbl 0161.42802
[7] Friedlander:2002aa E. M. Friedlander and A. Suslin, The spectral sequence relating algebraic \(K\)-theory to motivic cohomology, Ann. Sci. \'Ecole Norm. Sup. (4)  35 (2002), no. 6, 773-875. · Zbl 1047.14011
[8] Gille:2007hb S. Gille, The general d\'evissage theorem for Witt groups of schemes, Arch. Math. (Basel)  88 (2007), no. 4, 333-343. · Zbl 1175.19001
[9] Gille:2003ad S. Gille and A. Nenashev, Pairings in triangular Witt theory, J. Algebra  261 (2003), no. 2, 292-309. · Zbl 1016.18007
[10] Hornbostel:2005ph J. Hornbostel, \(A\sp 1\)-representability of Hermitian \(K\)-theory and Witt groups, Topology  44 (2005), no. 3, 661-687. · Zbl 1078.19004
[11] Jardine:2000aa F. Jardine, Motivic symmetric spectra, Doc. Math.  5 (2000), 445-552. · Zbl 0969.19004
[12] Karoubi:1975aa M. Karoubi, Localisation de formes quadratiques. I, Ann. Sci. \'Ecole Norm. Sup. (4)  7 (1975), 359-403. · Zbl 0325.18011
[13] Karoubi:1980aa M. Karoubi, Le th\'eor\`“eme fondamental de la \(K\)-th\'”eorie hermitienne, Ann. of Math.(2)  112 (1980), no. 2, 259-282. · Zbl 0483.18008
[14] Morel:1999ab F. Morel and V. Voevodsky, \( \mathbfA^1\)-homotopy theory of schemes, Inst. Hautes \'Etudes Sci. Publ. Math.  90 (1999), 45-143.
[15] Nenashev:2007rm A. Nenashev, Gysin maps in Balmer-Witt theory, J. Pure Appl. Algebra  211 (2007), no. 1, 203-221. · Zbl 1140.11024
[16] Panin:2003rz I. Panin, Oriented cohomology theories of algebraic varieties, Special issue in honor of Hyman Bass on his seventieth birthday. Pt. III, \(K\)-Theory  30 (2003), no. 3, 265-314. · Zbl 1047.19001
[17] Panin:2009rz I. Panin, Oriented cohomology theories of algebraic varieties. \rm II (after I. Panin and A. Smirnov\/), Homology Homotopy Appl.  11 (2009), no. 1, 349-405. · Zbl 1169.14016
[18] Panin:2009aa I. Panin, K. Pimenov and O. R\`“ondigs, On Voevodsky”s algebraic \(K\)-theory spectrum, Algebraic topology, vol. 4, Abel Symp., Springer, Berlin, 2009, pp. 279-330. · Zbl 1179.14022
[19] Panin:2010fk I. Panin and C. Walter, Quaternionic Grassmannians and Borel classes in algebraic geometry, Preprint, 2018.
[20] Panin:2010fk2 I. Panin and C. Walter, On the algebraic cobordism spectra, MSL and MSp Preprint, 2018. · Zbl 1442.19012
[21] Panin:2010fk3 I. Panin and C. Walter, On the relation of symplectic algebraic cobordism to Hermitian \(K\)-theory, Preprint, 2018. · Zbl 1442.19012
[22] Schlichting:2004aa M. Schlichting, Hermitian \(K\)-theory on a theorem of Giffen, \(K\)-Theory  32 (2004), no. 3, 253-267. · Zbl 1070.19006
[23] Schlichting:2006aa M. Schlichting, Hermitian \(K\)-theory, derived equivalences and Karoubi’s fundamental theorem, Draft, 2006. · Zbl 1360.19008
[24] Schlichting:2010uq M. Schlichting, The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes, Invent. Math.  179 (2010), no. 2, 349-433. · Zbl 1193.19005
[25] Voevodsky:1998kx V. Voevodsky, \( \mathbfA\sp 1\)-homotopy theory, Proc. Intern. Congress Math., Vol. I (Berlin, 1998), Doc. Math.  1998, Extra vol. I, 579-564. · Zbl 0907.19002
[26] Voevodsky:2007aa V. Voevodsky, O. R\`“ondigs, and P. A. stvr, Voevodsky”s Nordfjordeid lectures: Motivic homotopy theory, Motivic homotopy theory, Universitext, Springer, Berlin, 2007, pp. 147-221.
[27] Walter:2010ab C. Walter, The formal group law for Borel classes in Hermitian \(K\)-theory, 2010 (in preparation).
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