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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
Full Text:
##### References:
 [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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