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Motivic and real étale stable homotopy theory

  • Tom Bachmann (a1)


Let $S$ be a Noetherian scheme of finite dimension and denote by $\unicode[STIX]{x1D70C}\in [\unicode[STIX]{x1D7D9},\mathbb{G}_{m}]_{\mathbf{SH}(S)}$ the (additive inverse of the) morphism corresponding to $-1\in {\mathcal{O}}^{\times }(S)$ . Here $\mathbf{SH}(S)$ denotes the motivic stable homotopy category. We show that the category obtained by inverting $\unicode[STIX]{x1D70C}$ in $\mathbf{SH}(S)$ is canonically equivalent to the (simplicial) local stable homotopy category of the site $S_{\text{r}\acute{\text{e}}\text{t}}$ , by which we mean the small real étale site of $S$ , comprised of étale schemes over $S$ with the real étale topology. One immediate application is that $\mathbf{SH}(\mathbb{R})[\unicode[STIX]{x1D70C}^{-1}]$ is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the $\unicode[STIX]{x1D70C}$ -local sphere (over $\mathbb{R}$ ). As further applications we show that $D_{\mathbb{A}^{1}}(k,\mathbb{Z}[1/2])^{-}\simeq \mathbf{DM}_{W}(k)[1/2]$ (improving a result of Ananyevskiy–Levine–Panin), reprove Röndigs’ result that $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D7D9}[1/\unicode[STIX]{x1D702},1/2])=0$ for $i=1,2$ and establish some new rigidity results.



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[ALP17] Ananyevskiy, A., Levine, M. and Panin, I., Witt sheaves and the 𝜂-inverted sphere spectrum , J. Topol. 10 (2017), 370385.
[ABR12] Andradas, C., Bröcker, L. and Ruiz, J. M., Constructible sets in real geometry, vol. 33 (Springer, 2012).
[Ayo07] Ayoub, J., Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, vol. 315 (Société mathématique de France, 2007).
[Bac16] Bachmann, T., On the conservativity of the functor assigning to a motivic spectrum its motive, Duke Math. J., to appear. Preprint (2016), arXiv:1506.07375.
[Bal10] Balmer, P., Spectra, spectra, spectra–tensor triangular spectra versus Zariski spectra of endomorphism rings , Algebr. Geom. Topol. 10 (2010), 15211563.
[Bar10] Barwick, C., On left and right model categories and left and right Bousfield localizations , Homology, Homotopy Appl. 12 (2010), 245320.
[CF17] Calmès, B. and Fasel, J., Finite Chow–Witt correspondences, Preprint (2017),arXiv:1412.2989v2.
[CD09] Cisinski, D.-C. and Déglise, F., Local and stable homological algebra in Grothendieck abelian categories , Homology, Homotopy Appl. 11 (2009), 219260.
[CD12] Cisinski, D.-C. and Déglise, F., Triangulated categories of mixed motives, Preprint (2012),arXiv:0912.2110v3.
[CD13] Cisinski, D.-C. and Déglise, F., Étale motives , Compos. Math. 152 (2013), 556666.
[Del91] Delfs, H., Homology of locally semialgebraic spaces, Lecture Notes in Mathematics, vol. 1484 (Springer, 1991).
[Dug14] Dugger, D., Coherence for invertible objects and multigraded homotopy rings , Algebr. Geom. Topol. 14 (2014), 10551106.
[Gab92] Gabber, O., K-theory of Henselian local rings and Henselian pairs , Contemp. Math. 126 (1992), 5970.
[GK15] Gabber, O. and Kelly, S., Points in algebraic geometry , J. Pure Appl. Algebra 219 (2015), 46674680.
[Gil17] Gille, S., On quadratic forms over semilocal rings , Trans. Amer. Math. Soc. (2017), to appear.
[GSZ16] Gille, S., Scully, S. and Zhong, C., Milnor–Witt K-groups of local rings , Adv. Math. 286 (2016), 729753.
[GT84] Gillet, H. A. and Thomason, R. W., The K-theory of strict Hensel local rings and a theorem of Suslin , J. Pure Appl. Algebra 34 (1984), 241254.
[HO16] Heller, J. and Ormsby, K., Galois equivariance and stable motivic homotopy theory , Trans. Amer. Math. Soc. 368 (2016), 80478077.
[Hir09] Hirschhorn, P. S., Model categories and their localizations, vol. 99 (American Mathematical Society, 2009).
[Hov01] Hovey, M., Spectra and symmetric spectra in general model categories , J. Pure Appl. Algebra 165 (2001), 63127.
[HY07] Hornbostel, J. and Yagunov, S., Rigidity for Henselian local rings and A1 -representable theories , Math. Z. 255 (2007), 437449.
[Hoy17] Hoyois, M., Cdh descent in equivariant homotopy K-theory, Preprint (2017),arXiv:1604.06410v3.
[HKO11] Hu, P., Kriz, I. and Ormsby, K., Convergence of the motivic Adams spectral sequence , J. K-Theory 7 (2011), 573596.
[Jac17] Jacobson, J., Real cohomology and the powers of the fundamental ideal in the Witt ring , Ann. K-Theory 2 (2017), 357385.
[Jar15] Jardine, J. F., Local homotopy theory (Springer, 2015).
[KSW16] Karoubi, M., Schlichting, M. and Weibel, C., The Witt group of real algebraic varieties , J. Topol. 9 (2016), 12571302.
[Kne77] Knebusch, M., Symmetric bilinear forms over algebraic varieties, conference on quadratic forms , in Queen’s papers in pure and applied mathematics, vol. 46, ed. Orzech, G. (Queens University, Kingston, ON, 1977), 103283.
[Lur09] Lurie, J., Higher topos theory, vol. 170 (Princeton University Press, 2009).
[Lur16] Lurie, J., Higher algebra (2016), available at∼lurie/papers/HA.pdf.
[MS06] May, J. P. and Sigurdsson, J., Parametrized homotopy theory, vol. 132 (American Mathematical Society, Providence, RI, 2006).
[MH73] Milnor, J. W. and Husemoller, D., Symmetric bilinear forms, vol. 60 (Springer, 1973).
[Mor03] Morel, F., An introduction to A1 -homotopy theory , ICTP Trieste Lecture Note Ser. 15 (2003), 357441.
[Mor04] Morel, F., Sur les puissances de l’idéal fondamental de l’anneau de Witt , Comment. Math. Helv. 79 (2004), 689703.
[Mor05] Morel, F., The stable A1 -connectivity theorems , K-Theory 35 (2005), 168.
[Mor12] Morel, F., A1 -algebraic topology over a field, Lecture Notes in Mathematics (Springer, Berlin, Heidelberg, 2012).
[MV99] Morel, F. and Voevodsky, V., A1 -homotopy theory of schemes , Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45143 (English).
[Rön16] Röndigs, O., On the 𝜂-inverted sphere , TIFR Proc. Int. Colloq. K-theory (2016), to appear.
[RØ08] Röndigs, O. and Østvær, P., Rigidity in motivic homotopy theory , Math. Ann. 341 (2008), 651675 (English).
[RSØ16] Röndigs, O., Spitzweck, M. and Østvær, P. A., The first stable homotopy groups of motivic spheres, Preprint (2016), arXiv:1604.00365.
[Sch85] Scharlau, W., Quadratic and Hermitian forms, Grundlehren der mathematischen Wissenschaften, vol. 270 (Springer, 1985).
[Sch94] Scheiderer, C., Real and etale cohomology, Lecture Notes in Mathematics, vol. 1588 (Springer, Berlin, 1994).
[Sus83] Suslin, A., On the K-theory of algebraically closed fields , Invent. Math. 73 (1983), 241245.
[SV96] Suslin, A. and Voevodsky, V., Singular homology of abstract algebraic varieties , Invent. Math. 123 (1996), 6194.
[Sta17] The Stacks Project Authors, Stacks Project,, 2017.
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Motivic and real étale stable homotopy theory

  • Tom Bachmann (a1)


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