Framed motives of algebraic varieties (after V. Voevodsky)
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- by Grigory Garkusha and Ivan Panin
- J. Amer. Math. Soc. 34 (2021), 261-313
- DOI: https://doi.org/10.1090/jams/958
- Published electronically: December 3, 2020
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Abstract:
A new approach to stable motivic homotopy theory is given. It is based on Voevodsky’s theory of framed correspondences. Using the theory, framed motives of algebraic varieties are introduced and studied in the paper. They are the major computational tool for constructing an explicit quasi-fibrant motivic replacement of the suspension $\mathbb P^1$-spectrum of any smooth scheme $X\in Sm/k$. Moreover, it is shown that the bispectrum \begin{equation*} (M_{fr}(X),M_{fr}(X)(1),M_{fr}(X)(2),\ldots ), \end{equation*} each term of which is a twisted framed motive of $X$, has the motivic homotopy type of the suspension bispectrum of $X$. Furthermore, an explicit computation of infinite $\mathbb P^1$-loop motivic spaces is given in terms of spaces with framed correspondences. We also introduce big framed motives of bispectra and show that they convert the classical Morel–Voevodsky motivic stable homotopy theory into an equivalent local theory of framed bispectra. As a topological application, it is proved that the framed motive $M_{fr}(pt)(pt)$ of the point $pt=\operatorname {Spec} k$ evaluated at $pt$ is a quasi-fibrant model of the classical sphere spectrum whenever the base field $k$ is algebraically closed of characteristic zero.References
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Bibliographic Information
- Grigory Garkusha
- Affiliation: Department of Mathematics, Swansea University, Fabian Way, Swansea SA1 8EN, United Kingdom
- MR Author ID: 660286
- ORCID: 0000-0001-9836-0714
- Email: g.garkusha@swansea.ac.uk
- Ivan Panin
- Affiliation: St. Petersburg Branch of V. A. Steklov Mathematical Institute, Fontanka 27, 191023 St. Petersburg, Russia
- MR Author ID: 238161
- Email: paniniv@gmail.com
- Received by editor(s): March 11, 2018
- Received by editor(s) in revised form: January 15, 2020, January 23, 2020, April 21, 2020, and April 22, 2020
- Published electronically: December 3, 2020
- © Copyright 2020 American Mathematical Society
- Journal: J. Amer. Math. Soc. 34 (2021), 261-313
- MSC (2020): Primary 14F42, 14F45; Secondary 55Q10, 55P47
- DOI: https://doi.org/10.1090/jams/958
- MathSciNet review: 4188819
Dedicated: In memory of Vladimir Voevodsky