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Framed motives of algebraic varieties (after V. Voevodsky)

Authors: Grigory Garkusha and Ivan Panin
Journal: J. Amer. Math. Soc. 34 (2021), 261-313
MSC (2020): Primary 14F42, 14F45; Secondary 55Q10, 55P47
Published electronically: December 3, 2020
MathSciNet review: 4188819
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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.

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Additional 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

Ivan Panin
Affiliation: St. Petersburg Branch of V. A. Steklov Mathematical Institute, Fontanka 27, 191023 St. Petersburg, Russia
MR Author ID: 238161

Keywords: Motivic homotopy theory, framed correspondences, motivic infinite loop spaces
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
Dedicated: In memory of Vladimir Voevodsky
Article copyright: © Copyright 2020 American Mathematical Society