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

DOI:
https://doi.org/10.1090/jams/958

Published electronically:
December 3, 2020

MathSciNet review:
4188819

Full-text PDF

View in AMS MathViewer

Abstract | References | Similar Articles | Additional Information

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.

- A. Ananyevskiy, G. Garkusha, I. Panin,
*Cancellation theorem for framed motives of algebraic varieties*, arXiv:1601.06642. - Benjamin A. Blander,
*Local projective model structures on simplicial presheaves*, $K$-Theory**24**(2001), no. 3, 283–301. MR**1876801**, DOI 10.1023/A:1013302313123 - A. Druzhinin and I. Panin,
*Surjectivity of the etale excision map for homotopy invariant framed presheaves*, arXiv:1808.07765. - Bjørn Ian Dundas, Oliver Röndigs, and Paul Arne Østvær,
*Enriched functors and stable homotopy theory*, Doc. Math.**8**(2003), 409–488. MR**2029170** - B. I. Dundas, M. Levine, P. A. Østvær, O. Röndigs, and V. Voevodsky,
*Motivic homotopy theory*, Universitext, Springer-Verlag, Berlin, 2007. Lectures from the Summer School held in Nordfjordeid, August 2002. MR**2334212** - Grigory Garkusha and Ivan Panin,
*$K$-motives of algebraic varieties*, Homology Homotopy Appl.**14**(2012), no. 2, 211–264. MR**3007094**, DOI 10.4310/HHA.2012.v14.n2.a13 - Grigory Garkusha and Ivan Panin,
*On the motivic spectral sequence*, J. Inst. Math. Jussieu**17**(2018), no. 1, 137–170. MR**3742558**, DOI 10.1017/S1474748015000419 - G. Garkusha and I. Panin,
*Homotopy invariant presheaves with framed transfers*, Cambridge J. Math.**8**(2020), no. 1, 1–94. - G. Garkusha, A. Neshitov, and I. Panin,
*Framed motives of relative motivic spheres*, arXiv:1604.02732. - Daniel R. Grayson,
*Weight filtrations via commuting automorphisms*, $K$-Theory**9**(1995), no. 2, 139–172. MR**1340843**, DOI 10.1007/BF00961457 - A. Grothendieck,
*Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV*, Inst. Hautes Études Sci. Publ. Math.**32**(1967), 361 (French). MR**238860** - Mark Hovey,
*Spectra and symmetric spectra in general model categories*, J. Pure Appl. Algebra**165**(2001), no. 1, 63–127. MR**1860878**, DOI 10.1016/S0022-4049(00)00172-9 - Daniel C. Isaksen,
*Flasque model structures for simplicial presheaves*, $K$-Theory**36**(2005), no. 3-4, 371–395 (2006). MR**2275013**, DOI 10.1007/s10977-006-7113-z - J. F. Jardine,
*Simplicial presheaves*, J. Pure Appl. Algebra**47**(1987), no. 1, 35–87. MR**906403**, DOI 10.1016/0022-4049(87)90100-9 - J. F. Jardine,
*Motivic symmetric spectra*, Doc. Math.**5**(2000), 445–552. MR**1787949** - Marc Levine,
*A comparison of motivic and classical stable homotopy theories*, J. Topol.**7**(2014), no. 2, 327–362. MR**3217623**, DOI 10.1112/jtopol/jtt031 - Fabien Morel,
*$\Bbb A^1$-algebraic topology over a field*, Lecture Notes in Mathematics, vol. 2052, Springer, Heidelberg, 2012. MR**2934577** - Fabien Morel and Vladimir Voevodsky,
*$\textbf {A}^1$-homotopy theory of schemes*, Inst. Hautes Études Sci. Publ. Math.**90**(1999), 45–143 (2001). MR**1813224** - Alexander Neshitov,
*Framed correspondences and the Milnor-Witt $K$-theory*, J. Inst. Math. Jussieu**17**(2018), no. 4, 823–852. MR**3835524**, DOI 10.1017/S1474748016000190 - Graeme Segal,
*Categories and cohomology theories*, Topology**13**(1974), 293–312. MR**353298**, DOI 10.1016/0040-9383(74)90022-6 - Andrei Suslin and Vladimir Voevodsky,
*Singular homology of abstract algebraic varieties*, Invent. Math.**123**(1996), no. 1, 61–94. MR**1376246**, DOI 10.1007/BF01232367 - Andrei Suslin and Vladimir Voevodsky,
*Bloch-Kato conjecture and motivic cohomology with finite coefficients*, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 117–189. MR**1744945** - Vladimir Voevodsky,
*$\mathbf A^1$-homotopy theory*, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 579–604. MR**1648048** - V. Voevodsky,
*Notes on framed correspondences*, unpublished, 2001. Also available at https://www.math.ias.edu/vladimir/publications - Vladimir Voevodsky,
*Simplicial radditive functors*, J. K-Theory**5**(2010), no. 2, 201–244. MR**2640203**, DOI 10.1017/is010003026jkt097

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2020):
14F42,
14F45,
55Q10,
55P47

Retrieve articles in all journals with MSC (2020): 14F42, 14F45, 55Q10, 55P47

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

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

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