Garkusha, Grigory; Panin, Ivan Framed motives of algebraic varieties (after V. Voevodsky). (English) Zbl 1491.14034 J. Am. Math. Soc. 34, No. 1, 261-313 (2021). In this groundbreaking paper, the authors lay the foundation for motivic infinite loop space theory. Developing ideas of V. Voevodsky (outlined in unpublished notes), they define framed correspondences and prove that they provide computationally accessible models for infinite loop spaces of suspension spectra of smooth schemes.What does this mean? Motivic homotopy theory is a homotopy theory of smooth schemes introduced by F. Morel and V. Voevodsky [Publ. Math., Inst. Hautes Étud. Sci. 90, 45–143 (1999; Zbl 0983.14007)]. It has a stable variant in which smash product with the projective line \(\mathbb{P}^1\) becomes invertible. This forms a category of motivic spectra which is related to motivic spaces in the same way that topological spectra (the objects of classical stable homotopy theory) are related to topological spaces. In particular, we have an adjunction \((\Sigma^\infty_{\mathbb{P}^1},\Omega^{\infty}_{\mathbb{P}^1})\) between pointed motivic spaces and motivic spectra. As such, understanding the (nonnegative) homotopy groups (or sheaves) of \(\Omega^\infty_{\mathbb P^1}E\) is equivalent to understanding the (nonnegative, stable) homotopy groups (or sheaves) of \(E\).Motivic spectra represent cohomology theories of algebro-geometric interest – motivic cohomology, (homotopy) algebraic \(K\)-theory, Hermitian \(K\)-theory, and algebraic cobordism, to name a few – and thus it is very desirable to have good models for motivic infinite loop spaces. The authors verify that framed correspondences provide such models for \(\mathbb P^1\)-suspension spectra of simplicial smooth schemes under mild hypotheses. Indeed, a mild reinterpretation of Theorem 1.3 (see also Theorem 10.7) says that for a smooth scheme \(X\) over an infinite perfect field, the Nisnevich localization of the \(\mathbb{A}^1\)-localization of the group completion of the presheaf taking \(U\in Sm/k\) to the space of framed correspondences from \(U\) to \(X\) is an equivalence of simplicial presheaves on \(Sm/k\). This theorem is the computational basis for the \(\infty\)-categorical elaboration of motivic infinite loop space machines (in the spirit of Segal’s \(\Gamma\)-spaces) due to E. Elmanto et al. [Camb. J. Math. 9, No. 2, 431–549 (2021; Zbl 07422194)].The definition of framed correspondences is too technical for this review, and I also will not elaborate on the theory of ‘big framed motives’ introduced by the authors. An important warning, though, is in order: to the contrary of the authors’ claims, framed correspondences do not form the morphism sets of a category. One usually composes correspondences via pullback, but the data of a framed correspondence (speicifically the closed subset \(Z\) of \(\mathbb A^n_X\) from Definition 2.1) is not preserved by this construction. See [loc. cit., Paragraph 1.3.7] for an elaboration of this point, noting that those authors refer to Garkusha-Panin’s framed correspondences as equationally framed correspondences. This categorical frustration does not impact the arguments in the reviewed paper. Reviewer: Kyle Ormsby (Portland) Cited in 5 ReviewsCited in 17 Documents MSC: 14F42 Motivic cohomology; motivic homotopy theory 14F45 Topological properties in algebraic geometry 55Q10 Stable homotopy groups 55P47 Infinite loop spaces Keywords:motivic homotopy theory; framed correspondences; motivic infinite loop spaces Citations:Zbl 0983.14007; Zbl 07422194 PDFBibTeX XMLCite \textit{G. Garkusha} and \textit{I. Panin}, J. Am. Math. Soc. 34, No. 1, 261--313 (2021; Zbl 1491.14034) Full Text: DOI arXiv Link References: [1] AGP A. Ananyevskiy, G. Garkusha, I. Panin, Cancellation theorem for framed motives of algebraic varieties, 1601.06642. · Zbl 1471.14050 [2] Blander, Benjamin A., Local projective model structures on simplicial presheaves, \(K\)-Theory, 24, 3, 283-301 (2001) · Zbl 1073.14517 [3] DP A. Druzhinin and I. Panin, Surjectivity of the etale excision map for homotopy invariant framed presheaves, 1808.07765. [4] Dundas, Bj\o rn Ian; R\"{o}ndigs, Oliver; \O stv\ae r, Paul Arne, Enriched functors and stable homotopy theory, Doc. Math., 8, 409-488 (2003) · Zbl 1040.55002 [5] Dundas, B. I.; Levine, M.; \O stv\ae r, P. A.; R\"{o}ndigs, O.; Voevodsky, V., Motivic homotopy theory, Universitext, x+221 pp. (2007), Springer-Verlag, Berlin [6] Garkusha, Grigory; Panin, Ivan, \(K\)-motives of algebraic varieties, Homology Homotopy Appl., 14, 2, 211-264 (2012) · Zbl 1284.14029 [7] Garkusha, Grigory; Panin, Ivan, On the motivic spectral sequence, J. Inst. Math. Jussieu, 17, 1, 137-170 (2018) · Zbl 1390.19006 [8] GP4 G. Garkusha and I. Panin, Homotopy invariant presheaves with framed transfers, Cambridge J. Math. 8 (2020), no. 1, 1-94. · Zbl 1453.14066 [9] GPN G. Garkusha, A. Neshitov, and I. Panin, Framed motives of relative motivic spheres, 1604.02732. · Zbl 1484.14048 [10] Grayson, Daniel R., Weight filtrations via commuting automorphisms, \(K\)-Theory, 9, 2, 139-172 (1995) · Zbl 0826.19003 [11] Grothendieck, A., \'{E}l\'{e}ments de g\'{e}om\'{e}trie alg\'{e}brique. IV. \'{E}tude locale des sch\'{e}mas et des morphismes de sch\'{e}mas IV, Inst. Hautes \'{E}tudes Sci. Publ. Math., 32, 361 pp. (1967) · Zbl 0153.22301 [12] Hovey, Mark, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra, 165, 1, 63-127 (2001) · Zbl 1008.55006 [13] Isaksen, Daniel C., Flasque model structures for simplicial presheaves, \(K\)-Theory, 36, 3-4, 371-395 (2006) (2005) · Zbl 1116.18008 [14] Jardine, J. F., Simplicial presheaves, J. Pure Appl. Algebra, 47, 1, 35-87 (1987) · Zbl 0624.18007 [15] Jardine, J. F., Motivic symmetric spectra, Doc. Math., 5, 445-552 (2000) · Zbl 0969.19004 [16] Levine, Marc, A comparison of motivic and classical stable homotopy theories, J. Topol., 7, 2, 327-362 (2014) · Zbl 1333.14021 [17] Morel, Fabien, \( \mathbb{A}^1\)-algebraic topology over a field, Lecture Notes in Mathematics 2052, x+259 pp. (2012), Springer, Heidelberg · Zbl 1263.14003 [18] Morel, Fabien; Voevodsky, Vladimir, \( \mathbf{A}^1\)-homotopy theory of schemes, Inst. Hautes \'{E}tudes Sci. Publ. Math., 90, 45-143 (2001) (1999) · Zbl 0983.14007 [19] Neshitov, Alexander, Framed correspondences and the Milnor-Witt \(K\)-theory, J. Inst. Math. Jussieu, 17, 4, 823-852 (2018) · Zbl 1407.14016 [20] Segal, Graeme, Categories and cohomology theories, Topology, 13, 293-312 (1974) · Zbl 0284.55016 [21] Suslin, Andrei; Voevodsky, Vladimir, Singular homology of abstract algebraic varieties, Invent. Math., 123, 1, 61-94 (1996) · Zbl 0896.55002 [22] Suslin, Andrei; Voevodsky, Vladimir, 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. 548, 117-189 (2000), Kluwer Acad. Publ., Dordrecht · Zbl 1005.19001 [23] Voevodsky, Vladimir, \( \mathbf{A}^1\)-homotopy theory, Doc. Math.. Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), Extra Vol. I, 579-604 (1998) · Zbl 0907.19002 [24] Voe2 V. Voevodsky, Notes on framed correspondences, unpublished, 2001. Also available at https://www.math.ias.edu/vladimir/publications [25] Voevodsky, Vladimir, Simplicial radditive functors, J. K-Theory, 5, 2, 201-244 (2010) · Zbl 1194.55021 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.