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Cobordism-framed correspondences and the Milnor \(K\)-theory. (English. Russian original) Zbl 1453.14070

St. Petersbg. Math. J. 32, No. 1, 183-198 (2021); translation from Algebra Anal. 32, No. 1, 244-264 (2020).
Summary: The 0th cohomology group is computed for a complex of groups of cobordism-framed correspondences. In the case of ordinary framed correspondences, an analogous computation was completed by A. Neshitov in his paper “Framed correspondences and the Milnor-Witt \(K\)-theory”. Neshitov’s result is, at the same time, a computation of the homotopy groups \(\pi_{i,i}(S^0)(\operatorname{Spec}(k))\), and the present work might be used subsequently as a basis for computing the homotopy groups \(\pi_{i,i}(MGL_{\bullet })(\operatorname{Spec}(k))\) of the spectrum \(MGL_{\bullet } \).

MSC:

14F42 Motivic cohomology; motivic homotopy theory
19D45 Higher symbols, Milnor \(K\)-theory
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References:

[1] H. Bass and J. Tate, The Milnor ring of a global field, Algebraic K-theory. II. “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972), Lecture Notes in Math., vol. 342, Springer, Berlin, 1973, pp. 349-446.
[2] A. Neshitov, Framed correspondences and the Milnor-Witt \(K\)-theory, J. Inst. Math. Jussieu 17 (2018), no. 4, 823-852. · Zbl 1407.14016
[3] G. Garkusha and I. Panin, Framed motives of algebraic varieties (after V. Voevodsky), arXiv:1409.4372 [math.KT], 2014. · Zbl 1491.14034
[4] V. Voevodsky, Notes on framed correspondences, unpublished, 2001. Available at http://www.math.ias.edu/Voevodsky/files/files-annotated/Dropbox/Unfinished_papers/Motives/Framed/framed.pdf.
[5] A. Suslin and V. 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. \newpage · Zbl 1005.19001
[6] R. Vakil, The rising sea: Foundations of algebraic geometry, http://math.stanford.edu/ vakil/216blog/FOAGnov1817public.pdf
[7] G. Garkusha and A. Neshitov, Fibrant resolutions for motivic Thom spectra, arXiv: 1804.07621.
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