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Cobordism-framed correspondences and the Milnor 𝐾-theory
St. Petersburg Mathematical Journal ( IF 0.7 ) Pub Date : 2021-01-11 , DOI: 10.1090/spmj/1643
A. Tsybyshev

Abstract: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)(\mathop {Spec}(k))$, and the present work might be used subsequently as a basis for computing the homotopy groups $ \pi _{i,i}(MGL_{\bullet })(\mathop {Spec}(k))$ of the spectrum $ MGL_{\bullet }$.
References [Enhancements On Off] (What's this?)
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[BT]
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. MR 0442061
[Nes]
A. Neshitov, Framed correspondences and the Milnor-Witt $ K$-theory, J. Inst. Math. Jussieu 17 (2018), no. 4, 823-852. MR 3835524
[GP]
G. Garkusha and I. Panin, Framed motives of algebraic varieties (after V. Voevodsky), arXiv:1409.4372 [math.KT], 2014.
[Voev]
V. Voevodsky, Notes on framed correspondences, unpublished, 2001.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.
[SV]
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. MR 1744945
[FOAG]
R. Vakil, The rising sea: Foundations of algebraic geometry, http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf
[GarNesh]
G. Garkusha and A. Neshitov, Fibrant resolutions for motivic Thom spectra, arXiv: 1804.07621.
Similar Articles Retrieve articles in St. Petersburg Mathematical Journal with MSC (2020): 19D45 Retrieve articles in all journals with MSC (2020): 19D45
Additional Information

A. Tsybyshev
Affiliation: St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia; and Chebyshev laboratory, St. Petersburg state university 14th Line 29B, Vasilyevsky Island, St. Petersburg 199178, Russia
Email: emperortsy@gmail.com

DOI: https://doi.org/10.1090/spmj/1643
Keywords: Framed correspondences, $A^1$-homotopy theory, algebraic cobordisms, Milnor $K$-theory
Received by editor(s): April 15, 2019
Published electronically: January 11, 2021
Additional Notes: The work was supported by the RFBR, grant no. 19-01-00513
Article copyright: © Copyright 2021 American Mathematical Society

中文翻译：
殖民主义框架的对应和米尔诺nor理论

摘要：第0次同调群是针对复杂的以协作框架为框架的对应群进行计算的。对于普通的框架式通信，A。涅希托夫在他的论文《框架式通信和Milnor-Witt$ K $理论》中完成了类似的计算。涅希托夫的结果同时是对同构基团的计算，本研究成果随后可作为计算光谱同构基团的基础。 $ \ pi _ {i，i}（S ^ 0）（\ mathop {Spec}（k））$ $ \ pi _ {i，i}（MGL _ {\ bullet}）（\ mathop {Spec}（k））$ $ MGL _ {\ bullet} $
参考文献[增强功能 上 关]（这是什么？）
  • [BT] H. Bass和J. Tate，全球领域的Milnor环，代数𝐾-理论，II：“经典”代数𝐾-理论以及与算术的联系（Pro。Conf。，西雅图，华盛顿，巴特尔纪念馆（Inst。，1972），Springer，柏林，1973，第349-446页。数学笔记，卷。342. MR 0442061，https://doi.org/10.1007/BFb0073733
  • [Nes] 亚历山大·涅希托夫（Alexander Neshitov），《框架通信》和Milnor-Witt𝐾-理论，J。Inst。数学。Jussieu 17（2018），No. 4，823–852。MR 3835524，https: //doi.org/10.1017/S1474748016000190
  • [GP] G. Garkusha和I. Panin ，代数形式的动机（在V. Voevodsky之后），arXiv：1409.4372 [math.KT]，2014年。
  • [Voev] V. Voevodsky，关于带框架的信函的注释，未出版，2001年。可在http://www.math.ias.edu/Voevodsky/files/files-annotated/Dropbox/Unfinished_pa​​pers/Motives/Framed/framed.pdf中获得。
  • [SV] Andrei Suslin和Vladimir Voevodsky，具有有限系数的Bloch-Kato猜想和动力同调，代数循环的算术和几何学（Banff，AB，1998年）。老师 C数学。物理 科学，卷。548，克鲁维尔学院 出版，多德雷赫特，2000年，第117-189页。MR 1744945
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[BT]
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. MR 0442061
[Nes]
A. Neshitov, Framed correspondences and the Milnor-Witt $ K$-theory, J. Inst. Math. Jussieu 17 (2018), no. 4, 823-852. MR 3835524
[GP]
G. Garkusha and I. Panin, Framed motives of algebraic varieties (after V. Voevodsky), arXiv:1409.4372 [math.KT], 2014.
[Voev]
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.
[SV]
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. MR 1744945
[FOAG]
R. Vakil, The rising sea: Foundations of algebraic geometry, http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf
[GarNesh]
G. Garkusha and A. Neshitov, Fibrant resolutions for motivic Thom spectra, arXiv: 1804.07621.
类似的文章检索文章圣彼得堡数学期刊与MSC（2020）：19D45检索与MSC的所有期刊（2020年）的文章：19D45
附加信息

：A. Tsybyshev
隶属：Steklov数学研究所圣彼得堡分院，圣彼得堡Fontanka 27，191023，俄罗斯；圣彼得堡国立大学14号线29B和Chebyshev实验室，俄罗斯圣彼得堡Vasilyevsky岛199178，
电子邮件： emperortsy@gmail.com

DOI：https
：//doi.org/10.1090/spmj/1643关键字：带框的信函，美元A ^ 1 $-同伦理论，代数cobordisms，Milnor $ K $-理论
编辑者已收到：2019年4月15日，以
电子方式发布：2021年1月11日
附加说明：该工作得到RFBR的支持，批准号为。19-01-00513
文章版权：©版权所有2021美国数学协会
更新日期：2021-02-02
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