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CHERN CLASSES WITH MODULUS

Part of: Higher algebraic $K$-theory Cycles and subschemes

Published online by Cambridge University Press:  01 February 2019

RYOMEI IWASA  and
WATARU KAI Open the ORCID record for WATARU KAI [Opens in a new window]
RYOMEI IWASA
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark email ryomei@math.ku.dk
WATARU KAI
Affiliation:
Mathematical Institute, Tohoku University, Aza-Aoba 6-3, Sendai 980-8578, Japan email kaiw@tohoku.ac.jp
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Abstract

In this paper, we construct Chern classes from the relative $K$-theory of modulus pairs to the relative motivic cohomology defined by Binda–Saito. An application to relative motivic cohomology of henselian dvr is given.

MSC classification

Primary: 19D10: Algebraic $K$-theory of spaces 14C35: Applications of methods of algebraic $K$-theory
Type
Article
Information
Nagoya Mathematical Journal , Volume 236: Celebrating the 60th Birthday of Shuji Saito , December 2019 , pp. 84 - 133
Copyright
© 2019 Foundation Nagoya Mathematical Journal  

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References

Artin, M., Algebraic approximation of structures over complete local rings , Publ. Math. Inst. Hautes Études Sci. 36 (1969), 23–58.10.1007/BF02684596Google Scholar
Asakura, M. and Sato, K., Chern class and Riemann–Roch theorem for cohomology theory without homotopy invariance, preprint, 2013, arXiv:1301.5829v10 [mathAG].Google Scholar
Beilinson, A., Relative continuous K-theory and cyclic homology , Münster J. Math. 7(1) (2014), 51–81.Google Scholar
Binda, F. and Saito, S., Relative cycles with moduli and regulator maps , J. Inst. Math. Jussieu (2017), 1–61.Google Scholar
Binda, F. and Krishna, A., Zero cycles with modulus and zero cycles on singular varieties , Compos. Math. 154(1) (2018), 120–187.10.1112/S0010437X17007503Google Scholar
Bloch, S., Algebraic cycles and higher K-theory , Adv. Math. 61(3) (1986), 267–304.10.1016/0001-8708(86)90081-2Google Scholar
Bloch, S., The moving lemma for higher Chow groups , J. Algebraic Geom. 3(3) (1994), 537–568.Google Scholar
Bloch, S., Some notes on elementary properties of higher chow groups, including functoriality properties and cubical chow groups (not for publication). Short notes available on his web page. http://www.math.uchicago.edu/∼bloch/publications.html.Google Scholar
Bloch, S. and Esnault, H., An additive version of higher Chow groups , Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), 463–477.10.1016/S0012-9593(03)00015-6Google Scholar
Bousfield, A. K. and Kan, D. M., Homotopy limits, completions and Localizations, Lecture Notes in Math. 304 , Springer, New York, 1972.10.1007/978-3-540-38117-4Google Scholar
Friedlander, E. M. and Lawson, H. B., Moving algebraic cycles of bounded degree , Invent. Math. 132(1) (1998), 91–119.10.1007/s002220050219Google Scholar
Gillet, H., Riemann–Roch theorems for higher algebraic K-theory , Adv. Math. 40(3) (1981), 203–289.10.1016/S0001-8708(81)80006-0Google Scholar
Goodwillie, T., Relative algebraic K-theory and cyclic homology , Ann. of Math. (2) 124(2) (1986), 347–402.10.2307/1971283Google Scholar
Hirschhorn, P., Model categories and their localizations, Math. Surveys Monogr. 99 , American Mathematical Society, Providence, RI, 2003.Google Scholar
Jardine, J. F., Local homotopy theory, Springer Monographs in Mathematics, Springer, New York, 2015.10.1007/978-1-4939-2300-7Google Scholar
Kai, W., A moving lemma for algebraic cycles with modulus and contravariance, To appear in International Mathematics Research Notices, preprint, 2015, arXiv:1507.07619v4 [math.AG].Google Scholar
Krishna, A. and Levine, M., Additive higher Chow groups of schemes , J. Reine Angew. Math. 619 (2008), 75–140.Google Scholar
Krishna, A. and Park, J., A module structure and a vanishing theorem for cycles with modulus , Math. Res. Lett. 24(4) (2017), 1147–1176.10.4310/MRL.2017.v24.n4.a10Google Scholar
Krishna, A. and Park, J., Algebraic cycles and crystalline cohomology, preprint, 2015, arXiv:1504.08181v5 [math.AG].Google Scholar
Krishna, A., On 0-cycles with modulus , Algebra Number Theory 9(10) (2015), 2397–2415.10.2140/ant.2015.9.2397Google Scholar
Kahn, B., Saito, S. and Yamazaki, T., Motives with modulus, preprint, 2016, arXiv:1511.07124v4 [math.AG].Google Scholar
Levine, M., Mixed Motives, Math. Surveys Monogr. 57 , American Mathematical Society, Providence, RI, 1998.10.1090/surv/057Google Scholar
Loday, J.-L., Cyclic homology, 2nd ed., Grundlehren Math. Wiss. 301 , Springer, Berlin, Heidelberg, 1998.10.1007/978-3-662-11389-9Google Scholar
Morrow, M., Pro unitality and pro excision in algebraic K-theory and cyclic homology , J. Reine Angew. Math. 736 (2018), 95–139.10.1515/crelle-2015-0007Google Scholar
Park, J., Regulators on Additive Higher Chow Groups , Amer. J. Math. 131(1) (2009), 257–276.10.1353/ajm.0.0035Google Scholar
Rülling, K. and Saito, S., Higher Chow groups with modulus and relative Milnor K-theory , Trans. Amer. Math. Soc. 370 (2018), 987–1043.10.1090/tran/7018Google Scholar
Rülling, K., The generalized de Rham-Witt complex over a field is a complex of zero-cycles , J. Algebraic Geom. 16(1) (2007), 109–169.10.1090/S1056-3911-06-00446-2Google Scholar
Thomason, R. W. and Trobaugh, T., “ Higher algebraic K-theory of schemes and of derived categories ”, in The Grothendieck Festschrift, Progr. Math. 88, III, Brikhäuser, 1990, 247–435.10.1007/978-0-8176-4576-2_10Google Scholar
Weibel, C., Le caractère de Chern en homologie cyclique périodique , C. R. Acad. Sci. Paris Sér. I Math. 317(9) (1993), 867–871.Google Scholar
Weibel, C., An introduction to homological algebra, Cambridge Stud. Adv. Math. 38 , Cambridge University Press, Cambridge, 1994.10.1017/CBO9781139644136Google Scholar
Berthelot, P., Grothendieck, A. and Illusie, L., Théorie des intersections et théorème de Riemann–Roch, Séminarie de Géométrie Algébrique, 1966/67, Lecture Notes in Math. 225 , Springer, New York, 1971.10.1007/BFb0066283Google Scholar
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