14. A Cycle Class Map from Chow Groups with Modulus to Relative $K$-Theory
Keywords:
Cycles with modulus, relative $K$-theory, cycle class map, non-$\mathbb{A}^1$-invariant motives
Abstract
Let $\ol{X}$ be a smooth quasi-projective $d$-dimensional variety over a field $k$ and let $D$ be an effective, non-reduced, Cartier divisor on it such that its support is strict normal crossing. In this note, we construct cycle class maps from (a variant of) the higher Chow group with modulus of the pair $(\ol{X};D)$ in the range $(d+n, n)$ to the relative $K$-groups $K_n(\ol{X}; D)$ for every $n\geq 0$.
References
\bibitem{BThesis}
{\sc F.~Binda}, {\em Motives and algebraic cycles with moduli conditions}, PhD
thesis, University of Duisburg-Essen, 2016.
https://duepublico.uni-duisburg-essen.de/servlets/DerivateServlet/Derivate-41964/DissBinda.pdf
\bibitem{BindaMotModulus}
{\sc F.~Binda}, {\em A motivic homotopy
theory without $\mathbb{A}^1$-invariance}, (2017).
\newblock Preprint, available on the author's webpage.
\bibitem{tor-div-rec}
{\sc F.~Binda, J.~Cao, W.~Kai, and R.~Sugiyama}, {\em Torsion and divisibility
for reciprocity sheaves and 0-cycles with modulus}, J. Algebra, 469 (2017),
pp.~437--463.
DOI 10.1016/j.jalgebra.2016.07.036;
zbl 1374.14007;
MR3563022;
arxiv 1503.02161v2
\bibitem{BK}
{\sc F.~Binda and A.~Krishna}, {\em Zero cycles with modulus and zero cycles on
singular varieties}, Compositio Mathematica, 154 (2018), pp.~120--187.
DOI 10.1112/S0010437X17007503;
zbl 06810466;
MR3719246;
arxiv 1512.04847
\bibitem{BS}
{\sc F.~Binda and S.~Saito}, {\em Relative cycles with moduli and regulator
maps}, J. of the Inst. of Mathematics of Jussieu, to appear (2018),
pp.~1--61.
arxiv 1412.0385
\bibitem{Bloch86}
{\sc S.~Bloch}, {\em Algebraic cycles and higher {$K$}-theory}, Adv. in Math.,
61 (1986), pp.~267--304.
DOI 10.1016/0001-8708(86)90081-2;
zbl 0608.14004;
MR0852815
\bibitem{BEAdditive}
{\sc S.~Bloch and H.~Esnault}, {\em The additive dilogarithm}, Doc. Math.,
(2003), pp.~131--155 (electronic).
\newblock Kazuya Kato's fiftieth birthday.
https://www.elibm.org/article/10011526
zbl 1052.11048;
MR2046597;
arxiv 0210138
\bibitem{Crisman2018}
{\sc K.-D. Crisman and J.~Park}, {\em Zero-cycles with modulus associated to
hyperplane arrangements on affine spaces}, manuscripta mathematica, 155
(2018), pp.~15--45.
DOI 10.1007/s00229-017-0931-x;
zbl 06830033;
MR3742770
\bibitem{FS}
{\sc E.~M. Friedlander and A.~Suslin}, {\em The spectral sequence relating
algebraic {$K$}-theory to motivic cohomology}, Ann. Sci. \'Ecole Norm. Sup.
(4), 35 (2002), pp.~773--875.
DOI 10.1016/S0012-9593(02)01109-6;
zbl 1047.14011;
MR1949356
\bibitem{Gupta-Krishna}
{\sc R.~Gupta and A.~Krishna}, {\em Relative {K}-theory and {C}how groups with
modulus}, (2018).
\newblock Preprint.
\bibitem{Ivorra-Rulling}
{\sc F.~Ivorra and K.~R{\"u}lling}, {\em K-groups of reciprocity functors}, J.
Algebraic Geom., 26 (2017), pp.~199--278.
DOI 10.1090/jag/678;
zbl 1360.19007;
MR3606996;
arxiv 1209.1217
\bibitem{IwasaKai}
{\sc R.~Iwasa and W.~Kai}, {\em Chern classes with modulus}, (2016).
\newblock arXiv:1611.07882.
\bibitem{KSY2}
{\sc B.~Kahn, S.~Saito, and T.~Yamazaki}, {\em Motives with modulus}, (2015).
\newblock arXiv:1511.07124.
\bibitem{KSY}
{\sc B.~Kahn, S.~Saito, and T.~Yamazaki}, {\em Reciprocity
sheaves}, Compositio Mathematica, 152 (2016), pp.~1851---1898.
\newblock With two appendices by {K}ay {R}{\"u}lling.
DOI 10.1112/S0010437X16007466;
zbl 06781996;
MR3568941;
arxiv 1402.4201
\bibitem{KSY-RecAndMotives}
{\sc B.~Kahn, S.~Saito, and T.~Yamazaki}, {\em Reciprocity sheaves
and motives with modulus}, (2017).
\newblock arXiv:1707.07398.
\bibitem{KahnYamazakiDuke}
{\sc B.~Kahn and T.~Yamazaki}, {\em Voevodsky's motives and {W}eil
reciprocity}, Duke Math. J., 162 (2013), pp.~2751--2796.
DOI 10.1215/00127094-2381379;
zbl 1309.19009;
MR3127813;
arxiv 1108.2764
\bibitem{MR862639}
{\sc K.~Kato and S.~Saito}, {\em Global class field theory of arithmetic
schemes}, in Applications of algebraic {$K$}-theory to algebraic geometry and
number theory, {P}art {I}, {II} ({B}oulder, {C}olo., 1983), vol.~55 of
Contemp. Math., Amer. Math. Soc., Providence, RI, 1986, pp.~255--331.
zbl 0614.14001;
MR0862639
\bibitem{KS1}
{\sc M.~Kerz and S.~Saito}, {\em Chow group of 0-cycles with modulus and
higher-dimensional class field theory}, Duke Math. J., 165 (2016),
pp.~2811--2897.
DOI 10.1215/00127094-3644902;
zbl 06656236;
MR3557274;
arxiv 1304.4400
\bibitem{KrishnaOnCycles}
{\sc A.~Krishna}, {\em On 0-cycles with modulus}, Algebra Number Theory, 9
(2015), pp.~2397--2415.
DOI 10.2140/ant.2015.9.2397;
zbl 1356.14010;
MR3437766;
arxiv 1504.03125
\bibitem{KL}
{\sc A.~Krishna and M.~Levine}, {\em Additive higher {C}how groups of schemes},
J. Reine Angew. Math., 619 (2008), pp.~75--140.
DOI 10.1515/CRELLE.2008.041;
zbl 1158.14009;
MR2414948;
arxiv math/0702138
\bibitem{KP}
{\sc A.~Krishna and J.~Park}, {\em Moving lemma for additive higher {C}how
groups}, Algebra Number Theory, 6 (2012), pp.~293--326.
DOI 10.2140/ant.2012.6.293;
zbl 1263.14012;
MR2950155;
arxiv 0909.3155
\bibitem{KP3}
{\sc A.~Krishna and J.~Park}, {\em A module structure
and a vanishing theorem for cycles with modulus}, (2015).
\newblock arXiv:1412.7396v2 (to appear in Math. Res. Lett.).
DOI 10.4310/MRL.2017.v24.n4.a10;
zbl 06854542
\bibitem{LecomteAdams}
{\sc F.~Lecomte}, {\em Op{\'e}rations d'{A}dams en {$K$}-th{\'e}orie
alg{\'e}brique}, $K$-Theory, 13 (1998), pp.~179--207.
DOI 10.1023/A:1007780931760;
zbl 0954.19003;
MR1611639
\bibitem{LevineBlochrevisited}
{\sc M.~Levine}, {\em Bloch's higher {C}how groups revisited}, Ast\'erisque,
(1994), pp.~10, 235--320.
\newblock $K$-theory (Strasbourg, 1992).
zbl 0817.19004;
MR1317122
\bibitem{LevineLambdaOp}
{\sc M.~Levine}, {\em Lambda-operations,
{$K$}-theory and motivic cohomology}, in Algebraic {$K$}-theory ({T}oronto,
{ON}, 1996), vol.~16 of Fields Inst. Commun., Amer. Math. Soc., Providence,
RI, 1997, pp.~131--184.
zbl 0883.19001;
MR1466974
\bibitem{LevineLoc}
{\sc M.~Levine}, {\em Techniques of
localization in the theory of algebraic cycles}, J. Algebraic Geom., 10
(2001), pp.~299--363.
zbl 1077.14509;
MR1811558
\bibitem{LevineSmooth}
{\sc M.~Levine}, {\em Smooth motives}, in
Motives and algebraic cycles, vol.~56 of Fields Inst. Commun., Amer. Math.
Soc., Providence, RI, 2009, pp.~175--231.
zbl 1183.14016;
MR2562459;
arxiv 0807.2265
\bibitem{Miyazaki}
{\sc H.~Miyazaki}, {\em Cube invariance of higher chow groups with modulus},
(2017).
\newblock arXiv:1604.06155v3.
\bibitem{munson2015cubical}
{\sc B.~A. Munson and I.~Voli{\'c}}, {\em Cubical homotopy theory}, vol.~25,
Cambridge University Press, 2015.
zbl 1352.55001;
MR3559153
\bibitem{ParkAMJ}
{\sc J.~Park}, {\em Regulators on additive higher {C}how groups}, Amer. J.
Math., 131 (2009), pp.~257--276.
DOI 10.1353/ajm.0.0035;
zbl 1176.14001;
MR2488491;
arxiv math/0605702
\bibitem{Rul}
{\sc K.~R{\"u}lling}, {\em The generalized de {R}ham-{W}itt complex over a
field is a complex of zero-cycles}, J. Algebraic Geom., 16 (2007),
pp.~109--169.
DOI 10.1090/S1056-3911-06-00446-2;
zbl 1122.14006;
MR2257322
\bibitem{RulSaito}
{\sc K.~{R}{\"u}lling and S.~Saito}, {\em Higher {C}how groups with modulus and
relative {M}ilnor $k$-theory}, Trans. of the Amer. Math. Society, (to
appear).
\newblock arXiv:1504.02669.
DOI 10.1090/tran/7018;
zbl 06814518;
MR3729494
\bibitem{Sugiyama:2017aa}
{\sc R.~Sugiyama}, {\em A remark on the tensor products of sc-reciprocity
sheaves}, (2017).
\newblock arXiv:1704.03638.
\bibitem{TT}
{\sc R.~W. Thomason and T.~Trobaugh}, {\em Higher algebraic {$K$}-theory of
schemes and of derived categories}, in The {G}rothendieck {F}estschrift,
{V}ol.\ {III}, vol.~88 of Progr. Math., Birkh\"auser Boston, Boston, MA,
1990, pp.~247--435.
zbl 0731.14001;
MR1106918
\bibitem{VSF}
{\sc V.~Voevodsky, A.~Suslin, and E.~M. Friedlander}, {\em Cycles, transfers,
and motivic homology theories}, vol.~143 of Annals of Mathematics Studies,
Princeton University Press, Princeton, NJ, 2000.
DOI 10.1515/9781400837120;
zbl 1021.14006;
MR1764197
{\sc F.~Binda}, {\em Motives and algebraic cycles with moduli conditions}, PhD
thesis, University of Duisburg-Essen, 2016.
https://duepublico.uni-duisburg-essen.de/servlets/DerivateServlet/Derivate-41964/DissBinda.pdf
\bibitem{BindaMotModulus}
{\sc F.~Binda}, {\em A motivic homotopy
theory without $\mathbb{A}^1$-invariance}, (2017).
\newblock Preprint, available on the author's webpage.
\bibitem{tor-div-rec}
{\sc F.~Binda, J.~Cao, W.~Kai, and R.~Sugiyama}, {\em Torsion and divisibility
for reciprocity sheaves and 0-cycles with modulus}, J. Algebra, 469 (2017),
pp.~437--463.
DOI 10.1016/j.jalgebra.2016.07.036;
zbl 1374.14007;
MR3563022;
arxiv 1503.02161v2
\bibitem{BK}
{\sc F.~Binda and A.~Krishna}, {\em Zero cycles with modulus and zero cycles on
singular varieties}, Compositio Mathematica, 154 (2018), pp.~120--187.
DOI 10.1112/S0010437X17007503;
zbl 06810466;
MR3719246;
arxiv 1512.04847
\bibitem{BS}
{\sc F.~Binda and S.~Saito}, {\em Relative cycles with moduli and regulator
maps}, J. of the Inst. of Mathematics of Jussieu, to appear (2018),
pp.~1--61.
arxiv 1412.0385
\bibitem{Bloch86}
{\sc S.~Bloch}, {\em Algebraic cycles and higher {$K$}-theory}, Adv. in Math.,
61 (1986), pp.~267--304.
DOI 10.1016/0001-8708(86)90081-2;
zbl 0608.14004;
MR0852815
\bibitem{BEAdditive}
{\sc S.~Bloch and H.~Esnault}, {\em The additive dilogarithm}, Doc. Math.,
(2003), pp.~131--155 (electronic).
\newblock Kazuya Kato's fiftieth birthday.
https://www.elibm.org/article/10011526
zbl 1052.11048;
MR2046597;
arxiv 0210138
\bibitem{Crisman2018}
{\sc K.-D. Crisman and J.~Park}, {\em Zero-cycles with modulus associated to
hyperplane arrangements on affine spaces}, manuscripta mathematica, 155
(2018), pp.~15--45.
DOI 10.1007/s00229-017-0931-x;
zbl 06830033;
MR3742770
\bibitem{FS}
{\sc E.~M. Friedlander and A.~Suslin}, {\em The spectral sequence relating
algebraic {$K$}-theory to motivic cohomology}, Ann. Sci. \'Ecole Norm. Sup.
(4), 35 (2002), pp.~773--875.
DOI 10.1016/S0012-9593(02)01109-6;
zbl 1047.14011;
MR1949356
\bibitem{Gupta-Krishna}
{\sc R.~Gupta and A.~Krishna}, {\em Relative {K}-theory and {C}how groups with
modulus}, (2018).
\newblock Preprint.
\bibitem{Ivorra-Rulling}
{\sc F.~Ivorra and K.~R{\"u}lling}, {\em K-groups of reciprocity functors}, J.
Algebraic Geom., 26 (2017), pp.~199--278.
DOI 10.1090/jag/678;
zbl 1360.19007;
MR3606996;
arxiv 1209.1217
\bibitem{IwasaKai}
{\sc R.~Iwasa and W.~Kai}, {\em Chern classes with modulus}, (2016).
\newblock arXiv:1611.07882.
\bibitem{KSY2}
{\sc B.~Kahn, S.~Saito, and T.~Yamazaki}, {\em Motives with modulus}, (2015).
\newblock arXiv:1511.07124.
\bibitem{KSY}
{\sc B.~Kahn, S.~Saito, and T.~Yamazaki}, {\em Reciprocity
sheaves}, Compositio Mathematica, 152 (2016), pp.~1851---1898.
\newblock With two appendices by {K}ay {R}{\"u}lling.
DOI 10.1112/S0010437X16007466;
zbl 06781996;
MR3568941;
arxiv 1402.4201
\bibitem{KSY-RecAndMotives}
{\sc B.~Kahn, S.~Saito, and T.~Yamazaki}, {\em Reciprocity sheaves
and motives with modulus}, (2017).
\newblock arXiv:1707.07398.
\bibitem{KahnYamazakiDuke}
{\sc B.~Kahn and T.~Yamazaki}, {\em Voevodsky's motives and {W}eil
reciprocity}, Duke Math. J., 162 (2013), pp.~2751--2796.
DOI 10.1215/00127094-2381379;
zbl 1309.19009;
MR3127813;
arxiv 1108.2764
\bibitem{MR862639}
{\sc K.~Kato and S.~Saito}, {\em Global class field theory of arithmetic
schemes}, in Applications of algebraic {$K$}-theory to algebraic geometry and
number theory, {P}art {I}, {II} ({B}oulder, {C}olo., 1983), vol.~55 of
Contemp. Math., Amer. Math. Soc., Providence, RI, 1986, pp.~255--331.
zbl 0614.14001;
MR0862639
\bibitem{KS1}
{\sc M.~Kerz and S.~Saito}, {\em Chow group of 0-cycles with modulus and
higher-dimensional class field theory}, Duke Math. J., 165 (2016),
pp.~2811--2897.
DOI 10.1215/00127094-3644902;
zbl 06656236;
MR3557274;
arxiv 1304.4400
\bibitem{KrishnaOnCycles}
{\sc A.~Krishna}, {\em On 0-cycles with modulus}, Algebra Number Theory, 9
(2015), pp.~2397--2415.
DOI 10.2140/ant.2015.9.2397;
zbl 1356.14010;
MR3437766;
arxiv 1504.03125
\bibitem{KL}
{\sc A.~Krishna and M.~Levine}, {\em Additive higher {C}how groups of schemes},
J. Reine Angew. Math., 619 (2008), pp.~75--140.
DOI 10.1515/CRELLE.2008.041;
zbl 1158.14009;
MR2414948;
arxiv math/0702138
\bibitem{KP}
{\sc A.~Krishna and J.~Park}, {\em Moving lemma for additive higher {C}how
groups}, Algebra Number Theory, 6 (2012), pp.~293--326.
DOI 10.2140/ant.2012.6.293;
zbl 1263.14012;
MR2950155;
arxiv 0909.3155
\bibitem{KP3}
{\sc A.~Krishna and J.~Park}, {\em A module structure
and a vanishing theorem for cycles with modulus}, (2015).
\newblock arXiv:1412.7396v2 (to appear in Math. Res. Lett.).
DOI 10.4310/MRL.2017.v24.n4.a10;
zbl 06854542
\bibitem{LecomteAdams}
{\sc F.~Lecomte}, {\em Op{\'e}rations d'{A}dams en {$K$}-th{\'e}orie
alg{\'e}brique}, $K$-Theory, 13 (1998), pp.~179--207.
DOI 10.1023/A:1007780931760;
zbl 0954.19003;
MR1611639
\bibitem{LevineBlochrevisited}
{\sc M.~Levine}, {\em Bloch's higher {C}how groups revisited}, Ast\'erisque,
(1994), pp.~10, 235--320.
\newblock $K$-theory (Strasbourg, 1992).
zbl 0817.19004;
MR1317122
\bibitem{LevineLambdaOp}
{\sc M.~Levine}, {\em Lambda-operations,
{$K$}-theory and motivic cohomology}, in Algebraic {$K$}-theory ({T}oronto,
{ON}, 1996), vol.~16 of Fields Inst. Commun., Amer. Math. Soc., Providence,
RI, 1997, pp.~131--184.
zbl 0883.19001;
MR1466974
\bibitem{LevineLoc}
{\sc M.~Levine}, {\em Techniques of
localization in the theory of algebraic cycles}, J. Algebraic Geom., 10
(2001), pp.~299--363.
zbl 1077.14509;
MR1811558
\bibitem{LevineSmooth}
{\sc M.~Levine}, {\em Smooth motives}, in
Motives and algebraic cycles, vol.~56 of Fields Inst. Commun., Amer. Math.
Soc., Providence, RI, 2009, pp.~175--231.
zbl 1183.14016;
MR2562459;
arxiv 0807.2265
\bibitem{Miyazaki}
{\sc H.~Miyazaki}, {\em Cube invariance of higher chow groups with modulus},
(2017).
\newblock arXiv:1604.06155v3.
\bibitem{munson2015cubical}
{\sc B.~A. Munson and I.~Voli{\'c}}, {\em Cubical homotopy theory}, vol.~25,
Cambridge University Press, 2015.
zbl 1352.55001;
MR3559153
\bibitem{ParkAMJ}
{\sc J.~Park}, {\em Regulators on additive higher {C}how groups}, Amer. J.
Math., 131 (2009), pp.~257--276.
DOI 10.1353/ajm.0.0035;
zbl 1176.14001;
MR2488491;
arxiv math/0605702
\bibitem{Rul}
{\sc K.~R{\"u}lling}, {\em The generalized de {R}ham-{W}itt complex over a
field is a complex of zero-cycles}, J. Algebraic Geom., 16 (2007),
pp.~109--169.
DOI 10.1090/S1056-3911-06-00446-2;
zbl 1122.14006;
MR2257322
\bibitem{RulSaito}
{\sc K.~{R}{\"u}lling and S.~Saito}, {\em Higher {C}how groups with modulus and
relative {M}ilnor $k$-theory}, Trans. of the Amer. Math. Society, (to
appear).
\newblock arXiv:1504.02669.
DOI 10.1090/tran/7018;
zbl 06814518;
MR3729494
\bibitem{Sugiyama:2017aa}
{\sc R.~Sugiyama}, {\em A remark on the tensor products of sc-reciprocity
sheaves}, (2017).
\newblock arXiv:1704.03638.
\bibitem{TT}
{\sc R.~W. Thomason and T.~Trobaugh}, {\em Higher algebraic {$K$}-theory of
schemes and of derived categories}, in The {G}rothendieck {F}estschrift,
{V}ol.\ {III}, vol.~88 of Progr. Math., Birkh\"auser Boston, Boston, MA,
1990, pp.~247--435.
zbl 0731.14001;
MR1106918
\bibitem{VSF}
{\sc V.~Voevodsky, A.~Suslin, and E.~M. Friedlander}, {\em Cycles, transfers,
and motivic homology theories}, vol.~143 of Annals of Mathematics Studies,
Princeton University Press, Princeton, NJ, 2000.
DOI 10.1515/9781400837120;
zbl 1021.14006;
MR1764197
Published
2018-05-24
Issue
Section
Unassigned Articles
Copyright (c) 2018 Federico Binda
This work is licensed under a Creative Commons Attribution 4.0 International License.
- Copyright on any article is retained by the author(s). Regarding copyright transfers please see below.
- Authors grant the publisher of DOCUMENTA MATHEMATICA a license to publish the article and identify itself as the original publisher.
- Authors grant the publisher of DOCUMENTA MATHEMATICA commercial rights to produce hardcopy volumes of the journal for sale to libraries and individuals.
- Authors grant any third party the right to use the article freely as long as its original authors and citation details are identified.
- The article is distributed under the Creative Commons Attribution License (CC BY 4.0). Unless otherwise stated, associated published material is distributed under the same license.
