Skip to main content

Zero cycles with modulus and zero cycles on singular varieties

  • Federico Binda (a1) and Amalendu Krishna (a2)

Given a smooth variety $X$ and an effective Cartier divisor $D\subset X$ , we show that the cohomological Chow group of 0-cycles on the double of $X$ along $D$ has a canonical decomposition in terms of the Chow group of 0-cycles $\text{CH}_{0}(X)$ and the Chow group of 0-cycles with modulus $\text{CH}_{0}(X|D)$ on $X$ . When $X$ is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of $\text{CH}_{0}(X|D)$ . As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that $\text{CH}_{0}(X|D)$ is torsion-free and there is an injective cycle class map $\text{CH}_{0}(X|D){\hookrightarrow}K_{0}(X,D)$ if $X$ is affine. For a smooth affine surface $X$ , this is strengthened to show that $K_{0}(X,D)$ is an extension of $\text{CH}_{1}(X|D)$ by $\text{CH}_{0}(X|D)$ .

Hide All
[BPW96] Barbieri-Viale, L., Pedrini, C. and Weibel, C., Roitman’s theorem for singular complex projective surfaces , Duke Math. J. 84 (1996), 155190.
[Bas68] Bass, H., Algebraic K-theory (W. A. Benjamin, New York, NY, 1968).
[Bin16] Binda, F., Motives and algebraic cycles with moduli conditions, PhD thesis, University of Duisburg-Essen (2016).
[Bin18] Binda, F., Torsion 0-cycles with modulus on affine varieties , J. Pure Appl. Algebra 222 (2018), 6174.
[BS16] Binda, F. and Saito, S., Relative cycles with moduli and regulator maps, Preprint (2016),arXiv:1412.0385v2.
[BS99] Biswas, J. and Srinivas, V., Roitman’s theorem for singular projective varieties , Compos. Math. 119 (1999), 213237.
[Blo86] Bloch, S., Algebraic cycles and higher K-theory , Adv. Math. 61 (1986), 267304.
[BKL76] Bloch, S., Kas, A. and Lieberman, D., Zero cycles on surfaces with p g = 0 , Compos. Math. 33 (1976), 135145.
[ESV99] Esnault, H., Srinivas, V. and Viehweg, E., The universal regular quotient of the Chow group of points on projective varieties , Invent. Math. 135 (1999), 595664.
[EV98] Esnault, H. and Viehweg, E., Deligne–Beĭlinson cohomology , in Beĭlinson’s conjectures on special values of L-functions, Perspectives in Mathematics, vol. 4 (Academic Press, Boston, MA, 1988), 4391.
[FS02] Friedlander, E. M. and Suslin, A., The spectral sequence relating algebraic K-theory to motivic cohomology , Ann. Sci. Éc. Norm. Supér. (4) 35 (2002), 773875.
[Ful98] Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, second edition (Springer, Berlin, 1998).
[GW83] Geller, S. C. and Weibel, C. A., K 1(A, B, I) , J. Reine Angew. Math. 342 (1983), 1234.
[GW10] Görtz, U. and Wedhorn, T., Algebraic geometry I, Schemes, with examples and exercises, Advanced Lectures in Mathematics (Vieweg+Teubner, Wiesbaden, 2010).
[Har77] Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, NY, 1977).
[Har10] Hartshorne, R., Deformation theory, Graduate Texts in Mathematics, vol. 257 (Springer, New York, NY, 2010).
[Jou83] Jouanolou, J.-P., Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42 (Birkhäuser Boston, Boston, MA, 1983).
[Kai16] Kai, W., A moving lemma for algebraic cycles with modulus and contravariance, Preprint (2016), arXiv:1507.07619v3.
[KR12] Kato, K. and Russell, H., Albanese varieties with modulus and Hodge theory , Ann. Inst. Fourier (Grenoble) 62 (2012), 783806.
[KS16] Kerz, M. and Saito, S., Chow group of 0-cycles with modulus and higher-dimensional class field theory , Duke Math. J. 165 (2016), 28112897.
[KL79] Kleiman, S. L. and Altman, A. B., Bertini theorems for hypersurface sections containing a subscheme , Comm. Algebra 7 (1979), 775790.
[Kri09] Krishna, A., Zero cycles on singular surfaces , J. K-Theory 4 (2009), 101143.
[Kri15a] Krishna, A., On 0-cycles with modulus , Algebra Number Theory 9 (2015), 23972415.
[Kri15b] Krishna, A., Zero cycles on affine varieties, Preprint (2015), arXiv:1511.04221v1.
[KL08] Krishna, A. and Levine, M., Additive higher Chow groups of schemes , J. Reine Angew. Math. 619 (2008), 75140.
[KP14] Krishna, A. and Park, J., A module structure and a vanishing theorem for cycles with modulus , Math. Res. Lett., to appear, Preprint (2014), arXiv:1412.7396v2.
[KS07] Krishna, A. and Srinivas, V., Zero cycles on singular varieties , in Algebraic cycles and motives, London Mathematical Society Lecture Note Series, vol. 343 (Cambridge University Press, Cambridge, 2007), 264277.
[Lev85a] Levine, M., Bloch’s formula for singular surfaces , Topology 24 (1985), 165174.
[Lev85b] Levine, M., A geometric theory of the Chow ring for singular varieties, unpublished manuscript, 1985.
[Lev87] Levine, M., Zero-cycles and K-theory on singular varieties , in Algebraic geometry: Bowdoin 1985, Brunswick, Maine, 1985, Proceedings of Symposia in Pure Mathematics, vol. 46 (American Mathematical Society, Providence, RI, 1987), 451462.
[Lev92] Levine, M., Deligne–Beĭlinson cohomology for singular varieties , in Algebraic K-theory, commutative algebra, and algebraic geometry, Santa Margherita Ligure, 1989, Contemporary Mathematics, vol. 126 (American Mathematical Society, Providence, RI, 1992), 113146.
[Lev94] Levine, M., Bloch’s higher Chow groups revisited , Astérisque 10 (1994), 235320.
[LW85] Levine, M. and Weibel, C., Zero cycles and complete intersections on singular varieties , J. Reine Angew. Math. 359 (1985), 106120.
[Mal09] Mallick, V. M., Roitman’s theorem for singular projective varieties in arbitrary characteristic , J. K-Theory 3 (2009), 501531.
[Mil82] Milne, J. S., Zero cycles on algebraic varieties in nonzero characteristic: Rojtman’s theorem , Compos. Math. 47 (1982), 271287.
[Mil71] Milnor, J., Introduction to algebraic K-theory, Annals of Mathematics Studies, vol. 72 (Princeton University Press, Princeton, NJ, 1971).
[Par09] Park, J., Regulators on additive higher Chow groups , Amer. J. Math. 131 (2009), 257276.
[PW94] Pedrini, C. and Weibel, C., Divisibility in the Chow group of zero-cycles on a singular surface , Astérisque 226 (1994), 371409.
[Roi80] Rojtman, A. A., The torsion of the group of 0-cycles modulo rational equivalence , Ann. of Math. (2) 111 (1980), 553569.
[Rue07] Rülling, K., The generalized de Rham–Witt complex over a field is a complex of zero-cycles , J. Algebraic Geom. 16 (2007), 109169.
[RS16] Rülling, K. and Saito, S., Higher Chow groups with modulus and relative Milnor K-theory , Trans. Amer. Math. Soc., to appear, doi:10.1090/tran/7018.
[Rus13] Russell, H., Albanese varieties with modulus over a perfect field , Algebra Number Theory 7 (2013), 853892.
[Sei50] Seidenberg, A., The hyperplane sections of normal varieties , Trans. Amer. Math. Soc. 69 (1950), 357386.
[Ser60] Serre, J.-P., Morphismes universels et differéntielles de troisiéme espéce , Séminaire Claude Chevalley 4(11) (1958–1959).
[Ser88] Serre, J.-P., Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117 (Springer, New York, NY, 1988).
[Sri00] Srinivas, V., Zero cycles on singular varieties , in The arithmetic and geometry of algebraic cycles, Banff, AB, 1998, NATO Science Series C: Mathematics, Physics and Science, vol. 548 (Kluwer Academic Publishers, Dordrecht, 2000), 347382.
[Sri08] Srinivas, V., Algebraic K-theory, paperback reprint of the 1996 second edition (Birkhäuser, Boston, MA, 2008).
[SV96] Suslin, A. and Voevodsky, V., Singular homology of abstract algebraic varieties , Invent. Math. 123 (1996), 6194.
[Voi04] Voisin, C., Remarks on filtrations on Chow groups and the Bloch conjecture , Ann. Mat. Pura Appl. (4) 183 (2004), 421438.
[Wei54] Weil, A., Sur les critères d’équivalence en géométrie algébrique , Math. Ann. 128 (1954), 95127.
[Zar58] Zariski, O., Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, vol. 4 (The Mathematical Society of Japan, Tokyo, 1958).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed