Hostname: page-component-cb9f654ff-5kfdg Total loading time: 0 Render date: 2025-08-31T11:00:54.325Z Has data issue: false hasContentIssue false
  • English
  • Français

De Rham–Witt sheaves via algebraic cycles

Part of: (Co)homology theory Cycles and subschemes Arithmetic rings and other special rings $K$-theory in geometry

Published online by Cambridge University Press:  19 August 2021

Amalendu Krishna  and
Jinhyun Park
Amalendu Krishna
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba, Mumbai, India amal@math.tifr.res.in
Jinhyun Park
Affiliation:
Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu, Daejeon, 34141, Republic of Korea (South) jinhyun@mathsci.kaist.ac.kr, jinhyun@kaist.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the additive higher Chow groups of regular schemes over a field induce a Zariski sheaf of pro-differential graded algebras, the Milnor range of which is isomorphic to the Zariski sheaf of big de Rham–Witt complexes. This provides an explicit cycle-theoretic description of the big de Rham–Witt sheaves. Several applications are derived.

Keywords

algebraic cyclesde Rham–Witt complexcrystalline cohomology

MSC classification

Primary: 14C25: Algebraic cycles
Secondary: 13F35: Witt vectors and related rings 14F30: $p$-adic cohomology, crystalline cohomology 19E15: Algebraic cycles and motivic cohomology

Information

Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 10 , October 2021 , pp. 2089 - 2132
Check for updates
Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

With an appendix by Kay Rülling

J.P. was partially supported by the National Research Foundation of Korea (NRF; grant numbers 2015R1A2A2A01004120 and 2018R1A2B6002287) funded by the Korean government (MSIP), and TJ Park Junior Faculty Fellowship funded by POSCO TJ Park Foundation.

References

André, M., Homologie des algèbres commutatives, Grundlehren der mathematischen Wissenschaften, vol. 206 (Springer, Berlin, Heidelberg, 1974).CrossRefGoogle Scholar
Berthelot, P., Cohomologie cristalline des schémas de caractéristique $p>0$, Lecture Notes in Mathematics, vol. 407 (Springer-Verlag, Berlin, New York, 1974).Google Scholar
Bloch, S., On the tangent space to Quillen $K$-theory, in Algebraic $K$-theory, I: higher $K$-theories (Proceedings of the conference, Battelle Memorial Institute, Seattle, WA, 1972), Lecture Notes in Mathematics, vol. 341 (Springer-Verlag, Berlin, 1973), 205–210.Google Scholar
Bloch, S., Algebraic $K$-theory and crystalline cohomology, Publ. Math. Inst. Hautes Études Sci. 47 (1977), 187–268.CrossRefGoogle Scholar
Bloch, S., Algebraic cycles and higher $K$-theory, Adv. Math. 61 (1986), 267–304.CrossRefGoogle Scholar
Bloch, S. and Esnault, H., An additive version of higher Chow groups, Ann. Sci. Éc. Norm. Supér. 36 (2003), 463–477.CrossRefGoogle Scholar
Bloch, S. and Esnault, H., The additive dilogarithm, Doc. Math. Extra Vol. (2003), 131–155 (Kazuya Kato's fiftieth birthday).Google Scholar
Bloch, S. and Kato, K., $p$-adic étale cohomology, Publ. Math. Inst. Hautes Études Sci. 63 (1986), 107–152.CrossRefGoogle Scholar
Bloch, S. and Lichtenbaum, S., A spectral sequence for motivic cohomology. Preprint (1995).Google Scholar
Costeanu, V., On the $2$-typical de Rham–Witt complex, Doc. Math. 13 (2008), 413–452.Google Scholar
Grothendieck, A., Éléments de Géométrie Algébrique IV, Étude locale des schémas et des morphismes de schémas, (Seconde Partie), Publ. Math. Inst. Hautes Études Sci. 24 (1965), 231.Google Scholar
Ekedahl, T., On the multiplicative properties of the de Rham–Witt complex. I, Ark. Math. 22 (1984), 185–239.CrossRefGoogle Scholar
Elbaz-Vincent, P. and Müller-Stach, S., Milnor $K$-theory of rings, higher Chow groups and applications, Invent. Math. 148 (2002), 177–206.CrossRefGoogle Scholar
Friedlander, E. and Suslin, A., The spectral sequence relating algebraic $K$-theory to motivic cohomology, Ann. Sci. Éc. Norm. Supér. 35 (2002), 773–875.CrossRefGoogle Scholar
Fulton, W., Intersection theory, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer, Berlin, 1998).CrossRefGoogle Scholar
Geisser, T. and Hesselholt, L., The de Rham–Witt complex and $p$-adic vanishing cycles, J. Amer. Math. Soc. 19 (2005), 1–36.CrossRefGoogle Scholar
Geisser, T. and Levine, M., The $K$-theory of fields in characteristic $p$, Invent. Math. 139 (2000), 459–493.CrossRefGoogle Scholar
Görtz, U. and Wedhorn, T., Algebraic geometry I, schemes with examples and exercises, first edition, Advanced Lectures in Mathematics (Vieweg + Teubner, Wiesbaden, 2010).CrossRefGoogle Scholar
Gros, M., Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique, Mém. Soc. Math. Fr., $2^{e}$ série 21 (1985), 1–87.Google Scholar
Gupta, R. and Krishna, A., Relative K-theory via 0-cycles in finite characteristics, Ann. K-theory, to appear. Preprint (2019), arXiv:1910.06630.Google Scholar
Gupta, R. and Krishna, A., Zero-cycles with modulus and relative K-theory, Ann. K-theory 5 (2020), 757–819.CrossRefGoogle Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977).CrossRefGoogle Scholar
Hesselholt, L., K-theory of truncated polynomial algebras, in Handbook of K-theory, vol. 1 (Springer, New York, 2005), 71–110.CrossRefGoogle Scholar
Hesselholt, L., The big de Rham–Witt complex, Acta Math. 214 (2015), 135–207.CrossRefGoogle Scholar
Hesselholt, L. and Madsen, I., On the K-theory of nilpotent endomorphisms, in Homotopy methods in algebraic topology (Boulder, CO, 1999), eds J. P. C. Greenlees, et al. , Contemporary Mathematics, vol. 271 (American Mathematical Society, Providence, RI, 2001), 127–140.CrossRefGoogle Scholar
Hesselholt, L. and Madsen, I., On the de Rham–Witt complex in mixed characteristic, Ann. Sci. Éc. Norm. Supér. 37 (2004), 1–43.CrossRefGoogle Scholar
Illusie, L., Complexe de de Rham–Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Supér. 12 (1979), 501–661.CrossRefGoogle Scholar
Kato, K., A generalization of local class field theory by using $K$-groups. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 603–683.Google Scholar
Kerz, M., The Gersten conjecture for Milnor $K$-theory, Invent. Math. 175 (2009), 1–33.CrossRefGoogle Scholar
Kerz, M., Milnor K-theory of local rings with finite residue fields, J. Algebr. Geom. 19 (2010), 173–191.CrossRefGoogle 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., Moving lemma for additive higher Chow groups, Algebra Number Theory 6 (2012), 293–326.CrossRefGoogle Scholar
Krishna, A. and Park, J., Mixed motives over $k{t}/(t^{m}+1)$, J. Inst. Math. Jussieu 11 (2012), 611–657.CrossRefGoogle Scholar
Krishna, A. and Park, J., DGA-structure on additive higher Chow groups, Int. Math. Res. Not. 2015 (2015), 1–54.CrossRefGoogle Scholar
Krishna, A. and Park, J., On additive higher Chow groups of affine schemes, Doc. Math. 21 (2016), 49–89.Google Scholar
Krishna, A. and Park, J., A module structure and a vanishing theorem for cycles with modulus, Math. Res. Lett. 24 (2017), 1147–1176.CrossRefGoogle Scholar
Krishna, A. and Park, J., A moving lemma for cycles with very ample modulus, Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (2017), 1521–1549.Google Scholar
Krishna, A. and Park, J., A moving lemma for relative $0$-cycles, Algebra Number Theory 14 (2020), 991–1054.CrossRefGoogle Scholar
Levine, M., The homotopy coniveau tower, J. Topol. 1 (2008), 217–267.CrossRefGoogle Scholar
Loday, J.-L., Cyclic homology, second edition, Grundlehren der mathematischen Wissenschaften, vol. 301 (Springer, Berlin, 1998).CrossRefGoogle Scholar
Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, Cambridge, 1986).Google Scholar
Morrow, M., $K$-theory and logarithmic Hodge-Witt sheaves of formal schemes in characteristic $p$, Ann. Sci. École Norm. Supér. 52 (2019), 1537–1601.CrossRefGoogle Scholar
Park, J., Regulators on additive higher Chow groups, Amer. J. Math. 131 (2009), 257–276.CrossRefGoogle Scholar
Popescu, D., General Néron desingularization and approximation, Nagoya Math. J. 10 (1986), 85–115.CrossRefGoogle Scholar
Quillen, D., On the (co)-homology of commutative rings, in Applications of categorical algebra, New York, 1968, Proceedings of Symposia in Pure Mathematics, vol. 17 (American Mathematical Society, Providence, RI, 1970), 65–87.Google Scholar
Quillen, D., Higher algebraic K-theory I, in Algebraic K-theory, I: higher K-theories (Proceedings of the Conference, Battelle Memorial Institute, Seattle, WA, 1972), Lecture Notes in Mathematics, vol. 341 (Springer, Berlin, 1973), 77–139.Google Scholar
Rülling, K., The generalized de Rham–Witt complex over a field is a complex of zero-cycles, J. Algebr. Geom. 16 (2007), 109–169.CrossRefGoogle Scholar
Rülling, K., Erratum to “The generalized de Rham–Witt complex over a field is a complex of zero-cycles”, J. Algebr. Geom. 16 (2007), 793–795.CrossRefGoogle Scholar
Rülling, K. and Saito, S., Higher Chow groups with modulus and relative Milnor $K$-theory, Trans. Amer. Math. Soc. 370 (2018), 987–1043.CrossRefGoogle Scholar
The Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu.Google Scholar
Swan, R., Néron-Popescu desingularization, in Algebra and geometry, Taipei, 1995, Lecture Notes Algebraic Geometry, vol. 2 (International Press, Cambridge, MA, 1998), 135–192.Google Scholar
Totaro, B., Milnor $K$-theory is the simplest part of algebraic $K$-theory, K-theory 6 (1992), 177–189.CrossRefGoogle Scholar