An analogue of the Milnor conjecture for the de Rham-Witt complex in characteristic 2

Dotto, Emanuele

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e87

Résumé

We describe the modulo $2$ de Rham-Witt complex of a field of characteristic $2$, in terms of the powers of the augmentation ideal of the $\mathbb {Z}/2$-geometric fixed points of real topological restriction homology ${\mathrm {TRR}}$. This is analogous to the conjecture of Milnor, proved in [Kat82] for fields of characteristic $2$, which describes the modulo $2$ Milnor K-theory in terms of the powers of the augmentation ideal of the Witt group of symmetric forms. Our proof provides a somewhat explicit description of these objects, as well as a calculation of the homotopy groups of the geometric fixed points of ${\mathrm {TRR}}$ and of real topological cyclic homology, for all fields.

Détail

DOI : 10.1017/fms.2025.40

Bibliographie

[AN21] Antieau, B. and Nikolaus, T., ‘Cartier modules and cyclotomic spectra’, J. Amer. Math. Soc. 34(1) (2021), 1–78. Google Scholar

[Ara20] Arason, J. Kr., ‘On differential forms over fields of characteristic 2’, Online Note, 2020. Google Scholar

[BDS16] Brun, M., Dundas, B. I. and Stolz, M., ‘Equivariant structure on smash powers’, Preprint, 2016, arXiv: . Google Scholar | arXiv

[BF84] Burghelea, D. and Fiedorowicz, Z., ‘Hermitian algebraic -theory of topological spaces’, in Algebraic -theory, Number Theory, Geometry and Analysis (Bielefeld, 1982) (Lecture Notes in Math.) vol. 1046 (Springer, Berlin, 1984), 32–46. Google Scholar

[BGT14] Blumberg, A. J., Gepner, D. and Tabuada, G., ‘Uniqueness of the multiplicative cyclotomic trace’, Adv. Math. 260 (2014), 191–232. Google Scholar

[BHM93] Bökstedt, M., Hsiang, W. C. and Madsen, I., ‘The cyclotomic trace and algebraic -theory of spaces’, Invent. Math. 111(3) (1993), 465–539. Google Scholar

[Bök86] Bökstedt, M., ‘Topological Hochschild homology’, Preprint, 1986. Google Scholar

[CDH+20a] Calmès, B., Dotto, E., Harpaz, Y., Hebestreit, F., Land, M., Moi, K., Nardin, D., Nikolaus, T. and Steimle, W., ‘Hermitian K-theory for stable -categories II: Cobordism categories and additivity’, Preprint, 2020, arXiv: . Google Scholar | arXiv

[CDH+20b] Calmès, B., Dotto, E., Harpaz, Y., Hebestreit, F., Land, M., Moi, K., Nardin, D., Nikolaus, T. and Steimle, W., ‘Hermitian K-theory for stable -categories III: Grothendieck-Witt groups of rings’, Preprint, 2020, arXiv: . Google Scholar | arXiv

[CDH+23] Calmès, B., Dotto, E., Harpaz, Y., Hebestreit, F., Land, M., Moi, K., Nardin, D., Nikolaus, T. and Steimle, W., ‘Hermitian K-theory for stable -categories I: foundations’, Selecta Math. (N.S.) 29(1) (2023), Paper No. 10, 269.10.1007/s00029-022-00758-2 Google Scholar | DOI

[CMM21] Clausen, D., Mathew, A. and Morrow, M., ‘ -theory and topological cyclic homology of henselian pairs’, J. Amer. Math. Soc. 34(2) (2021), 411–473. Google Scholar

[Cos08] Costeanu, V., ‘On the 2-typical de Rham-Witt complex’, Doc. Math. 13 (2008), 413–452.10.4171/dm/251 Google Scholar | DOI

[DKNP23] Dotto, E., Krause, A., Nikolaus, T. and Patchkoria, I., ‘Witt vectors with coefficients and TR’, Preprint, 2023, arXiv: . Google Scholar | arXiv

[DMP24] Dotto, E., Moi, K. and Patchkoria, I., ‘On the geometric fixed points of real topological cyclic homology’, J. Lond. Math. Soc. (2) 109(2) (2024), Paper No. e12862, 68.10.1112/jlms.12862 Google Scholar | DOI

[DMPR21] Dotto, E., Moi, K., Patchkoria, I. and Reeh, S. P., ‘Real topological Hochschild homology’, J. Eur. Math. Soc. (JEMS) 23(1) (2021), 63–152.10.4171/jems/1007 Google Scholar | DOI

[DO19] Dotto, E. and Ogle, C., ‘-theory of Hermitian Mackey functors, real traces, and assembly’, Ann. K-Theory 4(2) (2019), 243–316.10.2140/akt.2019.4.243 Google Scholar | DOI

[DPM22] Dotto, E., Patchkoria, I. and Jonsson Moi, K., ‘Witt vectors, polynomial maps, and real topological Hochschild homology’, Ann. Sci. Éc. Norm. Supér. (4) 55(2) (2022), 473–535.10.24033/asens.2500 Google Scholar | DOI

[FL91] Fiedorowicz, Z. and Loday, J.-L., ‘Crossed simplicial groups and their associated homology’, Trans. Amer. Math. Soc. 326(1) (1991), 57–87.10.1090/S0002-9947-1991-0998125-4 Google Scholar | DOI

[GH99] Geisser, T. and Hesselholt, L., ‘Topological cyclic homology of schemes’, in Algebraic -theory (Seattle, WA, 1997) (Proc. Sympos. Pure Math.) vol. 67 (Amer. Math. Soc., Providence, RI, 1999), 41–87.10.1090/pspum/067/1743237 Google Scholar | DOI

[Gro67] Grothendieck, A., ‘Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV’, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361. Google Scholar

[Hes04] Hesselholt, L., ‘Topological Hochschild homology and the de Rham-Witt complex for -algebras’, in Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic -theory (Contemp. Math.) vol. 346 (Amer. Math. Soc., Providence, RI, 2004), 253–259. Google Scholar

[Hes15] Hesselholt, L., ‘The big de Rham-Witt complex’, Acta Math. 214(1) (2015), 135–207.10.1007/s11511-015-0124-y Google Scholar | DOI

[HHR16] Hill, M. A., Hopkins, M. J. and Ravenel, D. C., ‘On the nonexistence of elements of Kervaire invariant one’, Ann. of Math. (2) 184(1) (2016), 1–262.10.4007/annals.2016.184.1.1 Google Scholar | DOI

[HM97] Hesselholt, L. and Madsen, I., ‘On the -theory of finite algebras over Witt vectors of perfect fields’, Topology 36(1) (1997), 29–101.10.1016/0040-9383(96)00003-1 Google Scholar | DOI

[HM04] Hesselholt, L. and Madsen, I., ‘On the De Rham-Witt complex in mixed characteristic’, Ann. Sci. École Norm. Sup. (4) 37(1) (2004), 1–43.10.1016/j.ansens.2003.06.001 Google Scholar | DOI

[HNS21] Harpaz, Y., Nikolaus, T. and Shah, J., ‘Real topological cyclic homology and normal L-theory’, to appear, 2021. Google Scholar

[Høg16] Høgenhaven, A., ‘Real topological cyclic homology of spherical group rings’, Preprint, 2016, arXiv: . Google Scholar | arXiv

[Ill79] Illusie, L., ‘Complexe de de Rham-Witt et cohomologie cristalline’, Ann. Sci. École Norm. Sup. (4) 12(4) (1979), 501–661.10.24033/asens.1374 Google Scholar | DOI

[Kat82] Kato, K., ‘Symmetric bilinear forms, quadratic forms and Milnor -theory in characteristic two’, Invent. Math. 66(3) (1982), 493–510.10.1007/BF01389226 Google Scholar | DOI

[Lod87] Loday, J.-L., ‘Homologies diédrale et quaternionique’, Adv. in Math. 66(2) (1987), 119–148.10.1016/0001-8708(87)90032-6 Google Scholar | DOI

[Mad94] Madsen, I., ‘Algebraic -theory and traces’, in Current Developments in Mathematics, 1995 (Cambridge, MA) (Int. Press, Cambridge, MA, 1994), 191–321. Google Scholar

[Mil71] Milnor, J., ‘Symmetric inner products in characteristic ’, in Prospects in Mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) (Ann. of Math. Stud.) vol. 70 (Princeton Univ. Press, Princeton, NJ, 1971), 59–75. Google Scholar

[Mil70] Milnor, J., ‘Algebraic -theory and quadratic forms’, Invent. Math. 9 (1969/70), 318–344.10.1007/BF01425486 Google Scholar | DOI

[Mor05] Morel, F., ‘Milnor’s conjecture on quadratic forms and mod 2 motivic complexes’, Rend. Sem. Mat. Univ. Padova 114 (2005), 63–101. Google Scholar

[NS18] Nikolaus, T. and Scholze, P., ‘On topological cyclic homology’, Acta Math. 221(2) (2018), 203–409.10.4310/ACTA.2018.v221.n2.a1 Google Scholar | DOI

[OVV07] Orlov, D., Vishik, A. and Voevodsky, V., ‘An exact sequence for with applications to quadratic forms’, Ann. of Math. (2) 165(1) (2007), 1–13.10.4007/annals.2007.165.1 Google Scholar | DOI

[Sto11] Stolz, M., ‘Equivariant structure on smash powers of commutative ring spectra’, PhD thesis, University of Bergen, 2011. Google Scholar

[Voe03] Voevodsky, V., ‘Motivic cohomology with /2-coefficients’, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104.10.1007/s10240-003-0010-6 Google Scholar | DOI