On non-commutative twisting in étale and motivic cohomology

Hornbostel, Jens; Kings, Guido

Annales de l'Institut Fourier, Volume 56 (2006) no. 4, pp. 1257-1279.

This article confirms a consequence of the non-abelian Iwasawa main conjecture. It is proved that under a technical condition the étale cohomology groups $H^1({𝒪}_K[1/S],H^i(\overline{X},{\mathbb{Q}}_p\left(j\right)))$, where $X\to Spec{𝒪}_K[1/S]$ is a smooth, projective scheme, are generated by twists of norm compatible units in a tower of number fields associated to $H^i(\overline{X},{\mathbb{Z}}_p\left(j\right))$. Using the “Bloch-Kato-conjecture” a similar result is proven for motivic cohomology with finite coefficients.

Cet article confirme une conséquence de la conjecture principale de la théorie d’Iwasawa non abélienne. On démontre que, sous une condition technique, les groupes de cohomologie étale $H^1({𝒪}_K[1/S],H^i(\overline{X},{\mathbb{Q}}_p\left(j\right)))$, où $X\to Spec{𝒪}_K[1/S]$ est un schéma projectif lisse, sont engendrés par des unités tordues compatible par rapport aux normes dans une tour de corps de nombres associés à $H^i(\overline{X},{\mathbb{Z}}_p\left(j\right))$. On établit un résultat similaire pour la cohomologie motivique à coefficients finis en utilisant la conjecture de Bloch-Kato.

[1] Artin, M.; Grothendieck, A.; Verdier, J.-L. Théorie des topos et cohomologie étale des schémas, t.3, Lect. Notes. Math., Volume 305, Springer, 1973 | MR

[2] Bloch, S.; Kato, K. $L$-functions and Tamagawa numbers of motives, “The Grothendieck Festschrift”, Vol.I, Progress in Math., Volume 86, Birkhäuser, Boston, 1990, pp. 333-400 | MR | Zbl

[3] Coates, J.; Sujatha, R. Fine Selmer groups of elliptic curves over $p$-adic Lie extensions, Math. Annalen, Volume 331 (2005), pp. 809-839 | DOI | MR | Zbl

[4] Dwyer, W. G.; Friedlander, E. M. Algebraic and étale $K$-theory, Trans. Amer. Math. Soc., Volume 292 (1985), pp. 247-280 | MR | Zbl

[5] Fontaine, J.-M.; Perrin-Riou, B. Cohomologie galoisienne et valeurs des fontions $L$, Proceedings of Symposia in Pure Mathematics, part I, Volume 55 (1994), pp. 599-706 | MR | Zbl

[6] Geisser, T. Motivic Cohomology over Dedekind rings, Math. Z., Volume 248 (2004), pp. 773-794 | DOI | MR | Zbl

[7] Geisser, T.; Levine, M. The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, J. reine angew. Math., Volume 530 (2001), pp. 55-103 | DOI | MR | Zbl

[8] Gillet, H. Riemann-Roch theorems for higher algebraic $K$-theory, Adv. in Math., Volume 40 (1981), pp. 203-289 | DOI | MR | Zbl

[9] Huber, A.; Kings, G. Degeneration of $\ell$-adic Eisenstein classes and of the elliptic polylog, Invent. Math., Volume 135 (1999), pp. 545-594 | DOI | MR | Zbl

[10] Huber, A.; Kings, G. Equivariant Bloch-Kato conjecture and non-abelian Iwasawa main conjecture, Proceedings ICM, Volume II (2002), pp. 149-162 | MR | Zbl

[11] Huber, A.; Wildeshaus, J. Classical motivic polylogarithm according to Beilinson and Deligne, Doc. Math., Volume 3 (1998), pp. 27-133 | MR | Zbl

[12] Jannsen, U.; Ihara On the $\ell$-adic cohomology of varieties over number fields and its Galois cohomology, Galois groups over $\mathbb{Q}$, MSRI Publication, 1989 | Zbl

[13] Kahn, B. $K$-theory of semi-local rings with finite coefficients and étale cohomology, K-Theory, Volume 25 (2002), pp. 99-138 | DOI | MR | Zbl

[14] Kato, K. Iwasawa theory and $p$-adic Hodge theory, Kodai Math. J., Volume 16 (1993), pp. 1-31 | DOI | MR | Zbl

[15] Kato, K. $p$-adic Hodge theory and values of zeta functions of modular forms. Cohomologies $p$-adiques et applications arithmétiques III, Astérisque, Volume 295, Soc. Math. Fr., 2004, pp. 117-290 | Numdam | MR | Zbl

[16] Kings, G. The Tamagawa number conjecture for CM elliptic curves, Invent. Math., Volume 143 (2001), pp. 571-627 | DOI | MR | Zbl

[17] Lazard, M. Groupes analytiques $p$-adiques, Pub. Math. IHÉS, Volume 26 (1965), pp. 389-603 | Numdam | MR | Zbl

[18] Levine, M. $K$-theory and motivic cohomology of schemes (http://www.math.uiuc.edu/K-theory/336)

[19] McCallum, W. G.; Sharifi, R. T. A cup product in the Galois cohomology of number fields, Duke Math. J., Volume 120 (2004), pp. 269-310 | MR | Zbl

[20] Milne, J. S. Étale Cohomology, Princeton University Press, 1980 | MR | Zbl

[21] Morel, F.; Voevodsky, V. ${\mathbf{A}}^1$-homotopy theory of schemes, Pub. Math. IHÉS, Volume 90 (1999), pp. 45-143 | Numdam | MR | Zbl

[22] Neukirch, J.; Schmidt, A.; Wingberg, K. Cohomology of Number Fields, Grundlehren der math. Wiss., Volume 323, Springer, 2000 | MR | Zbl

[23] Perrin-Riou, B. $p$-adic $L$-functions and $p$-adic representations, SMF/AMS Texts and Monographs, Volume 3, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2000 | MR | Zbl

[24] Quillen, D. Finite generation of the groups $K_i$ of algebraic integers, LNM, 341, Springer, 1973 | MR | Zbl

[25] Serre, J.-P. Cohomologie galoisienne, Lecture Notes in Math., 5 e éd., Springer, 1994 | MR | Zbl

[26] Soulé, C. $K$-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. Math., Volume 55 (1979), pp. 251-295 | DOI | MR | Zbl

[27] Soulé, C. Operations on étale $K$-theory. Applications, Lecture Notes in Math., Volume 966, Springer, 1982, pp. 271-303 | MR | Zbl

[28] Soulé, C. Opérations en K-théorie algébrique, Canad. J.Math., Volume 37 (1985), pp. 488-550 | DOI | MR | Zbl

[29] Soulé, C. $p$-adic $K$-theory of elliptic curves, Duke, Volume 54 (1987), pp. 249-269 | DOI | MR | Zbl

[30] Suslin, A. A. Higher Chow Groups and Étale Cohomology, Cycles, Transfers and Motivic Homology Theories, Annals of Math. Studies, Volume 143, Princeton University Press, 2000 | MR | Zbl

[31] Suslin, A. A.; Voevodsky, V. Relative cycles and Chow sheaves, Cycles, Transfers and Motivic Homology Theories, Annals of Math. Studies, Volume 143, Princeton University Press, 2000 | MR | Zbl

[32] Thomason, R. W. Algebraic K-theory and étale cohomology, Ann. Sci. École Norm. Sup., Volume 13 (1985), pp. 437-552 | Numdam | MR | Zbl

[33] Thomason, R. W. Bott stability in Algebraic $K$-theory, in “Applications of Algebraic $K$-theory”, Contemp. Math., Volume 55 (1986), pp. 389-406 | MR | Zbl

[34] Voevodsky, V. Motivic cohomology with $Z/\ell$-coefficients (http://www.math.uiuc.edu/K-theory/639) | Numdam | Zbl

[35] Voevodsky, V. Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not. (2002), pp. 351-355 | DOI | MR | Zbl

[36] Weibel, C.A. An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Volume 38, Cambridge, 1994 | MR | Zbl