On non-commutative twisting in étale and motivic cohomology

Hornbostel, Jens; Kings, Guido

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

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.

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.

Classification : 11R23, 14G40, 14F42, 11R32
Keywords: Étale cohomology, motivic cohomology, non-commutative Iwasawa-theory
Mot clés : cohomologie étale, cohomologie motivique, théorie d’Iwasawa non-commutative

[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