Operations in étale and motivic cohomology
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- by Bert Guillou and Chuck Weibel PDF
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Abstract:
We classify all étale cohomology operations on $H_{\operatorname {et}}^n(-,\mu _\ell ^{\otimes i})$, showing that they were all constructed by Epstein. We also construct operations $P^a$ on the mod-$\ell$ motivic cohomology groups $H^{p,q}$, differing from Voevodsky’s operations. We use them to classify all motivic cohomology operations on $H^{p,1}$ and $H^{1,q}$ and suggest a general classification.References
- Lawrence Breen, Extensions du groupe additif, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 39–125 (French). MR 516914
- P. Brosnan and R. Joshua, Cohomology operations in motivic, étale and de Rham-Witt cohomology, preprint, 2007.
- Patrick Brosnan and Roy Joshua, Comparison of motivic and simplicial operations in mod-$l$-motivic and étale cohomology, Feynman amplitudes, periods and motives, Contemp. Math., vol. 648, Amer. Math. Soc., Providence, RI, 2015, pp. 29–55. MR 3415409, DOI 10.1090/conm/648/12997
- Séminaire Henri Cartan de l’Ecole Normale Supérieure, 1954/1955. Algèbres d’Eilenberg-MacLane et homotopie, Secrétariat Mathématique, 11 rue Pierre Curie, Paris, 1955 (French). MR 0087934
- Henri Cartan, Sur les groupes d’Eilenberg-Mac Lane. II, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 704–707 (French). MR 65161, DOI 10.1073/pnas.40.8.704
- Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77 (French). MR 498552
- A. Grothendieck and P. Deligne, La classe de cohomologie associée à un cycle, Cohomologie étale, Lecture Notes in Math., vol. 569, Springer, Berlin, 1977, pp. 129–153 (French). MR 3727436
- Samuel Eilenberg and Saunders MacLane, On the groups $H(\Pi ,n)$. III, Ann. of Math. (2) 60 (1954), 513–557. MR 65163, DOI 10.2307/1969849
- D. B. A. Epstein, Steenrod operations in homological algebra, Invent. Math. 1 (1966), 152–208. MR 199240, DOI 10.1007/BF01389726
- Eric M. Friedlander, Étale homotopy of simplicial schemes, Annals of Mathematics Studies, No. 104, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 676809
- Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
- V. A. Hinich and V. V. Schechtman, On homotopy limit of homotopy algebras, $K$-theory, arithmetic and geometry (Moscow, 1984–1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 240–264. MR 923138, DOI 10.1007/BFb0078370
- C. Haesemeyer and C. Weibel, The norm residue theorem in motivic cohomology, Princeton Univ. Press, 2019. Available at math.rutgers.edu/$\sim$weibel.
- J. F. Jardine, Steenrod operations in the cohomology of simplicial presheaves, Algebraic $K$-theory: connections with geometry and topology (Lake Louise, AB, 1987) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 279, Kluwer Acad. Publ., Dordrecht, 1989, pp. 103–116. MR 1045847, DOI 10.1007/978-94-009-2399-7_{5}
- J. F. Jardine, Higher spinor classes, Mem. Amer. Math. Soc. 110 (1994), no. 528, vi+88. MR 1211372, DOI 10.1090/memo/0528
- R. Joshua, Motivic $E_\infty$ algebras and the motivic DGA, preprint, 2012. Available at http://www.math.ohio-state.edu/$\sim$joshua/pub.html.
- Igor Kříž and J. P. May, Operads, algebras, modules and motives, Astérisque 233 (1995), iv+145pp (English, with English and French summaries). MR 1361938
- Marc Levine, Inverting the motivic Bott element, $K$-Theory 19 (2000), no. 1, 1–28. MR 1740880, DOI 10.1023/A:1007874218371
- W. S. Massey, Products in exact couples, Ann. of Math. (2) 59 (1954), 558–569. MR 60829, DOI 10.2307/1969719
- J. Peter May, A general algebraic approach to Steenrod operations, The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970) Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 153–231. MR 0281196
- J. P. May, Operads and sheaf cohomology, preprint, 2004. Available at http://www.math.uchicago.edu/~may/PAPERS/Esheaf.pdf.
- John McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. MR 1793722
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
- Fabien Morel and Vladimir Voevodsky, $\textbf {A}^1$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001). MR 1813224
- Carlo Mazza, Vladimir Voevodsky, and Charles Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006. MR 2242284
- Michèle Raynaud, Modules projectifs universels, Invent. Math. 6 (1968), 1–26 (French). MR 236164, DOI 10.1007/BF01389829
- N. E. Steenrod, Cohomology operations, Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. MR 0145525
- N. Steenrod and D. Epstein, Errata for Cohomology Operations, 5 pp., circa 1965.
- Jean-Pierre Serre, Cohomologie modulo $2$ des complexes d’Eilenberg-MacLane, Comment. Math. Helv. 27 (1953), 198–232 (French). MR 60234, DOI 10.1007/BF02564562
- Andrei Suslin and Vladimir Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 117–189. MR 1744945
- J.-L. Verdier, Cohomologie dans les topos, SGA 4, Lecture Notes in Math. 270, Springer, Berlin, 1972, pp. 1–82.
- Vladimir Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 1–57. MR 2031198, DOI 10.1007/s10240-003-0009-z
- Vladimir Voevodsky, Motivic cohomology with $\textbf {Z}/2$-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104. MR 2031199, DOI 10.1007/s10240-003-0010-6
- Vladimir Voevodsky, Motivic Eilenberg-MacLane spaces, Publ. Math. Inst. Hautes Études Sci. 112 (2010), 1–99. MR 2737977, DOI 10.1007/s10240-010-0024-9
- Vladimir Voevodsky, On motivic cohomology with $\mathbf Z/l$-coefficients, Ann. of Math. (2) 174 (2011), no. 1, 401–438. MR 2811603, DOI 10.4007/annals.2011.174.1.11
- Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136
- C. Weibel, The norm residue isomorphism theorem, J. Topol. 2 (2009), no. 2, 346–372. MR 2529300, DOI 10.1112/jtopol/jtp013
Additional Information
- Bert Guillou
- Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
- MR Author ID: 682731
- Email: bertguillou@uky.edu
- Chuck Weibel
- Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08901
- MR Author ID: 181325
- Email: weibel@math.rutgers.edu
- Received by editor(s): January 26, 2017
- Received by editor(s) in revised form: April 18, 2018
- Published electronically: April 4, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 1057-1090
- MSC (2010): Primary 14F42, 55S05; Secondary 14F20
- DOI: https://doi.org/10.1090/tran/7657
- MathSciNet review: 3968795