In this paper we construct symmetric powers in the motivic homotopy categories of morphisms and finite correspondences associated with f-admissible subcategories in the categories of schemes of finite type over a field. Using this construction we provide a description of the motivic Eilenberg-MacLane spaces representing motivic cohomology on some f-admissible categories including the category of semi-normal quasi-projective schemes and, over fields which admit resolution of singularities, on some admissible subcategories including the category of smooth schemes. This description is then used to give a complete computation of the algebra of bistable motivic cohomological operations on smooth schemes over fields of characteristic zero and to obtain partial results on unstable operations which are required for the proof of the Bloch-Kato conjecture.
@article{PMIHES_2010__112__1_0, author = {Voevodsky, Vladimir}, title = {Motivic Eilenberg-MacLane spaces}, journal = {Publications Math\'ematiques de l'IH\'ES}, volume = {112}, year = {2010}, pages = {1-99}, doi = {10.1007/s10240-010-0024-9}, mrnumber = {2737977}, zbl = {1227.14025}, language = {en}, url = {http://dml.mathdoc.fr/item/PMIHES_2010__112__1_0} }
Voevodsky, Vladimir. Motivic Eilenberg-MacLane spaces. Publications Mathématiques de l'IHÉS, Tome 112 (2010) pp. 1-99. doi : 10.1007/s10240-010-0024-9. http://gdmltest.u-ga.fr/item/PMIHES_2010__112__1_0/
2. Finite correspondences and transfers over a regular base, in: Algebraic Cycles and Motives. Vol. 1, London Math. Soc. Lecture Note Ser. 343 (2007), Cambridge Univ. Press, Cambridge | MR 2385302 | Zbl 1149.14014
,3. Voevodsky’s lectures on motivic cohomology 2000/2001, in: Algebraic Topology, Abel Symposia 4 (2009), Springer, Berlin | MR 2597743 | Zbl 1183.14028
,4. Topological hypercovers and -realizations, Math. Z. 246 (2004), p. 667-689 | MR 2045835 | Zbl 1055.55016
, ,5. Bivariant cycle cohomology, in: Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud. 143 (2000), Princeton Univ. Press, Princeton | MR 1764201 | Zbl 1019.14011
, ,6. Infinite Abelian Groups. Vol. II, Pure and Applied Mathematics 36-II (1973), Academic Press, New York | MR 349869 | Zbl 0257.20035
,7. On seminormal schemes, Compos. Math. 40 (1980), p. 325-365 | Numdam | MR 571055 | Zbl 0412.14024
, ,8. Etude Locale des Schemas et des Morphismes de Schemas (EGA 4), Publ. Math. IHES,20,24,28,32, 1964–1967. | Numdam | MR 173675 | Zbl 0135.39701
,9. Triangulations of algebraic sets, in: Algebraic Geometry, (1975), American Mathematical Society, Providence | MR 374131 | Zbl 0332.14001
,10. Model Categories and Their Localizations, Mathematical Surveys and Monographs 99 (2003), American Mathematical Society, Providence | MR 1944041 | Zbl 1017.55001
,11. Model Categories, Mathematical Surveys and Monographs 63 (1999), American Mathematical Society, Providence | MR 1650134 | Zbl 0909.55001
,12. Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (1971), Springer, Berlin | MR 1712872 | Zbl 0232.18001
,13. Weak equivalences and quasifibrations, in: Groups of Self-Equivalences and Related Topics, Lecture Notes in Math. 1425 (1990), Springer, Berlin | MR 1070579 | Zbl 0713.55010
,14. Lecture Notes on Motivic Cohomology, Clay Mathematics Monographs 2 (2006), American Mathematical Society, Providence | MR 2242284 | Zbl 1115.14010
, , ,15. Etale Cohomology, (1980), Princeton Univ. Press, Princeton | MR 559531 | Zbl 0433.14012
,16. A 1-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (2001), p. 45-143 | Numdam | MR 1813224 | Zbl 0983.14007
, ,17. Cohomology mod p of symmetric products of spheres, J. Inst. Polytech. Osaka City Univ., Ser. A 9 (1958), p. 1-18 | MR 121788 | Zbl 0084.39102
,18. Triangulated Categories, Ann. of Math. Studies 148 (2001), Princeton Univ. Press, Princeton | MR 1812507 | Zbl 0974.18008
,19. Karoubi’s construction for motivic cohomology operations, Am. J. Math. 130 (2008), p. 713-762 | MR 2418926 | Zbl 1147.14007
,20. A theorem on semi-simplicial monoid complexes, Ann. Math. (2) 70 (1959), p. 379-394 | MR 123321 | Zbl 0088.39101
,21. Homotopical Algebra, Lecture Notes in Math. 43 (1973), Springer, Berlin | MR 223432 | Zbl 0168.20903
,22. Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955), p. 1-42 | Numdam | MR 82175 | Zbl 0075.30401
,23. Cohomology Operations, (1962), Princeton Univ. Press, Princeton | MR 145525 | Zbl 0521.55001
, ,24. Higher Chow groups and etale cohomology, in: Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud. 143 (2000), Princeton Univ. Press, Princeton | MR 1764203 | Zbl 1019.14001
,25. Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), p. 61-94 | MR 1376246 | Zbl 0896.55002
, ,26. Relative cycles and Chow sheaves, in: Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud. 143 (2000), Princeton Univ. Press, Princeton | MR 1764199 | Zbl 1019.14004
, ,27. On seminormality, J. Algebra 67 (1980), p. 210-229 | MR 595029 | Zbl 0473.13001
,28. Letter to A. Beilinson. 1993. www.math.uiuc.edu/K-theory/33 .
,29. Homology of schemes, Sel. Math. (N.S.) 2 (1996), p. 111-153 | MR 1403354 | Zbl 0871.14016
,30. Cohomological theory of presheaves with transfers, in: Cycles, transfers, and motivic homology theories, Ann. of Math. Stud. 143 (2000), Princeton Univ. Press, Princeton | MR 1764200 | Zbl 1019.14010
,31. Triangulated categories of motives over a field, in: Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud. 143 (2000), Princeton Univ. Press, Princeton | MR 1764202 | Zbl 1019.14009
,32. Motivic cohomology with Z/l-coefficients, 2003. www.math.uiuc.edu/K-theory/639 .
,33. Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. 98 (2003), p. 1-57 | Numdam | MR 2031198 | Zbl 1057.14027
,34. On the zero slice of the sphere spectrum, Tr. Mat. Inst. Steklova 246 (2004), p. 106-115 | MR 2101286 | Zbl 1182.14012
,35. Cancellation theorem, Doc. Math., to appear. arXiv:math/0202012 , 2009. | MR 2804268 | Zbl 1202.14022
,36. Motives over simplicial schemes, J. K-theory, to appear. arXiv:0805.4431 , 2009. | MR 2600283 | Zbl 1194.14029
,37. Motivic cohomology with Z/l-coefficients, Ann. Math., submitted. arXiv:0805.4430 , 2009. | MR 2811603
,38. Simplicial radditive functors, J. K-theory, to appear. arXiv:0805.4434 , 2009. | MR 2640203 | Zbl 1194.55021
,39. Unstable motivic homotopy categories in Nisnevich and cdh-topologies, arXiv:0805.4576 , doi: 10.1016/j.jpaa.2009.11.005 , 2009. | MR 2593671 | Zbl 1187.14025
,40. Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud. 143 (2000), Princeton Univ. Press, Princeton | MR 1764197 | Zbl 1019.14008
, , ,41. Patching the norm residue isomorphism theorem, 2007. www.math.uiuc.edu/K-theory/844 .
,