Algebraic Morava K-theory spectra over perfect fields
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 2, p. 369-390
In the paper [2] we constructed (co)homology theories on the category of smooth schemes which share some of the some of the defining properties of the (co)homology theories induced by the Morava k-theory spactra in classical homotopy theory. Some proofs used the topological realization functor (cf. [8]). The existence of that functor requires the base field k to be embedded in . In this manuscript we investigate up to what extent we can obtain the same results under the sole assumption of perfectness of the base field. The results proved here guarantee the existence of spectra Φ i satisfying the same properties as in [2], provided that the algebra of all the bistable motivic cohomology operations verifies an assumption involving the Milnor operation Q t .
Classification:  14F42,  55P42,  14A15
@article{ASNSP_2009_5_8_2_369_0,
     author = {Borghesi, Simone},
     title = {Algebraic Morava $K$-theory spectra over perfect fields},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {2},
     year = {2009},
     pages = {369-390},
     zbl = {1179.14019},
     mrnumber = {2548251},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2009_5_8_2_369_0}
}
Borghesi, Simone. Algebraic Morava $K$-theory spectra over perfect fields. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 2, pp. 369-390. http://www.numdam.org/item/ASNSP_2009_5_8_2_369_0/

[1] S. Borghesi, “Algebraic Morava K-theories and the Higher Degree Formula”, PhD Thesis, Northwestern University, 2000, http://www.math.uiuc.edu/K-theory/0412/. | MR 2700816

[2] S. Borghesi, Algebraic Morava K-theories, Invent. Math. 151 (2003), 381–413. | MR 1953263 | Zbl 1030.55003

[3] A. A. Elmendorf, I. Kriz, M. A. Mandell and J. P. May, “Rings, Modules, and Algebras in Stable Homotopy Theory”, Mathematical Surveys and Monographs, Vol. 47, American Mathematical Society, Providence, RI, 1997, xii–249. | MR 1417719 | Zbl 0894.55001

[4] J. F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445–552. | MR 1787949 | Zbl 0969.19004

[5] J. Milnor, The Steenrod Algebra and its Dual, Ann. of Math. (2) 67 (1958), 150–171. | MR 99653 | Zbl 0080.38003

[6] J. Milnor and J. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. | MR 174052 | Zbl 0163.28202

[7] J. Milnor and J. Stasheff, ‘Characteristic Classes”, Princeton University Press and University of Tokyo Press, Princeton, New Jersey, 1974. | MR 440554 | Zbl 0071.00414

[8] F. Morel and V. Voevodsky, 𝔸 1 -homotopy theory of schemes, Publ. Math. Inst. Hautes Étude Sci. 90 (1999), 45–143. | Numdam | MR 1813224 | Zbl 0983.14007

[9] P. Hu, S-modules in the Category of Schemes”, Mem. Amer. Math. Soc., Vol. 161, 2003, no. 767, viii–125. | MR 1950209 | Zbl 1024.55006

[10] A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients, In: “The arithmetic and geometry of algebraic cycles” (Banff, AB, 1998), 117–189, Nato Sci. Ser. C Math. Phys. Sci., Vol. 548, Kluwer Acad. Publ., Dordrecht, 2000. | MR 1744945 | Zbl 1005.19001

[11] R. M. Switzer, “Algebraic Topology-Homotopy and Homology”, Springer-Verlag, New York-Heidelberg, 1975, xii–526. | MR 385836 | Zbl 0305.55001

[12] V. Voevodsky, Lectures on motivic cohomology 2000/2001 (written by Pierre Deligne). http://www.math.uiuc.edu/K-theory/0527/ | Zbl 0941.19001

[13] V. Voevodsky, Reduced power operation in motivic cohomology, Publ. Math. Inst. Hautes Étude Sci. 98 (2003), 1–57. | Numdam | MR 2031198 | Zbl 1057.14027

[14] V. Voevodsky, Motivic cohomology with /2 coefficients, Publ. Math. Inst. Hautes Étude Sci. 98 (2003), 59–104. | Numdam | MR 2031199 | Zbl 1057.14028

[15] V. Voevodsky, On motivic cohomology with /l-coefficients, preprint, http://www.math.uiuc.edu/K-theory/0639/. | MR 2811603 | Zbl 1236.14026

[16] V. Voevodsky, Motivic Eilenberg-MacLane spaces, http://www.math.uiuc.edu/K-theory/0864/. | Numdam | Zbl 1227.14025