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Motives of Azumaya algebras

Part of: $K$-theory in geometry Cycles and subschemes

Published online by Cambridge University Press:  14 April 2010

Bruno Kahn  and
Marc Levine
Bruno Kahn
Affiliation:
Institut de Mathématiques de Jussieu, 175–179 rue du Chevaleret, 75013 Paris, France ([email protected])
Marc Levine
Affiliation:
Department of Mathematics, Northeastern University, Boston, MA 02115, USA ([email protected])
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Abstract

We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson–Lichtenbaum conjecture, we apply our results to show the vanishing of SK2(A) for a central simple algebra A of square-free index (prime to the characteristic). This proves a conjecture of Merkurjev.

Keywords

Bloch-Lichtenbaum spectral sequencealgebraic cyclesMorel–Voevodsky stable homotopy categoryslice filtrationAzumaya algebrasSeveri–Brauer schemes

MSC classification

Secondary: 14C25: Algebraic cycles 19E08: $K$-theory of schemes
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 3 , July 2010 , pp. 481 - 599
Copyright
Copyright © Cambridge University Press 2010

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References

1.Amitsur, S. A., Generic splitting fields of central simple algebras, Annals Math. 68 (1955), 8–43.CrossRefGoogle Scholar
2.Arason, J. K., Cohomologische Invarianten Quadratischer Formen, J. Alg. 36 (1975), 448–491.CrossRefGoogle Scholar
3.Barbieri-Viale, L. and Kahn, B., On the derived category of 1-motives, I, preprint (2007; arXiv:0706.1498v1 [math.AG]).Google Scholar
4.Bass, H., Algebraic K-theory (Benjamin, New York, 1968).Google Scholar
5.Beilinson, A. and Vologodsky, V., A DG guide to Voevodsky's motives, Geom. Funct. Analysis 17(6) (2008), 1709–1787.CrossRefGoogle Scholar
6.Bloch, S., Algebraic cycles and higher K-theory, Adv. Math. 61(3) (1986), 267–304.CrossRefGoogle Scholar
7.Bloch, S., Some notes on elementary properties of higher Chow groups, Recent papers of Spencer Bloch, No. 18 (available at www.math.uchicago.edu/.bloch/publications.html).Google Scholar
8.Bloch, S. and Lichtenbaum, S., A spectral sequence for motivic cohomology, preprint (1995).Google Scholar
9.Cartan, H. and Eilenberg, S., Homological algebra, Princeton Mathematical Series, Volume 19 (Princeton University Press, 1956).Google Scholar
10.Chernousov, V. and Merkurjev, A., Connectedness of classes of fields and zero-cycles on projective homogeneous varieties, Compositio Math. 142(6) (2006), 1522–1548.CrossRefGoogle Scholar
11.Cisinski, D.-C. and Déglise, F., Local and stable homological algebra in Grothendieck abelian categories, Homology Homotopy Applicat. 11 (2009), 219–260.CrossRefGoogle Scholar
12.de Meyer, F. R., Projective modules over central separable algebras, Can. J. Math. 21 (1969), 39–43.CrossRefGoogle Scholar
13.Déglise, F., Finite correspondences and transfers over a regular base, in Algebraic cycles and motives, Volume 1, pp. 138–205, London Mathematical Society Lecture Notes Series, Volume 343 (Cambridge University Press, 2007).Google Scholar
14.Dold, A., Homology of symmetric products and other functors of complexes, Annals Math. (2) 68 (1958), 54–80.CrossRefGoogle Scholar
15.Dundas, B. I., Röndigs, O. and Østvær, P. A., Motivic functors, Documenta Math. 8 (2003), 489–525.CrossRefGoogle Scholar
16.Friedlander, E. and Suslin, A., The spectral sequence relating algebraic K-theory to motivic cohomology, Annales Scient. Éc. Norm. Sup. 35(6) (2002), 773–875.CrossRefGoogle Scholar
17.Friedlander, E., Suslin, A. and Voevodsky, V., Cycles, transfers and motivic homology theories, Annals of Mathematics Studies, Volume 143 (Princeton University Press, 2000).Google Scholar
18.Geisser, T. and Levine, M., The Bloch–Kato conjecture and a theorem of Suslin–Voevodsky, J. Reine Angew. Math. 530 (2001), 55–103.Google Scholar
19.Hirschhorn, P. S., Model categories and their localizations, Mathematical Surveys and Monographs, Volume 99 (American Mathematical Society, Providence, RI, 2003).Google Scholar
20.Hovey, M., Model categories, Mathematical Surveys and Monographs, Volume 63 (American Mathematical Society, Providence, RI, 1999).Google Scholar
21.Hovey, M., Spectra and symmetric spectra in general model categories, J. Pure Appl. Alg. 165 (2001), 63–127.CrossRefGoogle Scholar
22.Huber, A. and Kahn, B., The slice filtration and mixed Tate motives, Compositio Math. 142 (2006), 907–936.CrossRefGoogle Scholar
23.Ivorra, F., Réalization l-adic des motifs mixte, Thése de doctorat, Université de Paris 6 (2005; available at http://people.math.jussieu.fr/.fivorra/These.pdf).Google Scholar
24.Jardine, J. F., Motivic symmetric spectra, Documenta Math. 5 (2000), 445–553.CrossRefGoogle Scholar
25.Kahn, B., The Geisser–Levine method revisited and algebraic cycles over a finite field, Math. Annalen 324(3) (2002), 581–617.CrossRefGoogle Scholar
26.Kahn, B., Cohomologie non ramifiée des quadriques, in Geometric methods in the algebraic theory of quadratic forms, Lecture Notes in Mathematics, Volume 1835, pp. 1–23 (Springer, 2004).Google Scholar
27.Kahn, B. and Sujatha, R., Birational motives (revised version), in preparation.Google Scholar
28.Kan, D. M., Functors involving c.s.s. complexes, Trans. Am. Math. Soc. 87 (1958), 330–346.CrossRefGoogle Scholar
29.Knus, M., Quadratic and Hermitian forms over rings, Grundlehren der mathematischen Wissenschaften, Volume 294 (Springer, 1991).Google Scholar
30.Levine, M., Bloch's higher Chow groups revisited, Astérisque 226(10) (1994), 235–320.Google Scholar
31.Levine, M., Techniques of localization in the theory of algebraic cycles, J. Alg. Geom. 10 (2001), 299–363.Google Scholar
32.Levine, M., Chow's moving lemma in -homotopy theory, K-Theory 37(1) (2006), 129–209.CrossRefGoogle Scholar
33.Levine, M., The homotopy coniveau tower, J. Topology 1 (2008), 217–267.CrossRefGoogle Scholar
34.Levine, M., K-theory and motivic cohomology of schemes, I, preprint (2004; available at www.math.neu.edu/.levine/publ/Publ.html).Google Scholar
35.Merkurjev, A. S., The group SK 2 for quaternion algebras, Izv. Akad. Nauk SSSR Ser. Mat. 52(2) (1988), 310–335, 447 (in Russian; translation in Math. USSR Izv. 32(2) (1989), 313–337).Google Scholar
36.Merkurjev, A. S., K-theory of simple algebras, in K-theory and algebraic geometry: connections with quadratic forms and division algebras, Proceedings of Symposia in Pure Mathematics, Volume 58(I), pp. 65–83 (American Mathematical Society, Providence, RI, 1995).Google Scholar
37.Merkurjev, A. S., On the norm residue homomorphism for fields, in Mathematics in St Petersburg, American Mathematical Society Translations Series 2, Volume 174, pp. 49–71 (American Mathematical Society, Providence, RI, 1996).Google Scholar
38.Merkurjev, A. S. and Suslin, A. A., K-cohomology of Severi–Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46(5) (1982), 1011–1046, 1135–1136.Google Scholar
39.Merkurjev, A. S. and Suslin, A. A., Motivic cohomology of the simplicial motive of a Rost variety, J. Pure Appl. Alg., in press.Google Scholar
40.Morel, F., On the motivic π0 of the sphere spectrum, in Axiomatic, enriched and motivic homotopy theory, NATO Science Series II: Mathematics, Physics and Chemistry, Volume 131, pp. 219–260 (Kluwer, Dordrecht, 2004).Google Scholar
41.Morel, F. and Voevodsky, V., -homotopy theory of schemes, Publ. Math. IHES 90 (1999), 45–143.CrossRefGoogle Scholar
42.Neeman, A., Triangulated categories, Annals of Mathematics Studies, Volume 148 (Princeton University Press, 2001).Google Scholar
43.Nisnevich, Ye., The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory, in Algebraic K-theory: connections with geometry and topology (Lake Louise, AB, 1987), NATO Advanced Science Institute Series C: Mathematical and Physical Sciences, Volume 279, pp. 241–342 (Kluwer, Dordrecht, 1989).Google Scholar
44.Østvær, P. A. and Röndigs, O., Motives and modules over motivic cohomology, C. R. Acad. Sci. Paris Sér. I 342(10) (2006), 751–754.Google Scholar
45.Østvær, P. A. and Röndigs, O., Modules over motivic cohomology, Adv. Math. 219 (2008), 689–727.Google Scholar
46.Panin, I., Applications of K-theory in algebraic geometry, PhD thesis, LOMI, Leningrad (1984).Google Scholar
47.Pelaez, P., Mixed motives and the slice filtration, C. R. Acad. Sci. Paris Sér. I 347(9–10) (2009), 541–544.Google Scholar
48.Quillen, D., Higher algebraic K-theory, I, in Algebraic K-theory, Volume I, Lecture Notes in Mathematics, Volume 341, pp. 85–147 (Springer, 1973).Google Scholar
49.Riou, J., Catégorie homotopique stable d'un site suspendu avec intervalle, Bull. Soc. Math. France 135 (2007), 495–547.CrossRefGoogle Scholar
50.Riou, J., Théorie homotopique des S-schémas, Mémoire de DEA (available at www.math.u-psud.fr/.riou/dea/).Google Scholar
51.Rost, M., Injectivity of K2(D) → K2(F) for quaternion algebras, preprint (1986).Google Scholar
52.Sherman, C. C., K-cohomology of regular schemes, Commun. Alg. 7(10) (1979), 999–1027.CrossRefGoogle Scholar
53.Suslin, A., Torsion in K 2 of fields, K-Theory 1 (1987), 5–29.CrossRefGoogle Scholar
54.Suslin, A., SK 1 of division algebras and Galois cohomology, in Algebraic K-theory, Advances in Soviet Mathematics, Volume 4, pp. 75–99 (American Mathematical Society, Providence, RI, 1991).Google Scholar
55.Suslin, A., SK 1 of division algebras and Galois cohomology revisited, in Proceedings of the St Petersburg Mathematical Society, Volume XII, pp. 125–147, American Mathematical Society Translations Series 2, Volume 219 (American Mathematical Society, Providence, RI, 2006).Google Scholar
56.Suslin, A. and Voevodsky, V., Bloch–Kato conjecture and motivic cohomology with finite coefficients, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Advanced Science Institute Series C: Mathematical and Physical Sciences, Volume 548, pp. 117–189 (Kluwer, Dordrecht, 2000).Google Scholar
57.Voevodsky, V., Cohomological theory of presheaves with transfers, in Cycles, transfers and motivic cohomology theories, Annals of Mathematics Studies, Volume 143 (Princeton University Press, 2000).Google Scholar
58.Voevodsky, V., Triangulated categories of motives over a field, in Cycles, transfers and motivic cohomology theories, Annals of Mathematics Studies, Volume 143 (Princeton University Press, 2000).Google Scholar
59.Voevodsky, V., A possible new approach to the motivic spectral sequence for algebraic K-theory, in Recent progress in homotopy theory (Baltimore, MD, 2000), Contemporary Mathematics, Volume 293, pp. 371–379 (American Mathematical Society, Providence, RI, 2002).Google Scholar
60.Voevodsky, V., Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not. 2002(7) (2002), 351–355.CrossRefGoogle Scholar
61.Voevodsky, V., Motivic cohomology with 蒄/2-coefficients, Publ. Math. IHES 98 (2003), 59–104.CrossRefGoogle Scholar
62.Voevodsky, V., Cancellation theorem, Documenta Math., in press.Google Scholar
63.Vorst, T., Polynomial extensions and excision for K 1, Math. Annalen 244(3) (1979), 193–204.CrossRefGoogle Scholar
64.Wang, S., On the commutator group of a simple algebra, kAm. J. Math. 72 (1950), 323–334.CrossRefGoogle Scholar
65.Weibel, C. A., Module structure on the K-theory of graded rings, J. Alg. 105 (1987), 465–483.CrossRefGoogle Scholar
66.Weibel, C. A., Fall 2006 lectures on the proof of the Bloch–Kato conjecture (available at www.math.rutgers.edu/.weibel/motivic2006.html).Google Scholar