The 𝔾m–equivariant motivic cohomology of Stiefel varieties

Williams, Ben¹

¹Department of Mathematics, University of Southern California, Kaprielian Hall, 3620 South Vermont Avenue, Los Angeles, CA 90089-2532, USA

Algebraic and Geometric Topology, Tome 13 (2013) no. 2, pp. 747-793

Résumé

We derive a version of the Rothenberg–Steenrod, fiber-to-base, spectral sequence for cohomology theories represented in model categories of simplicial presheaves. We then apply this spectral sequence to calculate the equivariant motivic cohomology of GLn with a general Gm–action; this coincides with the equivariant higher Chow groups. The motivic cohomology of PGLn and some of the equivariant motivic cohomology of a Stiefel variety, V m(An), with a general Gm–action is deduced as a corollary.

Détail

DOI : 10.2140/agt.2013.13.747

Classification : 19E15, 14C15, 18G55
Keywords: Equivariant, Motivic cohomology, Chow group, Fiber-to-base, Stiefel, Projective general linear group

Bibliographie

[1] S Bloch, Algebraic cycles and higher $K$-theory, Adv. in Math. 61 (1986) 267

[2] J M Boardman, Conditionally convergent spectral sequences, from: "Homotopy invariant algebraic structures" (editors J P Meyer, J Morava, W S Wilson), Contemp. Math. 239, Amer. Math. Soc. (1999) 49

[3] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972)

[4] D Dugger, D C Isaksen, Motivic cell structures, Algebr. Geom. Topol. 5 (2005) 615

[5] D Edidin, W Graham, Equivariant intersection theory, Invent. Math. 131 (1998) 595

[6] H Esnault, B Kahn, M Levine, E Viehweg, The Arason invariant and mod $2$ algebraic cycles, J. Amer. Math. Soc. 11 (1998) 73

[7] A Grothendieck, Torsion homologique et sections rationnelles, from: "Séminaire Claude Chevalley (2e année, 1958): Anneaux de Chow et applications", Secrétariat mathématique (1958)

[8] P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society (2003)

[9] D C Isaksen, Flasque model structures for simplicial presheaves, $K$–Theory 36 (2005) 371

[10] J F Jardine, Simplicial presheaves, J. Pure Appl. Algebra 47 (1987) 35

[11] J P May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 1 (1975)

[12] J P May, A concise course in algebraic topology, University of Chicago Press (1999)

[13] C Mazza, V Voevodsky, C Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs 2, American Mathematical Society (2006)

[14] F Morel, V Voevodsky, $\mathbf{A}^1$–homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. (1999) 45

[15] Y A Nisnevich, The completely decomposed topology on schemes and associated descent spectral sequences in algebraic $K$–theory, from: "Algebraic $K$–theory: Connections with geometry and topology" (editors J F Jardine, V P Snaith), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 279, Kluwer Acad. Publ. (1989) 241

[16] O Pushin, Higher Chern classes and Steenrod operations in motivic cohomology, $K$–Theory 31 (2004) 307

[17] M Rothenberg, N E Steenrod, The cohomology of classifying spaces of $H$–spaces, Bull. Amer. Math. Soc. 71 (1965) 872

[18] G Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. (1968) 105

[19] B Totaro, The Chow ring of a classifying space, from: "Algebraic $K$–theory" (editors W Raskind, C Weibel), Proc. Sympos. Pure Math. 67, Amer. Math. Soc. (1999) 249

[20] A Vistoli, On the cohomology and the Chow ring of the classifying space of $\mathrm{PGL}_p$, J. Reine Angew. Math. 610 (2007) 181

[21] V Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. (2003) 1

[22] V Voevodsky, Motivic Eilenberg–Maclane spaces, Publ. Math. Inst. Hautes Études Sci. (2010) 1

[23] C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge Univ. Press (1994)

[24] B Williams, The motivic cohomology of Stiefel varieties, Journal of $K$–Theory 10 (2012) 141