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Operations on algebraic \(K\)-theory and regulators via the \(A^1\)-homotopy theory. (Opérations sur la k-théorie algébrique et régulateurs via la théorie homotopique des schémas.) (French. English summary) Zbl 1113.19003
In this paper the author uses the \(A^{1}\)-homotopy theory (defined by F. Morel and V. Voevodsky) over regular schemes to reduce the construction of operations on algebraic K-theory and regulators to the classical case: \(K_{0} \) groups and Chow groups. The main result of this paper is:
Let S be a regular scheme. There are canonical bijections: \[ \text{End}_{H(S)}(\mathbb{Z}\times Gr) \widetilde{\rightarrow }\text{End}_{Sm/S^{\text{op}}\mathcal{S}ets}(K_{0}(-))\widetilde{\rightarrow}\prod_{n\in \mathbb{Z}}\underset{(d,r)\in \mathbb{N}^{2}}{\lim } K_{0}(Gr_{d,r})\widetilde{=}\prod_{n\in \mathbb Z}K_{0}(S)[[c_{1},c_{2},\dots]]. \] This result generalises the operations with several variables providing the following:
Let S be a regular scheme. The object \(\mathbb Z\times Gr\) is endowed with the structure of a special \(\lambda\)-ring inside the category \(H(S)\).
The author proves that these structures on \(\mathbb Z\times Gr\) arising from the classical ones on \(K_{0}(-)\) induce the same maps on \(K_{\ast }(-)\) as the other known constructions. This study is continued in the stable homotopy category \(SH(S)\) of \(P^{1}\)-spectra. The object BGL representing algebraic K-theory constructed in that category by V. Voevodsky gives tools to compute the endomorphism ring of BGL in \(SH(S)\). The \(\mathbb{Q}\)-localised version of this computation leads to an interesting result: “Let \(S\) be a regular scheme. There exists a canonical decomposition in \(SH(S)\) of the \(\mathbb{Q}\)-localisation BGL\(_{\mathbb{Q}}\) of BGL: \(BGL_{\mathbb{Q}}=\bigoplus_{i\in\mathbb Z}H^{(i)},\) where for any \(k\in \mathbb Z-\{0\}\) the Adams operation \( \Psi ^{k}\) acts on \(H^{(i)}\) by multiplication par \(k^{i}\)”. If \(k\) is a perfect field the set of morphisms from \(\mathbb{Z}\times Gr\) to motivic Eilenberg-MacLane spaces in \(H(k)\) and from BGL to motivic Eilenberg-MacLane spectra in \(SH(k)\) can be computed.

MSC:
19E08 \(K\)-theory of schemes
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
14F35 Homotopy theory and fundamental groups in algebraic geometry
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[1] Berthelot, P.; Grothendieck, A.; Illusie, L., Théorie des intersections et théorème de riemann – roch, () · Zbl 0218.14001
[2] Gillet, H., Riemann – roch theorems for higher algebraic K-theory, Advances in mathematics, 40, 3, 203-289, (1981) · Zbl 0478.14010
[3] Goerss, P.G.; Jardine, J.F., Simplicial homotopy theory, Progress in mathematics, vol. 174, (1999), Birkhaüser · Zbl 0949.55001
[4] Jardine, J.F., Motivic symmetric spectra, Documenta Mathematica, 5, 445-552, (2000) · Zbl 0969.19004
[5] Jouanolou, J.-P., Une suite exacte de mayer – vietoris en K-théorie algébrique, (), 293-316 · Zbl 0291.14006
[6] Lecomte, F., Simplicial schemes and Adams operations, (), 437-449 · Zbl 0969.55008
[7] Levine, M., Lambda-operations, K-theory and motivic cohomology, Fields institute communications, 16, 131-184, (1997) · Zbl 0883.19001
[8] Loday, J.-L., K-théorie algébrique et représentations de groupes, Annales scientifiques de l’école normale supérieure (quatrième série), 9, 3, 309-377, (1976) · Zbl 0362.18014
[9] Morel, F., Théorie homotopique des schémas, Astérisque, vol. 256, (1999), Société Mathématique de France · Zbl 0933.55021
[10] Morel, F.; Voevodsky, V., \(\mathbf{A}^1\)-homotopy theory of schemes, Publications mathématiques de l’I.H.E.S., 90, 45-143, (1999) · Zbl 0983.14007
[11] Quillen, D.G., Higher algebraic K-theory I, (), 85-147 · Zbl 0292.18004
[12] Riou, J., Opérations sur la K-théorie algébrique et régulateurs via la théorie homotopique des schémas, Thèse de l’Université Paris 7 - Denis Diderot, Juillet 2006
[13] Soulé, C., Opérations en K-théorie algébrique, Canadian journal of mathematics, 37, 488-550, (1985) · Zbl 0575.14015
[14] Voevodsky, V., \(\mathbf{A}^1\)-homotopy theory, Proceedings of the international congress of mathematicians (Berlin), vol. I, Documenta Mathematica, 1, 579-604, (1998), (Extra volume) · Zbl 0907.19002
[15] Waldhausen, F., Algebraic K-theory of spaces, (), 318-419
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