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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.

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
Full Text: DOI
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