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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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##### References:
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