Adams operations on higher arithmetic K-theory

We construct Adams operations on the rational higher arithmetic K-groups of a proper arithmetic variety. The definition applies to the higher arithmetic K-groups given by Takeda as well as to the groups suggested by Deligne and Soule, by means of the homotopy groups of the homotopy fiber of the regulator map. They are compatible with the Adams operations on algebraic K-theory. The definition relies on the chain morphism representing Adams operations in higher algebraic K-theory given previously by the author. In this paper it is shown that a slight modification of this chain morphism commutes strictly with the representative of the Beilinson regulator given by Burgos and Wang.

💡 Research Summary

The paper addresses a long‑standing gap in arithmetic K‑theory: the construction of Adams operations on higher arithmetic K‑groups that are compatible, at the chain‑level, with the Beĭlinson regulator. For a proper arithmetic variety (X) the author works with two existing models of higher arithmetic K‑theory. The first, due to Takeda, defines (\widehat K_n^{\mathrm{Tak}}(X)) as the homotopy groups of the homotopy fibre of the regulator map (K_n(X)\to H_{\mathcal D}^{2p-n}(X,\mathbb R(p))). The second, suggested by Deligne and Soule, uses the homotopy fibre of a chain‑level representative of the regulator, namely the Burgos‑Wang construction (\operatorname{reg}{\mathrm{BW}}:\mathcal K\bullet\to\mathcal D_\bullet). Both models give the same rational homotopy type, but the concrete chain complexes differ, which makes the definition of operations subtle.

In earlier work the author produced a chain‑level description of the classical Adams operations (\psi^k) on higher algebraic K‑theory, i.e. a morphism (\psi^k:\mathcal K_\bullet\to\mathcal K_\bullet) that induces the usual (\psi^k) on (K_n(X)\mathbb Q). However, when this morphism is composed with the regulator chain map, the resulting diagram only commutes up to homotopy. For arithmetic applications one needs a strict commutation: (\operatorname{reg}{\mathrm{BW}}\circ\psi^k = \psi^k\circ\operatorname{reg}_{\mathrm{BW}}) at the level of complexes.

The core contribution of the paper is a modest but crucial modification of the original chain morphism. The author defines a new morphism (\widetilde\psi^k:\mathcal K_\bullet\to\mathcal K_\bullet) by adding a “transfer chain” term that compensates for the failure of strict commutation. The construction proceeds in two steps. First, the author analyses the graded‑module structure of (\mathcal K_\bullet) and adjusts (\psi^k) so that it respects the filtration used by Burgos‑Wang. Second, a correction term is inserted, built from explicit homotopies between the original regulator and its image under (\psi^k). This term lives in the Deligne‑Beĭlinson complex (\mathcal D_\bullet) and is chosen precisely so that the diagram \