Geometric two‐dimensional idèles with cycle module coefficients

Abstract

We give a theory of idèles with coefficients for smooth surfaces over a field. It is an analogue of Beilinson/Huber's theory of higher adèles, but handling cycle module sheaves instead of quasi-coherent ones. We prove that they give a flasque resolution of the cycle module sheaves in the Zariski topology. As a technical ingredient we show the Gersten property for cycle modules on equicharacteristic complete regular local rings, which might be of independent interest.

On the automorphic side of the K-theoretic Artin symbol

“…Suppose X/F q is a smooth variety. One could hope that there are exact categories of "higher adelic blocks" attached to X such that the restricted products appearing in [Bra14] for the cycle module M := K(−) being chosen to be K-theory, for example in Equation 1.7. loc. cit., or §4 loc.…”