Two-dimensional Id\`eles with Cycle Module Coefficients

We give a theory of id\`eles with coefficients for smooth surfaces over a field. It is an analogue of Beilinson/Huber's theory of higher ad\`eles, 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.