  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.
