  We report on first principles calculations of superconductivity in a single
layer of lead on a silicon substrate including a full treatment of phononic and
RPA screened coulomb interactions within the parameter free framework of
Density Functional Theory for superconductors. A thorough investigation shows
that several approximations that are commonly valid in bulk systems fail in
this constrained 2D geometry. The calculated critical temperature turns out to
be much higher than the experimental value of 1.86K. We argue that the only
plausible explanation for the experimental Tc suppression is the onset of
fluctuations of the superconducting order parameter.
