  We investigate the issue of the nonadiabatic superconductivity in
uncorrelated systems. A local approximation is employed coherently with the
weak dependence on the involved momenta. Our results show that nonadiabatic
vertex corrections are never negligible, but lead to a strong suppression of
$T_c$ with respect to the conventional theory. This feature is understood in
terms of the momentum-frequency dependence of the vertex function. In contrast
to strongly correlated systems, where the small ${\bf q}$-selection probes the
positive part of vertex function, vertex corrections in uncorrelated systems
are essentially negative resulting in an effective reduction of the
superconducting pairing. Our analysis shows that vertex corrections in
nonadiabatic regime can be never disregarded independently of the degree of
electronic correlation in the system.
