  In this paper we prove that the 1D Schr\"odinger equation with derivative in
the nonlinear term is globally well-posed in $H^{s}$, for $s>\frac12$ for data
small in $L^{2}$. To understand the strength of this result one should recall
that for $s<\frac12$ the Cauchy problem is ill-posed, in the sense that uniform
continuity with respect to the initial data fails. The result follows from the
method of almost conserved energies, an evolution of the ``I-method'' used by
the same authors to obtain global well-posedness for $s>\frac23$. The same
argument can be used to prove that any quintic nonlinear defocusing
Schr\"odinger equation on the line is globally well-posed for large data in
$H^{s}$, for $s>\frac12$.
