  We establish the global well-posedness of the derivative nonlinear
Schr\"odinger equation with periodic boundary condition in the Sobolev space
$H^{\frac12}$, provided that the mass of initial data is less than $4\pi$. This
result matches the one by Miao, Wu, and Xu and its recent mass threshold
improvement by Guo and Wu in the non-periodic setting. Below $H^{\frac12}$, we
show that the uniform continuity of the solution map on bounded subsets of
$H^s$ does not hold, for any gauge equivalent equation.
