  We prove global existence of $H^2$ solutions to the Cauchy problem for the
generalized derivative nonlinear Schr\"{o}dinger equation on the 1-d torus.
This answers an open problem posed by Ambrose and Simpson (2015). The key is
the extraction of the terms that cause the problem in energy estimates and the
construction of suitable energies so as to cancel the problematic terms out by
effectively using integration by parts and the equation.
