  We consider the Cauchy problem of a system of quadratic derivative nonlinear
Schr\"odinger equations which was introduced by M. Colin and T. Colin (2004) as
a model of laser-plasma interaction. For the nonperiodic setting, the authors
proved some well-posedness results, which contain the scaling critical case for
$d\geq 2$. In the present paper, we prove the well-posedness of this system for
the periodic setting. In particular, well-posedness is proved at the scaling
critical regularity for $d\geq 3$ under some conditions for the coefficients of
the Laplacian. We also prove some ill-posedness results. As long as we use an
iteration argument, our well-posedness results are optimal except for some
critical cases.
