  We study the defocusing inhomogeneous mass-critical nonlinear Schr\"odinger
equation on $\mathbb R^2$ $$i u_t +\Delta u=g(nx) |u|^2 u$$ for initial data in
$L^2(\mathbb R^2)$. We obtain sufficient conditions on $g$ to ensure existence
and uniqueness of global solutions for $n$ sufficiently large, as well as
homogenization.
