This article is about the $Z^d-$periodic Green function $G_n(x, y)$ of the multiscale elliptic operator $Lu = −div (A(n·) \nabla u)$, where $A(x)$ is a $Z d-$periodic, coercive, and Hölder continuous matrix, and n is a large integer. We prove here pointwise estimates on $G_n(x, y)$, $\nabla_xG_n(x, y)$, $\nabla_y G_n(x, y)$ and $\nabla_{xy}G_n(x, y)$ in dimensions $d ≥ 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.