\newcommand{\eps}{\varepsilon} We study the homogenization of the linear equation $$ R(\eps^{-1}x){\partial u_\eps \over\partial t}- \textrm{div} (a(\eps^{-1}x) \cdot \nabla u_\eps) = f\ , $$ with appropriate initial/final conditions, where $R$ is a measurable bounded periodic function and $a$ is a bounded uniformly elliptic matrix, whose coefficients $a_{ij}$ are measurable periodic functions. \\ Since we admit that $R$ may vanish and change sign, the usual compactness of the solutions in $L^2$ may not hold if the mean value of $R$ is zero.