  Given a finitely generated module $M$ over a Noetherian local ring $R$, one
would like to know when the corresponding associated graded module has a pure
resolution over $G_{\mathfrak{m}}(R)$. In this article, we identify a complex
of free $G_{\mathfrak{m}}(R)$-modules, arising from given free resolution of
$M$ over $R$, which is a resolution of $G_{\mathfrak{m}}(M)$ if and only if
$G_{\mathfrak{m}}(M)$ is a pure $G_{\mathfrak{m}}(R)$-module. We also give
several applications of the purity of $G_{\mathfrak{m}}(M)$ in this paper.
