 Integral form of a group representation

For a p-modular system $(K,\mathcal{O},k)$, an integral form of a finite-dimensional $KG$-module $V$ is a $G$-stable finite free $\mathcal{O}$-submodule $W\subset V$ with $K{\otimes }_{\mathcal{O}}W\cong V$. Starting with a basis lattice $L$, the sum ${\sum }_{g\in G}gL$ supplies such a form: it is finitely generated and torsion-free, hence free over the discrete valuation ring $\mathcal{O}$. Reduction gives the $kG$-module $W/\pi W$, where $\pi$ is a uniformizer.