= Reduction of Hom from a projective group-algebra lattice
Let $W,W'$ be choices of <integral form of a group representation> over a complete <p-modular system>. Then $I=\operatorname{Hom}_{\mathcal OG}(W,W')$ is finite free, and $K\otimes_{\mathcal O}I=\operatorname{Hom}_{KG}(K\otimes W,K\otimes W')$: clearing denominators proves the spanning assertion. Also $\operatorname{Hom}_{\mathcal OG}(W,\pi W')=\pi I$. If $W$ is a <projective module>, applying the <Hom functor> to $0\to\pi W'\to W'\to W'/\pi W'\to0$ yields
$$
I/\pi I\cong\operatorname{Hom}_{kG}(W/\pi W,W'/\pi W').
$$
Consequently these ordinary and modular Hom spaces have the same <dimension of a vector space>. Projectivity is essential: for $G=C_2$ over $\mathbb Z_2$, the trivial and sign lattices have zero Hom between them, whereas their reductions in characteristic $2$ coincide and have a one-dimensional Hom space.
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