 Reduction of Hom from a projective group-algebra lattice

Let $W,W^{\prime }$ be choices of integral form of a group representation over a complete p-modular system. Then $I={\mathrm{Hom}}_{\mathcal{O}G}(W,W^{\prime })$ is finite free, and $K{\otimes }_{\mathcal{O}}I={\mathrm{Hom}}_{KG}(K\otimes W,K\otimes W^{\prime })$: clearing denominators proves the spanning assertion. Also ${\mathrm{Hom}}_{\mathcal{O}G}(W,\pi W^{\prime })=\pi I$. If $W$ is a projective module, applying the Hom functor to $0\to \pi W^{\prime }\to W^{\prime }\to W^{\prime }/\pi W^{\prime }\to 0$ yieldsConsequently 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.