Relative projectivity and vertices for integral group lattices

Fix a splitting $p$-modular system $(K,\mathcal{O},k)$, let $H$ be finite, let $Q\le H$, and let $M$ be an $\mathcal{O}H$-lattice in the sense of An OG-lattice is a finite free module over the valuation ring with G-action, and reduction modulo the maximal ideal produces a kG-module. The lattice $M$ is relatively $Q$-projective if it is an $\mathcal{O}H$-direct summand of

${\mathrm{Ind}}_Q^H{\mathrm{Res}}_Q^{H}M\mathrm{.}$

A vertex of a nonzero indecomposable $\mathcal{O}H$-lattice $M$ is a $p$-subgroup $Q\le H$ that is minimal under inclusion among the subgroups for which $M$ is relatively $Q$-projective. Vertices exist. Indeed, if $P$ is a Sylow $p$-subgroup of $H$, then $[H:P]$ is a unit of $\mathcal{O}$. For a left transversal $T$ of $P$ in $H$, the induction counit $\epsilon (t\otimes m)=tm$ is split by the $H$-linear map

$m⟼[H:P{]}^{-1}{\sum }_{t\in T}t\otimes t^{-1}m\mathrm{.}$

Thus every $M$ is relatively $P$-projective, and the finite set of $p$-subgroups with this property has a minimal member. All direct summands, inductions, and restrictions here are taken in the category of finite-free $\mathcal{O}$-lattices. The construction uses only finite sums and finite minimization, not the Axiom of Choice.