Source of an indecomposable module

For an indecomposable module $M$ with vertex of an indecomposable module $D$, a source is an indecomposable $RD$-module $S$ which is a direct summand of ${\mathrm{Res}}_D^{G}M$ and for which $M$ is a direct summand of ${\mathrm{Ind}}_D^{G}S$. It has vertex $D$ as a $D$-module. Sources at a fixed vertex are conjugate by its normalizer.

Over a modular field or a complete discrete valuation coefficient ring, an indecomposable module has trivial source when a source is the rank-one trivial module of its vertex. These are precisely the indecomposable direct summands of permutation modules. A general trivial source module is a direct sum of these indecomposables.

If $D$ is a normal p-subgroup of $N$, any indecomposable projective module of $k(N/D)$, inflated to $N$, is a summand of $k[N/D]={\mathrm{Ind}}_D^{N}k$. It has trivial source and vertex $D$. Its Brauer quotient of a module at $D$ is the entire module, since $D$ acts trivially and all proper-subgroup relative traces multiply by an index divisible by $p$. This prevents its vertex from being smaller than $D$.

For coset permutation modules, Mackey restriction formula and Frobenius reciprocity identify the homomorphism space with a free module having one orbit-sum basis vector per double coset. The same basis works over the residue field, so reduction of homomorphisms is surjective. Applied to endomorphisms of a permutation module, idempotent refinement theorem lifts each modular direct summand to a trivial source lattice. Lifting inverse isomorphisms between reductions shows uniqueness among trivial source lattices.