 Nagao module theorem

For a central idempotent $e$, a module $M=eM$, and $C_G(D)\le K\le N_G(D)$, put $b={\mathrm{Br}}_D(e)$. Then $M{\downarrow }_K=bM\oplus (1-b)M$, and every indecomposable summand of the second term has vertex not containing $D$. Indeed $e-b$ is a sum of $K$-orbit sums of group elements not centralizing $D$. Each is a relative trace from a centralizer not containing $D$. Applying these traces to the module and using the local endomorphism ring of an indecomposable summand proves relative projectivity for one such subgroup.