= Brauer morphism on a defect transfer ideal
{c}
For $N=N_G(D)$ and $C=C_G(D)$, the <Brauer morphism> restricts to a surjection
$$
\operatorname{Br}_D:I_D=\operatorname{Tr}_D^G((kG)^D)\longrightarrow J_D=\operatorname{Tr}_D^N(kC)=(kC)_D^N.
$$
This follows from <Brauer morphism and relative trace> and the fact that $kC$ is contained in $(kG)^D$. The target is an ideal in the commutative algebra $(kC)^N$. Applying <primitive idempotents under a surjection of commutative Artinian ideals> and the <trace criterion for defect groups> identifies block idempotents with defect group $D$ with the primitive idempotents of $J_D$. Applying the same argument to $N$ gives <Brauer first main theorem>. The whole kernel need not be nilpotent, since blocks with smaller defect can be killed.
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