= Modules in a block are projective relative to its defect group
{title2=$M\mid\operatorname{Ind}_D^G\operatorname{Res}_D^G M$}
= Green relative projectivity theorem
{c}
{synonym}
For a <block of a group algebra> with identity $e$ and <defect group of a block> $D$, the <trace criterion for defect groups> writes $e=\operatorname{Tr}_D^G(a)$ with $a\in(RG)^D$. On any module in the block, multiplication by $a$ is $D$-linear and its <relative trace> is the identity. The <D. Higman criterion> therefore makes the module relatively $D$-projective.
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