= Central defect blocks are matrix algebras
{title2=$B\cong\operatorname{Mat}_r(kD)$}
Let $k$ be an <algebraically closed field> and let a <block of a group algebra> $B$ have a central <defect group of a block> $D$. Its quotient by $J(kD)B$ is a defect-zero block, hence a full <matrix algebra> over $k$. The <algebra> $B$ is free over the central <local ring> $kD$. Lift a primitive idempotent from this matrix quotient; its corner is free of rank one over $kD$, and is therefore $kD$. Lifting matrix units gives $B\cong\operatorname{Mat}_r(kD)$.
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