A reduction theorem for non-vanishing of Hochschild cohomology of block algebras and Happel's property

In this short research note we obtain a reduction theorem for the non-vanishing of the first Hochschild cohomology of block algebras of finite groups with non-trivial defect groups. Along the way we investigate this problem for the blocks of some simple finite group algebras. Mimicking the case of blocks of finite group algebras we find some examples of category algebras that satisfy Happel’s property.