Hochschild cohomology and Linckelmann cohomology for blocks of finite groups

Let G be a finite group, image an algebraically closed field of finite characteristic p, and let B be a block of image. We show that the Hochschild and Linckelmann cohomology rings of B are isomorphic, modulo their radicals, in the cases where B is cyclic andB is arbitrary and G either a nilpotent group or a Frobenius group (p odd). (The second case is a consequence of a more general result.) We give some related results in the more general case that B has a Sylow p-subgroup P as a defect group, giving a precise local description of a quotient of the Hochschild cohomology ring. In case P is elementary abelian, this quotient is isomorphic to the Linckelmann cohomology ring of B, modulo radicals.