Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups

We give an explicit approach for Bockstein homomorphisms of the Hochschild cohomology of a group algebra and of a block algebra of a finite group and we show some properties. To give explicit definitions for these maps we use an additive decomposition and a Product Formula for the Hochschild cohomology of group algebras given by Siegel and Witherspoon in 1999. For k an algebraically closed field of characteristic p and G a finite group we prove an additive decomposition and a Product Formula for the cohomology algebra of a defect group of a block ideal of kG with coefficients in the source algebra of this block, and we define similar Bockstein homomorphisms.