  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.
