Bockstein homomorphism

In homological algebra, the Bockstein homomorphism, introduced by , is a connecting homomorphism associated with a short exact sequence
of abelian groups, when they are introduced as coefficients into a chain complex "C", and which appears in the homology groups as a homomorphism reducing degree by one,
To be more precise, "C" should be a complex of free, or at least torsion-free, abelian groups, and the homology is of the complexes formed by tensor product with "C" (some flat module condition should enter). The construction of β is by the usual argument (snake lemma).
A similar construction applies to cohomology groups, this time increasing degree by one. Thus we have
The Bockstein homomorphism formula_4 associated to the coefficient sequence
is used as one of the generators of the Steenrod algebra. This Bockstein homomorphism has the following two properties:
in other words, it is a superderivation acting on the cohomology mod "p" of a space.