of
abelian groups, when they are introduced as coefficients into a
chain complexC, 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 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.
Bockstein, Meyer (1942), "Universal systems of ∇-homology rings", C. R. (Doklady) Acad. Sci. URSS, New Series, 37: 243–245,
MR0008701
Bockstein, Meyer (1943), "A complete system of fields of coefficients for the ∇-homological dimension", C. R. (Doklady) Acad. Sci. URSS, New Series, 38: 187–189,
MR0009115