Bockstein square-zero identity (${\beta }^2=0$)

Exactness of the integral coefficient sequence gives $\hat{\beta }\rho =0$. The Bockstein factorization through integral cohomology therefore gives ${\beta }^2=\rho \hat{\beta }\rho \hat{\beta }=0$, for every modulus $n\ge 2$, including composite moduli.

The degree-one Bockstein homomorphism makes modulo-$n$ cohomology into a cochain complex. Its cohomology is the Bockstein cohomology. For prime $n$ this is a graded vector space; for composite $n$ it is a graded $\mathbb{Z}/n$-module. For ${\mathbb{RP}}^3$ at modulus two, its only nonzero groups are ${\mathbb{F}}_2$ in degrees zero and three. This is the cohomological counterpart of Bockstein homology.