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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.

A one-term summand $\mathbb{Z}[i]$ becomes a one-term ${\mathbb{F}}_p$ complex after reduction modulo $p$. Its Bockstein homomorphism vanishes, so it contributes one $\mathbb{Z}/p$ summand to Bockstein homology in degree $i$.