Bockstein spectral sequence

Remarks

The spectral sequence comes from an exact couple with $D = H_*(X; \Z)$ and $E = H_*(X; \F_p)$. The map $D \to D$ is multiplication by $p$, and the other two maps come from the long exact sequence of $0 \to \Z \xrightarrow{p} \Z \to \Z/p \to 0$.

Example: for $X = \R P^2$ and $p = 2$, one has $H_1(X; \Z) = \Z/2$ and $H_2(X; \Z) = 0$. Here $E^1_1 = E^1_2 = \F_2$ and $d^1 \colon E^1_2 \to E^1_1$ is an isomorphism, so $E^2_n = 0$ for $n = 1, 2$, which matches $E^\infty$ since the free part of $H_n(X; \Z)$ vanishes for $n = 1, 2$.

The finite generation hypothesis ensures that the spectral sequence stops at a finite page in each degree. For a summand such as $\Z[1/p]$ or $\Z/p^\infty$ the description in terms of $E^\infty$ changes.

Going deeper

For a student

The spectral sequence is used to detect higher torsion, for instance in the homology of Lie groups, loop spaces and Eilenberg–MacLane spaces. In cohomology it is multiplicative, and the higher Bocksteins interact with Steenrod operations, which gives restrictions on how torsion can appear. It is one of the simplest examples of a spectral sequence built from an exact couple.