Bockstein spectral sequence

In mathematics, the Bockstein spectral sequence is a spectral sequence relating the homology with mod "p" coefficients and the homology reduced mod "p". It is named after Meyer Bockstein.
Definition.
Let "C" be a chain complex of torsion-free abelian groups and "p" a prime number. Then we have the exact sequence:
Taking integral homology "H", we get the exact couple of "doubly graded" abelian groups:
where the grading goes: formula_3 and the same for formula_4
This gives the first page of the spectral sequence: we take formula_5 with the differential formula_6. The derived couple of the above exact couple then gives the second page and so forth. Explicitly, we have formula_7 that fits into the exact couple:
we get:
This tells the kernel and cokernel of formula_13. Expanding the exact couple into a long exact sequence, we get: for any "r",
When formula_15, this is the same thing as the universal coefficient theorem for homology.
Assume the abelian group formula_16 is finitely generated; in particular, only finitely many cyclic modules of the form formula_17 can appear as a direct summand of formula_16. Letting formula_19 we thus see formula_20 is isomorphic to formula_21.