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.

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: ${\displaystyle H_{*}(C)_{s,t}=H_{s+t}(C)}$ and the same for ${\displaystyle H_{*}(C\otimes \mathbb {Z} /p),\deg i=(1,-1),\deg j=(0,0),\deg k=(-1,0).}$

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