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: and the same for
This gives the first page of the spectral sequence: we take with the differential. The derived couple of the above exact couple then gives the second page and so forth. Explicitly, we have that fits into the exact couple:
where and . Now, taking of
we get:
This tells the kernel and cokernel of. Expanding the exact couple into a long exact sequence, we get: for any r,
When, this is the same thing as the universal coefficient theorem for homology.
Assume the abelian group is finitely generated; in particular, only finitely many cyclic modules of the form can appear as a direct summand of. Letting we thus see is isomorphic to.
