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:
p ) , deg i = ( 1 , − 1 ) , deg j = ( 0 , 0 ) , deg k = ( − 1 , 0 ) .
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
deg (
(the degrees of i, k are the same as before).
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
free part of
{\displaystyle ({\text{free part of }}H_{*}(C))\otimes \mathbb {Z} /p}
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