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

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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   (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  . Letting   we thus see   is isomorphic to  .

References

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  • McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, vol. 58 (2nd ed.), Cambridge University Press, ISBN 978-0-521-56759-6, MR 1793722
  • J. P. May, A primer on spectral sequences