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:

0⟶C⟶pC⟶modpC⊗ℤ/p⟶0.

Taking integral homology H, we get the exact couple of "doubly graded" abelian groups:

H*(C)⟶i=pH*(C)⟶jH*(C⊗ℤ/p)⟶k.

where the grading goes: H*(C)s,t=Hs+t(C) and the same for H*(C⊗ℤ/p),deg⁡i=(1,−1),deg⁡j=(0,0),deg⁡k=(−1,0).

This gives the first page of the spectral sequence: we take Es,t1=Hs+t(C⊗ℤ/p) with the differential 1d=j∘k. The derived couple of the above exact couple then gives the second page and so forth. Explicitly, we have Dr=pr−1H*(C) that fits into the exact couple:

Dr⟶i=pDr⟶rjEr⟶k

where rj=(mod p)∘p−r+1 and deg⁡(rj)=(−(r−1),r−1) (the degrees of i, k are the same as before). Now, taking Dnr⊗− of

0⟶ℤ⟶pℤ⟶ℤ/p⟶0,

we get:

0⟶Tor1ℤ(Dnr,ℤ/p)⟶Dnr⟶pDnr⟶Dnr⊗ℤ/p⟶0.

This tells the kernel and cokernel of Dnr⟶pDnr. Expanding the exact couple into a long exact sequence, we get: for any r,

0⟶(pr−1Hn(C))⊗ℤ/p⟶En,0r⟶Tor⁡(pr−1Hn−1(C),ℤ/p)⟶0.

When r=1, this is the same thing as the universal coefficient theorem for homology.

Assume the abelian group H*(C) is finitely generated; in particular, only finitely many cyclic modules of the form ℤ/ps can appear as a direct summand of H*(C). Letting r->∞ we thus see E∞ is isomorphic to (free part of H*(C))⊗ℤ/p.

## References

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

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