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:
0 → C → p C → mod p C ⊗ Z / p → 0 .
Taking integral homology H, we get the exact couple of "doubly graded" abelian groups:
H ∗ ( C ) → i = p H ∗ ( C ) → j H ∗ ( C ⊗ Z / p ) → k .
where the grading goes: H ∗ ( C ) s , t = H s + 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 E s , t 1 = H s + t ( C ⊗ Z / p ) with the differential 1 d = j ∘ k . The derived couple of the above exact couple then gives the second page and so forth. Explicitly, we have D r = p r − 1 H ∗ ( C ) that fits into the exact couple:
D r → i = p D r → r j E r → k where r j is ( mod p ) ∘ p − r + 1 and deg r j = ( − ( r − 1 ) , r − 1 ) (the degrees of i, k are the same as before). Now, taking D n r ⊗ − of 0 → Z → p Z → Z / p → 0 , we get:
0 → Tor 1 Z ( D n r , Z / p ) → D n r → p D n r → D n r ⊗ Z / p → 0 .
This tells the kernel and cokernel of D n r → p D n r . Expanding the exact couple into a long exact sequence, we get: for any r,
0 → ( p r − 1 H n ( C ) ) ⊗ Z / p → E n , 0 r → Tor ( p r − 1 H n − 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 / p s 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 .
