Bockstein spectral sequence explained

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 \longrightarrow C \overset{p}\longrightarrow C \overset{ mod p} \longrightarrow C ⊗ \Z/p \longrightarrow 0 .

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

H _{*(C)} \overset{i = p} \longrightarrow H _{*(C)} \overset{j} \longrightarrow H _{*(C} ⊗ \Z/p) \overset{k} \longrightarrow.

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 ^{1} _{s,t} = H _{s+t} (C ⊗ \Z/p) with the differential {} ^{1} d = j \circ 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} \overset{i = p}\longrightarrow D ^{r} \overset{{} ^{r} j} \longrightarrow E ^{r} \overset{k}\longrightarrow

where

{} ^{r} j = ( mod p) \circ p ^{-{r+1} } and \deg ({} ^{r} j) = ( - (r -1 ), r - 1 ) (the degrees of i, k are the same as before). Now, taking D ^{r} _{n} ⊗ - of 0 \longrightarrow \Z \overset{p}\longrightarrow \Z \longrightarrow \Z/p \longrightarrow 0 ,

we get:

0 \longrightarrow \operatorname{Tor} ^{\Z} _{1} (D ^{r,} _{n} \Z/p) \longrightarrow D ^{r} _{n} \overset{p}\longrightarrow D ^{r} _{n} \longrightarrow D ^{r} _{n} ⊗ \Z/p \longrightarrow 0 .

This tells the kernel and cokernel of

D ^{r} _{n} \overset{p}\longrightarrow D ^{r} _{n} . Expanding the exact couple into a long exact sequence, we get: for any r,0 \longrightarrow (p ^{r-1} H _{n(C))} ⊗ \Z/p \longrightarrow E ^{r} _{n, 0} \longrightarrow \operatorname{Tor}(p ^{r-1} H _{n-1} (C), \Z/p) \longrightarrow 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 \to inf ty we thus see E ^{infty} is isomorphic to ( freepartof H _{*(C))} ⊗ \Z/p .

References

• • J. P. May, A primer on spectral sequences