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:

$0 \to C \overset{p}\to C \overset{\text{mod } p}\to C \otimes \mathbb{Z}/p \to 0$.

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

$H_*(C) \overset{i = p}\to H_*(C) \overset{j} \to H_*(C \otimes \mathbb{Z}/p) \overset{k} \to$.

where the grading goes: $H_*(C)_{s, t} = H_{s+t}(C)$ and the same for $H_*(C \otimes \mathbb{Z}/p)$, $\operatorname{deg} i = (1, -1)$, $\operatorname{deg} j = (0, 0)$, $\operatorname{deg} k = (-1, 0)$.

This gives the first page of the spectral sequence: we take $E_{s, t}^1 = H_{s+t}(C \otimes \mathbb{Z}/p)$ 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:

$D^r \overset{i=p}\to D^r \overset{{}^r j} \to E^r \overset{k}\to$ 

where ${}^r j$ is $(\text{mod } p) \circ p^{-{r+1}}$ and $\operatorname{deg} {}^r j = (-(r-1), r - 1)$ (the degrees of i, k are the same as before). Now, taking $D_n^r \otimes -$ of $0 \to \mathbb{Z} \overset{p}\to \mathbb{Z} \to \mathbb{Z}/p \to 0$, we get:

$0 \to \operatorname{Tor}_1^{\mathbb{Z}}(D_n^r, \mathbb{Z}/p) \to D_n^r \overset{p}\to D_n^r \to D_n^r \otimes \mathbb{Z}/p \to 0$.

This tells the kernel and cokernel of $D^r_n \overset{p}\to D^r_n$. Expanding the exact couple into a long exact sequence, we get: for any r,

$0 \to (p^{r-1} H_n(C)) \otimes \mathbb{Z}/p \to E^r_{n, 0} \to \operatorname{Tor}(p^{r-1} H_{n-1}(C), \mathbb{Z}/p) \to 0$.

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 $\mathbb{Z}/p^s$ can appear as a direct summand of $H_*(C)$. Letting $r \to \infty$ we thus see $E^\infty$ is isomorphic to $(\text{free part of } H_*(C)) \otimes \mathbb{Z}/p$.

References

• McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics 58 (2nd ed.), Cambridge University Press, doi:10.2277/0521567599, ISBN 978-0-521-56759-6, MR 1793722
• J. P. May, A primer on spectral sequences