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 $D_{n}^{r}{\overset {p}\to }D_{n}^{r}$ . 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_{{n,0}}^{r}\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, ISBN 978-0-521-56759-6, MR 1793722, doi:10.2277/0521567599
J. P. May, A primer on spectral sequences

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