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:

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

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

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

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

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

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

where ${}^{r}j=({\text{mod }}p)\circ p^{-{r+1}}$ and $\deg({}^{r}j)=(-(r-1),r-1)$ (the degrees of i, k are the same as before). Now, taking $D_{n}^{r}\otimes -$ of

$0\longrightarrow \mathbb {Z} {\overset {p}{\longrightarrow }}\mathbb {Z} \longrightarrow \mathbb {Z} /p\longrightarrow 0,$

we get:

$0\longrightarrow \operatorname {Tor} _{1}^{\mathbb {Z} }(D_{n}^{r},\mathbb {Z} /p)\longrightarrow D_{n}^{r}{\overset {p}{\longrightarrow }}D_{n}^{r}\longrightarrow D_{n}^{r}\otimes \mathbb {Z} /p\longrightarrow 0$.

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

$0\longrightarrow (p^{r-1}H_{n}(C))\otimes \mathbb {Z} /p\longrightarrow E_{n,0}^{r}\longrightarrow \operatorname {Tor} (p^{r-1}H_{n-1}(C),\mathbb {Z} /p)\longrightarrow 0$.

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 $\mathbb {Z} /p^{s}$ 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, vol. 58 (2nd ed.), Cambridge University Press, ISBN 978-0-521-56759-6, MR 1793722
• J. P. May, A primer on spectral sequences