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:

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

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

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

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

This gives the first page of the spectral sequence: we take ${\displaystyle E_{s,t}^{1}=H_{s+t}(C\otimes \mathbb {Z} /p)}$ with the differential ${\displaystyle {}^{1}d=j\circ k}$. The derived couple of the above exact couple then gives the second page and so forth. Explicitly, we have ${\displaystyle D^{r}=p^{r-1}H_{*}(C)}$ that fits into the exact couple:

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

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

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

we get:

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

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

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

When ${\displaystyle r=1}$, this is the same thing as the universal coefficient theorem for homology.

Assume the abelian group ${\displaystyle H_{*}(C)}$ is finitely generated; in particular, only finitely many cyclic modules of the form ${\displaystyle \mathbb {Z} /p^{s}}$ can appear as a direct summand of ${\displaystyle H_{*}(C)}$. Letting ${\displaystyle r\to \infty }$ we thus see ${\displaystyle E^{\infty }}$ is isomorphic to ${\displaystyle ({\text{free part of }}H_{*}(C))\otimes \mathbb {Z} /p}$.

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