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:

$0⟶C\stackrel{p}{⟶}C\stackrel{\text{mod}p}{⟶}C\otimes \mathbb{Z}/p⟶0.$

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

$H_{\ast }(C)\stackrel{i=p}{⟶}{H}_{\ast }(C)\stackrel{j}{⟶}{H}_{\ast }(C\otimes \mathbb{Z}/p)\stackrel{k}{⟶}.$

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

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 $D^r=p^{r-1}{H}_{\ast }(C)$ that fits into the exact couple:

$D^r\stackrel{i=p}{⟶}{D}^r\stackrel{{}^{r}j}{⟶}{E}^r\stackrel{k}{⟶}$

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

$0⟶\mathbb{Z}\stackrel{p}{⟶}\mathbb{Z}⟶\mathbb{Z}/p⟶0,$

we get:

$0⟶{\mathrm{Tor}}_1^{\mathbb{Z}}(D_n^r,\mathbb{Z}/p)⟶D_n^r\stackrel{p}{⟶}{D}_n^r⟶D_n^r\otimes \mathbb{Z}/p⟶0$.

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

$0⟶(p^{r-1}{H}_n(C))\otimes \mathbb{Z}/p⟶E_{n,0}^r⟶\mathrm{Tor}(p^{r-1}{H}_{n-1}(C),\mathbb{Z}/p)⟶0$.

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

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

References

• Template:Citation
• J. P. May, A primer on spectral sequences

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