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

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Let C be a chain complex of torsion-free abelian groups and p a prime number. Then we have the exact sequence:

0 ⟶ C ⟶ p C ⟶ mod p C ⊗ Z / p ⟶ 0. {\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:

H ∗ ( C ) ⟶ i = p H ∗ ( C ) ⟶ j H ∗ ( C ⊗ Z / p ) ⟶ k . {\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:  H ∗ ( C ) s , t = H s + t ( C ) {\displaystyle H_{*}(C)_{s,t}=H_{s+t}(C)} and the same for  H ∗ ( C ⊗ Z / p ) , deg ⁡ i = ( 1 , − 1 ) , deg ⁡ j = ( 0 , 0 ) , deg ⁡ k = ( − 1 , 0 ) . {\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  E s , t 1 = H s + t ( C ⊗ Z / p ) {\displaystyle E_{s,t}^{1}=H_{s+t}(C\otimes \mathbb {Z} /p)} with the differential  1 d = j ∘ k {\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  D r = p r − 1 H ∗ ( C ) {\displaystyle D^{r}=p^{r-1}H_{*}(C)} that fits into the exact couple:

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

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

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

we get:

0 ⟶ Tor 1 Z ⁡ ( D n r , Z / p ) ⟶ D n r ⟶ p D n r ⟶ D n r ⊗ Z / p ⟶ 0 {\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  D n r ⟶ p D n r {\displaystyle D_{n}^{r}{\overset {p}{\longrightarrow }}D_{n}^{r}} . Expanding the exact couple into a long exact sequence, we get: for any r,

0 ⟶ ( p r − 1 H n ( C ) ) ⊗ Z / p ⟶ E n , 0 r ⟶ Tor ⁡ ( p r − 1 H n − 1 ( C ) , Z / p ) ⟶ 0 {\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  r = 1 {\displaystyle r=1} , this is the same thing as the universal coefficient theorem for homology.

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

## References

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* 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

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