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:

[math]\displaystyle{ 0 \longrightarrow C \overset{p}\longrightarrow C \overset{\text{mod} p} \longrightarrow C \otimes \Z/p \longrightarrow 0. }[/math]

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

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

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

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

[math]\displaystyle{ D^r \overset{i=p}\longrightarrow D^r \overset{{}^r j} \longrightarrow E^r \overset{k}\longrightarrow }[/math]

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

[math]\displaystyle{ 0 \longrightarrow \Z \overset{p}\longrightarrow \Z \longrightarrow \Z/p \longrightarrow 0, }[/math]

we get:

[math]\displaystyle{ 0 \longrightarrow \operatorname{Tor}_1^{\Z}(D_n^r, \Z/p) \longrightarrow D_n^r \overset{p}\longrightarrow D_n^r \longrightarrow D_n^r \otimes \Z/p \longrightarrow 0 }[/math].

This tells the kernel and cokernel of [math]\displaystyle{ D^r_n \overset{p}\longrightarrow D^r_n }[/math]. Expanding the exact couple into a long exact sequence, we get: for any r,

[math]\displaystyle{ 0 \longrightarrow (p^{r-1} H_n(C)) \otimes \Z/p \longrightarrow E^r_{n, 0} \longrightarrow \operatorname{Tor}(p^{r-1} H_{n-1}(C), \Z/p) \longrightarrow 0 }[/math].

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

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

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 
• J. P. May, A primer on spectral sequences

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