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 \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 \Z/p) \overset{k} \longrightarrow.

where the grading goes: H_*(C)_{s,t} = H_{s+t}(C) and the same for H_*(C \otimes \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 \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 \Z \overset{p}\longrightarrow \Z \longrightarrow \Z/p \longrightarrow 0,

we get:

: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.

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

: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.

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 \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 \Z/p.

References

  • {{Citation | last1=McCleary | first1=John | title=A User's Guide to Spectral Sequences | publisher=Cambridge University Press | edition=2nd | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-56759-6 | mr=1793722 | year=2001 | volume=58}}
  • J. P. May, [http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf A primer on spectral sequences]

Category:Spectral sequences

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