View source for Bockstein spectral sequence

In mathematics, the '''Bockstein spectral sequence''' is a [[spectral sequence]] relating the homology with mod&nbsp;''p'' coefficients and the homology reduced mod&nbsp;''p''.  It is named after [[Meyer Bockstein]].

== Definition ==
Let ''C'' be a chain complex of [[torsion-free abelian group]]s and ''p'' a [[prime number]]. Then we have the exact sequence:

:<math>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>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>H_*(C)_{s,t} = H_{s+t}(C)</math> and the same for <math>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>E_{s,t}^1 = H_{s+t}(C \otimes \Z/p)</math> with the differential <math>{}^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>D^r = p^{r-1}  H_*(C)</math> that fits into the exact couple:

:<math>D^r \overset{i=p}\longrightarrow D^r \overset{{}^r j} \longrightarrow E^r \overset{k}\longrightarrow </math>

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

:<math>0 \longrightarrow \Z \overset{p}\longrightarrow \Z \longrightarrow \Z/p \longrightarrow 0,</math> 

we get:

:<math>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>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>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>r = 1</math>, this is the same thing as the [[universal coefficient theorem]] for homology.

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

==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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