Bockstein spectral sequence

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

:$0\; \backslash longrightarrow\; C\; \backslash overset\{p\}\backslash longrightarrow\; C\; \backslash overset\{\backslash text\{mod\}\; p\}\; \backslash longrightarrow\; C\; \backslash otimes\; \backslash Z/p\; \backslash longrightarrow\; 0.$

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

:$H\_*(C)\; \backslash overset\{i\; =\; p\}\; \backslash longrightarrow\; H\_*(C)\; \backslash overset\{j\}\; \backslash longrightarrow\; H\_*(C\; \backslash otimes\; \backslash Z/p)\; \backslash overset\{k\}\; \backslash longrightarrow.$

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

This gives the first page of the spectral sequence: we take $E\_\{s,t\}^1\; =\; H\_\{s+t\}(C\; \backslash otimes\; \backslash Z/p)$ with the differential $\{\}^1\; d\; =\; j\; \backslash circ\; k$. 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:

:$D^r\; \backslash overset\{i=p\}\backslash longrightarrow\; D^r\; \backslash overset\{\{\}^r\; j\}\; \backslash longrightarrow\; E^r\; \backslash overset\{k\}\backslash longrightarrow$

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

:$0\; \backslash longrightarrow\; \backslash Z\; \backslash overset\{p\}\backslash longrightarrow\; \backslash Z\; \backslash longrightarrow\; \backslash Z/p\; \backslash longrightarrow\; 0,$

we get:

:$0\; \backslash longrightarrow\; \backslash operatorname\{Tor\}\_1^\{\backslash Z\}(D\_n^r,\; \backslash Z/p)\; \backslash longrightarrow\; D\_n^r\; \backslash overset\{p\}\backslash longrightarrow\; D\_n^r\; \backslash longrightarrow\; D\_n^r\; \backslash otimes\; \backslash Z/p\; \backslash longrightarrow\; 0$.

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

:$0\; \backslash longrightarrow\; (p^\{r-1\}\; H\_n(C))\; \backslash otimes\; \backslash Z/p\; \backslash longrightarrow\; E^r\_\{n,\; 0\}\; \backslash longrightarrow\; \backslash operatorname\{Tor\}(p^\{r-1\}\; H\_\{n-1\}(C),\; \backslash Z/p)\; \backslash longrightarrow\; 0$.

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 $\backslash Z/p^s$ can appear as a direct summand of $H\_*(C)$. Letting $r\; \backslash to\; \backslash infty$ we thus see $E^\backslash infty$ is isomorphic to $(\backslash text\{free\; part\; of\; \}\; H\_*(C))\; \backslash otimes\; \backslash Z/p$.

• {{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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