Bockstein spectral sequence

<h2 id="definition">Definition</h2>
Let C be a chain complex of <a href="/facts/Torsion-free_abelian_group/yLzQ3lu1">torsion-free abelian groups</a> and p a <a href="/facts/Prime_number/L20FLkgn">prime number</a>. 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 <a href="/facts/Exact_couple/cEqHl4A9">exact couple</a> 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 <a href="/facts/Derived_couple/9S1v2dRh">derived couple</a> 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 <a href="/facts/Universal_coefficient_theorem/N8ALJkMd">universal coefficient theorem</a> 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}
 
.

<ul><li>McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, vol. 58 (2nd ed.), <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-56759-6, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1793722">1793722</a></li>
<li>J. P. May, <a href="http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf">A primer on spectral sequences</a></li></ul>

Bockstein spectral sequence open-in-new

Bockstein spectral sequence