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A sequence of differential modules, each of which is the homology module of the preceding one. One usually studies spectral sequences of bigraded (less often graded or trigraded) modules, which are represented graphically in the form of tables in the plane superimposed on one another. More generally one can study spectral sequences of objects of an arbitrary [[Abelian category|Abelian category]] (e.g., bimodules, rings, algebras, co-algebras, Hopf algebras, etc.).
 
A sequence of differential modules, each of which is the homology module of the preceding one. One usually studies spectral sequences of bigraded (less often graded or trigraded) modules, which are represented graphically in the form of tables in the plane superimposed on one another. More generally one can study spectral sequences of objects of an arbitrary [[Abelian category|Abelian category]] (e.g., bimodules, rings, algebras, co-algebras, Hopf algebras, etc.).
  
All known spectral sequences can be obtained from exact couples. An exact couple $  ( D  ^ {1} , E  ^ {1} , i  ^ {1} , j  ^ {1} , k  ^ {1} ) $
+
All known spectral sequences can be obtained from exact couples. An exact couple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864901.png" /> is defined as an exact diagram of the form
is defined as an exact diagram of the form
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864902.png" /></td> </tr></table>
  
The homomorphism $  d  ^ {1} = j  ^ {1} k  ^ {1} $
+
The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864903.png" /> is a differential in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864904.png" />. From any exact couple one can construct the derived exact couple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864905.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864907.png" />. By iterating this construction one obtains the spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864908.png" />.
is a differential in $  E  ^ {1} $.  
 
From any exact couple one can construct the derived exact couple $  ( D  ^ {2} , E  ^ {2} , i  ^ {2} , j  ^ {2} , k  ^ {2} ) $,
 
for which $  D  ^ {2} = \mathop{\rm Im}  i  ^ {1} $
 
and $  E  ^ {2} = H( E  ^ {1} , d  ^ {1} ) $.  
 
By iterating this construction one obtains the spectral sequence $  E = \{ E  ^ {n} , d  ^ {n} \} $.
 
  
1) The Leray spectral sequence. A filtered chain complex of modules $  ( \{ K  ^ {p} \} , d) $
+
1) The Leray spectral sequence. A filtered chain complex of modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s0864909.png" /> determines an exact couple of bigraded modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649011.png" />. In the associated spectral sequence, the bidegree of the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649012.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649013.png" />, and
determines an exact couple of bigraded modules $  D _ {p,q}  ^ {1} = H _ {p+} q ( K  ^ {p} ) $,
 
$  E _ {p,q}  ^ {1} = H _ {p+} q ( K  ^ {p} / K  ^ {p-} 1 ) $.  
 
In the associated spectral sequence, the bidegree of the differential $  d  ^ {r} $
 
is equal to $  (- r, r- 1) $,  
 
and
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649014.png" /></td> </tr></table>
E _ {p,q}  ^ {r}  =
 
\frac{ \mathop{\rm Ker} ( d _ {p,q}  ^ {r-} 1 : E _ {p,q}  ^ {r-} 1 \rightarrow E _ {p-} r+ 1,q+ r- 2  ^ {r-} 1 ) }{
 
\mathop{\rm Im} ( d _ {p+} r- 1,q- r+ 2  ^ {r-} 1 : E _ {p+} r- 1,q- r+ 2  ^ {r-} 1 \rightarrow E _ {p,q}  ^ {r-} 1 ) }
 
  \simeq
 
$$
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649015.png" /></td> </tr></table>
\simeq 
 
\frac{ \mathop{\rm Im} ( H _ {p+} q ( K  ^ {p} / K  ^ {p-} r ) \rightarrow H _ {p+} q ( K  ^ {p} / K  ^ {p-} 1 )) }{ \mathop{\rm Im} ( \partial
 
:  H _ {p+} q+ 1 ( K  ^ {p+} r- 1 / K  ^ {p} ) \rightarrow H _ {p+} q ( K  ^ {p} / K  ^ {p-} 1 )) }
 
.
 
$$
 
  
The modules $  F _ {p,q} = \mathop{\rm Im} ( H _ {p+} q ( K  ^ {p} ) \rightarrow H _ {p+} q ( K)) $
+
The modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649016.png" /> form a filtration of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649017.png" />. The bigraded module
form a filtration of $  H _ {*} ( K) $.  
 
The bigraded module
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649018.png" /></td> </tr></table>
E _ {p,q}  ^  \infty  = F _ {p,q} / F _ {p-} 1,q+ 1  \simeq
 
$$
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649019.png" /></td> </tr></table>
\simeq 
 
\frac{ \mathop{\rm Im} ( H _ {p+} q ( K  ^ {p} ) \rightarrow H _ {p+} q ( K  ^ {p} / K  ^ {p-} 1 )) }{ \mathop{\rm Im} (
 
\partial  : H _ {p+} q+ 1 ( K / K  ^ {p} ) \rightarrow H _ {p+} q ( K  ^ {p} / K  ^ {p-} 1 )) }
 
  
$$
+
is called the associated graded module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649020.png" />. The filtration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649021.png" /> is called regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649022.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649024.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649026.png" />. For a regular filtration, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649027.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649028.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649029.png" />; such a spectral sequence is called a first-quadrant spectral sequence. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649030.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649031.png" />. In this case one says that the spectral sequence converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649032.png" />, and writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649033.png" />.
  
is called the associated graded module of  $  H _ {*} ( K) $.  
+
2) The Leray–Serre spectral sequence is a special case of the Leray spectral sequence above arising from a chain (or cochain) complex of a filtered topological space. E.g., the filtration of a [[CW-complex|CW-complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649034.png" /> by its skeletons gives the collapsing spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649035.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649036.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649038.png" />. The Leray–Serre spectral sequence is obtained from the filtration of the total space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649039.png" /> of the Serre fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649040.png" /> by the pre-images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649041.png" /> of the skeletons <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649042.png" /> of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649043.png" />. If the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649044.png" /> and base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649045.png" /> are path-connected, then for every coefficient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649046.png" /> this gives the spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649047.png" /> with differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649048.png" /> of bidegree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649049.png" /> for which
The filtration $  \{ K  ^ {p} \} $
 
is called regular if  $  K  ^ {p} = 0 $
 
when  $  p< 0 $,  
 
$  E _ {p,q}  ^ {1} = 0 $
 
when $  q< 0 $
 
and $  K = \cup K  ^ {p} $.  
 
For a regular filtration,  $  E _ {p,q}  ^ {r} = 0 $
 
when  $  p< 0 $
 
or  $  q< 0 $;
 
such a spectral sequence is called a first-quadrant spectral sequence. Moreover,  $  E _ {p,q}  ^ {r} \simeq E _ {p,q}  ^ {r+} 1 \simeq E _ {p,q}  ^  \infty  $
 
when  $  r > \max ( p, q+ 1) $.  
 
In this case one says that the spectral sequence converges to  $  H _ {*} ( K) $,
 
and writes  $  E _ {p,q}  ^ {r} \Rightarrow H _ {p+} q ( K) $.
 
  
2) The Leray–Serre spectral sequence is a special case of the Leray spectral sequence above arising from a chain (or cochain) complex of a filtered topological space. E.g., the filtration of a [[CW-complex|CW-complex]]  $  X $
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649050.png" /></td> </tr></table>
by its skeletons gives the collapsing spectral sequence  $  E _ {p,q}  ^ {r} \Rightarrow H _ {p+} q ( X) $,
 
for which  $  E _ {p,q}  ^ {r} = E _ {p,q}  ^  \infty  = 0 $
 
when  $  q \neq 0 $
 
and  $  E _ {n,0}  ^ {2} = E _ {n,0}  ^  \infty  = H _ {n} ( X) $.  
 
The Leray–Serre spectral sequence is obtained from the filtration of the total space  $  E $
 
of the Serre fibration  $  F \rightarrow  ^ {i} E \rightarrow  ^ {p} B $
 
by the pre-images  $  p  ^ {-} 1 ( B  ^ {n} ) $
 
of the skeletons  $  B  ^ {n} $
 
of the base  $  B $.  
 
If the fibre  $  F $
 
and base  $  B $
 
are path-connected, then for every coefficient group  $  G $
 
this gives the spectral sequence  $  E _ {p,q}  ^ {r} \Rightarrow H _ {p+} q ( E, G) $
 
with differentials  $  d  ^ {r} $
 
of bidegree  $  ( - r, r- 1) $
 
for which
 
  
$$
+
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649051.png" /> is a system of local coefficients over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649052.png" /> consisting of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649053.png" />. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649054.png" /> coincides with the composite
E _ {p,q}  ^ {1}  \simeq  C _ {p} ( B) \otimes H _ {q} ( F;  G)
 
\  \textrm{ and } \  E _ {p,q}  ^ {2}  \simeq  H _ {p} ( B;  {\mathcal H} _ {q} ( F;  G)),
 
$$
 
  
where  $  {\mathcal H} _ {q} ( F; G) $
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649055.png" /></td> </tr></table>
is a system of local coefficients over  $  B $
 
consisting of the groups  $  H _ {q} ( F; G) $.
 
The homomorphism  $  i _ {*} : H _ {n} ( F; G) \rightarrow H _ {n} ( E;  G) $
 
coincides with the composite
 
  
$$
+
and the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649056.png" /> coincides with the composite
H _ {n} ( F;  G)  = E _ {0,n}  ^ {2}  \rightarrow  E _ {0,n}  ^ {r}
 
= E _ {0,n}  ^  \infty  = F _ {0,n}  \subset  H _ {n} ( F;  G),
 
$$
 
  
and the homomorphism  $  p _ {*} : H _ {n} ( E; G) \rightarrow H _ {n} ( B; G) $
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649057.png" /></td> </tr></table>
coincides with the composite
 
  
$$
+
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649058.png" /> is sufficiently large. The differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649059.png" /> of the spectral sequence coincides with the [[Transgression|transgression]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649060.png" />.
H _ {n} ( E;  G)  = F _ {n,0}  \rightarrow  E _ {n,0}  ^  \infty  = \
 
E _ {n,0}  ^ {r}  \subset  \
 
E _ {n,0}  ^ {2}  = H _ {n} ( B;  G),
 
$$
 
  
where  $  r $
+
This homology Leray–Serre spectral sequence is dual to the cohomology Leray–Serre spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649061.png" />, with differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649062.png" /> of bidegree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649063.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649064.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649065.png" /> is a ring, then every term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649066.png" /> is a bigraded ring, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649067.png" /> is differentiation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649068.png" />, and the multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649069.png" /> is induced by that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649070.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649071.png" /> is a field and the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649072.png" /> is simply connected, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649073.png" />.
is sufficiently large. The differential  $  d _ {n,0}  ^ {n} $
 
of the spectral sequence coincides with the [[Transgression|transgression]]  $  \tau : H _ {n} ( B;  G) \rightarrow H _ {n-} 1 ( F;  G) $.
 
  
This homology Leray–Serre spectral sequence is dual to the cohomology Leray–Serre spectral sequence  $  E _ {r}  ^ {p,q} \Rightarrow H  ^ {p+} q ( E;  G) $,
+
3) The Atiyah–Hirzebruch (–Whitehead) spectral sequence is obtained by applying the generalized (co)homology functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649074.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649075.png" />) to the same filtration of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649076.png" />. In its cohomological version, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649078.png" />. In contrast to the Leray–Serre spectral sequence, the Atiyah–Hirzebruch spectral sequence for the trivial fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649079.png" /> is in general non-collapsing.
with differentials  $  d _ {r} $
 
of bidegree  $  ( r, - r+ 1) $,
 
for which  $  E _ {2}  ^ {p,q} \simeq H  ^ {p} ( B;  {\mathcal H} _ {q} ( F;  G)) $.
 
If  $  G $
 
is a ring, then every term  $  E _ {r} $
 
is a bigraded ring,  $  d _ {r} $
 
is differentiation in  $  E _ {r} $,
 
and the multiplication in  $  E _ {r+} 1 $
 
is induced by that in  $  E _ {r} $.
 
If  $  G $
 
is a field and the base  $  B $
 
is simply connected, then  $  E _ {2}  ^ {**} \simeq H  ^ {*} ( B;  G) \otimes H  ^ {*} ( F;  G) $.
 
 
 
3) The Atiyah–Hirzebruch (–Whitehead) spectral sequence is obtained by applying the generalized (co)homology functor $  h _ {*} $(
 
$  h  ^ {*} $)  
 
to the same filtration of the space $  E $.  
 
In its cohomological version, $  E _ {r}  ^ {p,q} \Rightarrow h  ^ {p+} q ( E) $,
 
$  E _ {2}  ^ {p,q} = H  ^ {p} ( B;  h  ^ {q} ( F  )) $.  
 
In contrast to the Leray–Serre spectral sequence, the Atiyah–Hirzebruch spectral sequence for the trivial fibration $  \mathop{\rm id} : X \rightarrow X $
 
is in general non-collapsing.
 
  
 
4) An Eilenberg–Moore spectral sequence is associated with any square of fibrations
 
4) An Eilenberg–Moore spectral sequence is associated with any square of fibrations
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649080.png" /></td> </tr></table>
  
 
In its cohomological version,
 
In its cohomological version,
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649081.png" /></td> </tr></table>
E _ {r}  \Rightarrow  H  ^ {*} ( E; R),
 
$$
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649082.png" /></td> </tr></table>
E _ {2}  ^ {p,q}  \simeq  \mathop{\rm Tor} _ {H  ^ {*}
 
( B;R) }  ^ {p,q} ( H  ^ {*} ( X; R); H  ^ {*} ( Y;  R)).
 
$$
 
  
If $  R $
+
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649083.png" /> is a field and the square consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649084.png" />-spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649085.png" />-mappings, then this is a spectral sequence in the category of bigraded Hopf algebras.
is a field and the square consists of $  H $-
 
spaces and $  H $-
 
mappings, then this is a spectral sequence in the category of bigraded Hopf algebras.
 
  
5) The Adams spectral sequence $  E _ {r}  ^ {s,t} $
+
5) The Adams spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649086.png" /> is defined for every prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649087.png" /> and all spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649089.png" /> (satisfying certain finiteness conditions). One has
is defined for every prime $  p\geq  2 $
 
and all spaces $  X $
 
and $  Y $(
 
satisfying certain finiteness conditions). One has
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649090.png" /></td> </tr></table>
E _ {2}  ^ {s,t}  \simeq  \mathop{\rm Ext} _ {A _ {p}  }  ^ {s,t} ( H
 
^ {*} ( X; \mathbf Z _ {p} ); H  ^ {*} ( Y ; \mathbf Z _ {p} )),
 
$$
 
  
where $  A _ {p} $
+
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649091.png" /> is the [[Steenrod algebra|Steenrod algebra]] modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649092.png" />. The bidegree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649093.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649094.png" />. This spectral sequence converges in the sense that, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649095.png" />, there is a monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649096.png" />, and so the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649097.png" /> is defined. There is a decreasing filtration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649098.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s08649099.png" /> of stable homotopy classes of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490100.png" /> such that
is the [[Steenrod algebra|Steenrod algebra]] modulo $  p $.  
 
The bidegree of $  d _ {r} $
 
is equal to $  ( r, r- 1) $.  
 
This spectral sequence converges in the sense that, when $  r> s $,  
 
there is a monomorphism $  E _ {r+} 1  ^ {s,t} \rightarrow E _ {r}  ^ {s,t} $,
 
and so the group $  E _  \infty  ^ {s,t} = \cap _ {r>} s E _ {r}  ^ {s,t} $
 
is defined. There is a decreasing filtration $  \{ F ^ { s } \} $
 
of the group $  \{ Y, X \} $
 
of stable homotopy classes of mappings $  Y \rightarrow X $
 
such that
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490101.png" /></td> </tr></table>
F ^ { s } \{ S  ^ {t-} s YX \} / F ^ { s+ 1 } \{ S  ^ {t-} s Y, X \}  \simeq  E _  \infty  ^ {s,t} ,
 
$$
 
  
and $  F ^ { \infty } = \cap _ {s\geq } 0 F ^ { s } $
+
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490102.png" /> consists of all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490103.png" /> of finite order prime with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490104.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490105.png" />, this spectral sequence enables one  "in principle"  to calculate the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490106.png" />-components of the stable homotopy groups of spheres. The Adams spectral sequence has been generalized by A.S. Mishchenko and S.P. Novikov to arbitrary [[Generalized cohomology theories|generalized cohomology theories]]. There are also extensions of the Adams spectral sequence that converge to non-stable homotopy groups.
consists of all elements of $  \{ Y, X \} $
 
of finite order prime with $  p $.  
 
When $  X= Y= S  ^ {0} $,  
 
this spectral sequence enables one  "in principle"  to calculate the $  p $-
 
components of the stable homotopy groups of spheres. The Adams spectral sequence has been generalized by A.S. Mishchenko and S.P. Novikov to arbitrary [[Generalized cohomology theories|generalized cohomology theories]]. There are also extensions of the Adams spectral sequence that converge to non-stable homotopy groups.
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.E. Mosher,  M.C. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.B. Fuks,  A.T. Fomenko,  V.L. Gutenmakher,  "Homotopic topology" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,  "Homologie singulière des espaces fibrés. Applications"  ''Ann. of Math.'' , '''54'''  (1951)  pp. 425–505</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.-T. Hu,  "Homotopy theory" , Acad. Press  (1959)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R. Godement,  "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  S.P. Novikov,  "The methods of algebraic topology from the viewpoint of cobordism theory"  ''Math. USSR Izv.'' , '''31'''  (1967)  pp. 827–913  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''31'''  (1967)  pp. 855–951</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J.F. Adams,  "Stable homotopy and generalised homology" , Univ. Chicago Press  (1974)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  L. Smith,  "Lectures on the Eilenberg–Moore spectral sequence" , ''Lect. notes in math.'' , '''134''' , Springer  (1970)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  D.C. Ravenel,  "A novices guide to the Adams–Novikov spectral sequence" , ''Geometric Applications of Homotopy Theory'' , '''2''' , Springer  (1978)  pp. 404–475</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.E. Mosher,  M.C. Tangora,  "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.B. Fuks,  A.T. Fomenko,  V.L. Gutenmakher,  "Homotopic topology" , Moscow  (1969)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,  "Homologie singulière des espaces fibrés. Applications"  ''Ann. of Math.'' , '''54'''  (1951)  pp. 425–505</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.-T. Hu,  "Homotopy theory" , Acad. Press  (1959)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R. Godement,  "Topologie algébrique et théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  S.P. Novikov,  "The methods of algebraic topology from the viewpoint of cobordism theory"  ''Math. USSR Izv.'' , '''31'''  (1967)  pp. 827–913  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''31'''  (1967)  pp. 855–951</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J.F. Adams,  "Stable homotopy and generalised homology" , Univ. Chicago Press  (1974)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  R.M. Switzer,  "Algebraic topology - homotopy and homology" , Springer  (1975)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  L. Smith,  "Lectures on the Eilenberg–Moore spectral sequence" , ''Lect. notes in math.'' , '''134''' , Springer  (1970)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  D.C. Ravenel,  "A novices guide to the Adams–Novikov spectral sequence" , ''Geometric Applications of Homotopy Theory'' , '''2''' , Springer  (1978)  pp. 404–475</TD></TR></table>
 +
 +
  
 
====Comments====
 
====Comments====
Let $  ( E  ^ {n} , d  ^ {n} ) $,
+
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490108.png" /> be a spectral sequence, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490109.png" /> is the homology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490110.png" />. A spectral sequence defines a series of modules of the initial term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490111.png" />, defined inductively as follows:
$  n = 2, 3 \dots $
 
be a spectral sequence, so that $  E  ^ {n+} 1 $
 
is the homology of $  ( E  ^ {n} , d  ^ {n} ) $.  
 
A spectral sequence defines a series of modules of the initial term $  E  ^ {2} $,  
 
defined inductively as follows:
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490112.png" /></td> </tr></table>
0 = B  ^ {1}  \subset  B  ^ {2}  \subset  B  ^ {3}  \subset  \dots
 
$$
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490113.png" /></td> </tr></table>
\dots \subset  C  ^ {3}  \subset  C  ^ {2}  \subset  C  ^ {1}  = E  ^ {2} ,
 
$$
 
  
with $  E  ^ {r+} 1 = C  ^ {r} / B  ^ {r} $,  
+
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490114.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490115.png" /> is the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490116.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490117.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490118.png" />. One now defines the infinity terms:
and $  C  ^ {r+} 1 /B  ^ {r} $
 
is the kernel of $  d  ^ {r} : E  ^ {r} \rightarrow E  ^ {r} $,  
 
while $  B  ^ {r+} 1 /B  ^ {r} $
 
is the image of $  d  ^ {r} $.  
 
One now defines the infinity terms:
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490119.png" /></td> </tr></table>
C  ^  \infty  = \cap _ { n } C  ^ {n} ,\  B  ^  \infty  = \
 
\cup _ { n } B  ^ {n} ,\  E  ^  \infty  = C  ^  \infty  / B  ^  \infty  .
 
$$
 
  
The terms $  E  ^ {r} $
+
The terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490120.png" /> are thought of as successive approximations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490121.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490122.png" /> is a spectral sequence of bigraded modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490124.png" />, all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490127.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490128.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490129.png" /> also carry corresponding natural bigraded structures.
are thought of as successive approximations of $  E  ^  \infty  $.  
 
If $  ( E  ^ {n} , d  ^ {n} ) $
 
is a spectral sequence of bigraded modules $  E  ^ {n} = \oplus E _ {p,q}  ^ {n} $,
 
$  d  ^ {r} : E _ {p,q}  ^ {r} \rightarrow E _ {p-} r,q+ r- 1  ^ {r} $,  
 
all the $  B  ^ {i} $,  
 
$  C  ^ {i} $,  
 
$  B  ^  \infty  $,  
 
$  C  ^  \infty  $,  
 
$  E  ^  \infty  $
 
also carry corresponding natural bigraded structures.
 
  
Sometimes there is an initial term $  E  ^ {1} $,  
+
Sometimes there is an initial term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490130.png" />, and then the same construction is carried out with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490131.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490132.png" />.
and then the same construction is carried out with $  E  ^ {1} $
 
instead of $  E  ^ {2} $.
 
  
For a first-quadrant spectral sequence, i.e. $  E _ {p,q}  ^ {2} = 0 $
+
For a first-quadrant spectral sequence, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490133.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490134.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490135.png" />, for given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490136.png" /> and large enough <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490137.png" /> one has that in
for $  p< 0 $
 
or $  q< 0 $,  
 
for given $  p, q $
 
and large enough $  r $
 
one has that in
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490138.png" /></td> </tr></table>
E _ {p+} r,q- r+ 1  ^ {r}  \rightarrow ^ { {d  ^ {r}} }  E _ {p,q}  ^ {r}
 
\rightarrow ^ { {d  ^ {r}} }  E _ {p-} r,q+ r- 1  ^ {r}
 
$$
 
  
both the outside modules are zero, so that $  E _ {p,q}  ^ {r} = E _ {p,q}  ^ {r+} 1 = E _ {p,q}  ^  \infty  $
+
both the outside modules are zero, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490139.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490140.png" /> large enough.
for $  r $
 
large enough.
 
  
For a first-quadrant spectral sequence one also always has that $  E _ {p,0}  ^ {r+} 1 $
+
For a first-quadrant spectral sequence one also always has that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490141.png" /> is a submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490142.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490143.png" /> is a quotient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490144.png" />, giving rise to sequences of monomorphisms and epimorphisms:
is a submodule of $  E _ {p,0}  ^ {r} $,  
 
and $  E _ {0,q}  ^ {r+} 1 $
 
is a quotient of $  E _ {0,q}  ^ {r} $,
 
giving rise to sequences of monomorphisms and epimorphisms:
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490145.png" /></td> </tr></table>
E _ {p,0}  ^  \infty  = E _ {p,0}  ^ {p+} 1  \rightarrow \dots \rightarrow  E _ {p,0}  ^ {3}  \rightarrow  E _ {p,0}  ^ {2} ,
 
$$
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490146.png" /></td> </tr></table>
E _ {0,q}  ^ {2}  \rightarrow  E _ {0,q}  ^ {3}  \rightarrow
 
\dots \rightarrow  E _ {0,q}  ^ {q+} 2  = E _ {0,q}  ^  \infty  ,
 
$$
 
  
 
which are known as the edge homomorphisms.
 
which are known as the edge homomorphisms.
  
Let $  ( A _ {p} ) $
+
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490147.png" /> be a filtration of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490148.png" /> by submodules
be a filtration of a module $  A $
 
by submodules
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490149.png" /></td> </tr></table>
\dots \subset  A _ {p-} 1  \subset  A _ {p}  \subset  A _ {p+} 1  \subset  \dots
 
$$
 
  
with associated graded module $  \mathop{\rm Gr} ( A) $:
+
with associated graded module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490150.png" />:
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490151.png" /></td> </tr></table>
\mathop{\rm Gr} ( A)  = \oplus _ { p } A _ {p} / A _ {p-} 1 .
 
$$
 
  
A spectral sequence $  ( E _ {p}  ^ {r} , d  ^ {r} ) $
+
A spectral sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490152.png" /> is said to converge to a graded module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490153.png" />, in symbols
is said to converge to a graded module $  H $,  
 
in symbols
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490154.png" /></td> </tr></table>
E _ {p}  ^ {r}  \Rightarrow  H ,
 
$$
 
  
if there is a filtration $  F _ {p} H $
+
if there is a filtration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490155.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490156.png" /> such that
of $  H $
 
such that
 
  
$$ \tag{* }
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490157.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
E _ {p}  ^  \infty  \simeq  F _ {p} H /F _ {p+} 1 H .
 
$$
 
  
In the usual cases the $  E _ {p}  ^ {r} $
+
In the usual cases the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490158.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086490/s086490159.png" /> are graded, and then both the filtration and the isomorphism (*) are to be compatible with the grading.
and $  H $
 
are graded, and then both the filtration and the isomorphism (*) are to be compatible with the grading.
 

Revision as of 14:53, 7 June 2020

A sequence of differential modules, each of which is the homology module of the preceding one. One usually studies spectral sequences of bigraded (less often graded or trigraded) modules, which are represented graphically in the form of tables in the plane superimposed on one another. More generally one can study spectral sequences of objects of an arbitrary Abelian category (e.g., bimodules, rings, algebras, co-algebras, Hopf algebras, etc.).

All known spectral sequences can be obtained from exact couples. An exact couple is defined as an exact diagram of the form

The homomorphism is a differential in . From any exact couple one can construct the derived exact couple , for which and . By iterating this construction one obtains the spectral sequence .

1) The Leray spectral sequence. A filtered chain complex of modules determines an exact couple of bigraded modules , . In the associated spectral sequence, the bidegree of the differential is equal to , and

The modules form a filtration of . The bigraded module

is called the associated graded module of . The filtration is called regular if when , when and . For a regular filtration, when or ; such a spectral sequence is called a first-quadrant spectral sequence. Moreover, when . In this case one says that the spectral sequence converges to , and writes .

2) The Leray–Serre spectral sequence is a special case of the Leray spectral sequence above arising from a chain (or cochain) complex of a filtered topological space. E.g., the filtration of a CW-complex by its skeletons gives the collapsing spectral sequence , for which when and . The Leray–Serre spectral sequence is obtained from the filtration of the total space of the Serre fibration by the pre-images of the skeletons of the base . If the fibre and base are path-connected, then for every coefficient group this gives the spectral sequence with differentials of bidegree for which

where is a system of local coefficients over consisting of the groups . The homomorphism coincides with the composite

and the homomorphism coincides with the composite

where is sufficiently large. The differential of the spectral sequence coincides with the transgression .

This homology Leray–Serre spectral sequence is dual to the cohomology Leray–Serre spectral sequence , with differentials of bidegree , for which . If is a ring, then every term is a bigraded ring, is differentiation in , and the multiplication in is induced by that in . If is a field and the base is simply connected, then .

3) The Atiyah–Hirzebruch (–Whitehead) spectral sequence is obtained by applying the generalized (co)homology functor () to the same filtration of the space . In its cohomological version, , . In contrast to the Leray–Serre spectral sequence, the Atiyah–Hirzebruch spectral sequence for the trivial fibration is in general non-collapsing.

4) An Eilenberg–Moore spectral sequence is associated with any square of fibrations

In its cohomological version,

If is a field and the square consists of -spaces and -mappings, then this is a spectral sequence in the category of bigraded Hopf algebras.

5) The Adams spectral sequence is defined for every prime and all spaces and (satisfying certain finiteness conditions). One has

where is the Steenrod algebra modulo . The bidegree of is equal to . This spectral sequence converges in the sense that, when , there is a monomorphism , and so the group is defined. There is a decreasing filtration of the group of stable homotopy classes of mappings such that

and consists of all elements of of finite order prime with . When , this spectral sequence enables one "in principle" to calculate the -components of the stable homotopy groups of spheres. The Adams spectral sequence has been generalized by A.S. Mishchenko and S.P. Novikov to arbitrary generalized cohomology theories. There are also extensions of the Adams spectral sequence that converge to non-stable homotopy groups.

References

[1] R.E. Mosher, M.C. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968)
[2] D.B. Fuks, A.T. Fomenko, V.L. Gutenmakher, "Homotopic topology" , Moscow (1969) (In Russian)
[3] J.-P. Serre, "Homologie singulière des espaces fibrés. Applications" Ann. of Math. , 54 (1951) pp. 425–505
[4] S. MacLane, "Homology" , Springer (1963)
[5] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[6] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[7] S.-T. Hu, "Homotopy theory" , Acad. Press (1959)
[8] R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958)
[9] S.P. Novikov, "The methods of algebraic topology from the viewpoint of cobordism theory" Math. USSR Izv. , 31 (1967) pp. 827–913 Izv. Akad. Nauk. SSSR Ser. Mat. , 31 (1967) pp. 855–951
[10] J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974)
[11] R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975)
[12] L. Smith, "Lectures on the Eilenberg–Moore spectral sequence" , Lect. notes in math. , 134 , Springer (1970)
[13] D.C. Ravenel, "A novices guide to the Adams–Novikov spectral sequence" , Geometric Applications of Homotopy Theory , 2 , Springer (1978) pp. 404–475


Comments

Let , be a spectral sequence, so that is the homology of . A spectral sequence defines a series of modules of the initial term , defined inductively as follows:

with , and is the kernel of , while is the image of . One now defines the infinity terms:

The terms are thought of as successive approximations of . If is a spectral sequence of bigraded modules , , all the , , , , also carry corresponding natural bigraded structures.

Sometimes there is an initial term , and then the same construction is carried out with instead of .

For a first-quadrant spectral sequence, i.e. for or , for given and large enough one has that in

both the outside modules are zero, so that for large enough.

For a first-quadrant spectral sequence one also always has that is a submodule of , and is a quotient of , giving rise to sequences of monomorphisms and epimorphisms:

which are known as the edge homomorphisms.

Let be a filtration of a module by submodules

with associated graded module :

A spectral sequence is said to converge to a graded module , in symbols

if there is a filtration of such that

(*)

In the usual cases the and are graded, and then both the filtration and the isomorphism (*) are to be compatible with the grading.

How to Cite This Entry:
Spectral sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_sequence&oldid=49434
This article was adapted from an original article by S.N. Malygin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article