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A spectral sequence is a collection of  -modules (or more generally, objects of an abelian category)
 for all
 ,  ,
 , equipped with maps
 such that
is a chain complex, and the  's are its homology, that is,
(Note: what I have defined above is a homology spectral sequence. Cohomology spectral sequences are identical, except that all the arrows go in the other direction.)
Most interesting spectral sequences are upper right quadrant, meaning that
 if  or  . If this is the case then for any  , both  and
 are 0 for sufficiently large  since the target or source is out of the upper right quadrant, so that for all 
 . This group is called
 .
A upper right quadrant spectral sequence
 is said to converge to a sequence  of  -modules if there is an exhaustive filtration
 of each  such that
This is typically written
 .
Typically spectral sequences are used in the following manner: we find an interpretation of  for a small value of  , typically 1, and of  , and then in cases where enough groups and differentials are 0, we can obtain information about one from the other.
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![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ \cdots & E^r_{p-r,q+r-1}\ar[l] &... ...& & E^r_{p+r,q-r+1}\ar[ll]_(0.575){d^r_{p+r,q-r+1}} & \ar[l]\cdots } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/4637/l2h/img7.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ \cdots & E^r_{p-r,q+r-1}\ar[l] &... ...& & E^r_{p+r,q-r+1}\ar[ll]_(0.575){d^r_{p+r,q-r+1}} & \ar[l]\cdots } } \end{xy}$")
