Editing Spectral sequence (section)

== Properties ==
=== Categorical properties ===
The set of cohomological spectral sequences form a category: a morphism of spectral sequences <math> f : E \to E' </math> is by definition a collection of maps <math> f_r : E_r \to E'_r </math> which are compatible with the differentials, i.e. <math> f_r \circ d_r = d'_r \circ f_r </math>, and with the given isomorphisms between the cohomology of the ''r''{{sup|th}} step and the {{tmath|(r+1)}}{{sup|th}} sheets of ''E'' and {{prime|''E''}}, respectively: <math> f_{r+1}(E_{r+1}) \,=\, f_{r+1}(H(E_r)) \,=\, H(f_r(E_r))  </math>. In the bigraded case, they should also respect the graduation: <math> f_r(E_r^{p,q}) \subset {E'_r}^{p,q}. </math>

=== Multiplicative structure ===
A [[cup product]] gives a [[ring (mathematics)|ring structure]] to a cohomology group, turning it into a [[cohomology ring]]. Thus, it is natural to consider a spectral sequence with a ring structure as well. Let <math>E^{p, q}_r</math> be a spectral sequence of cohomological type.  We say it has multiplicative structure if (i) <math>E_r</math> are (doubly graded) [[differential graded algebra]]s and (ii) the multiplication on <math>E_{r+1}</math> is induced by that on <math>E_r</math> via passage to cohomology.

A typical example is the cohomological [[Serre spectral sequence]] for a fibration <math>F \to E \to B</math>, when the coefficient group is a ring ''R''. It has the multiplicative structure induced by the cup products of fibre and base on the <math>E_{2}</math>-page.{{sfn|McCleary|2001|p={{pn|date=August 2021}}}} However, in general the limiting term <math>E_{\infty}</math> is not isomorphic as a graded algebra to H(''E''; ''R'').{{sfn|Hatcher|loc=Example 1.17}} The multiplicative structure can be very useful for calculating differentials on the sequence.{{sfn|Hatcher|loc=Example 1.18}}