Editing Universal coefficient theorem (section)

== Universal coefficient spectral sequence ==

There is a generalization of the universal coefficient theorem for (co)homology with [[Twisted Poincaré duality|twisted coefficients]].

For cohomology we have 

:<math>E^{p,q}_2=\operatorname{Ext}_{R}^q(H_p(C_*),G)\Rightarrow H^{p+q}(C_*;G),</math>

where <math>R</math> is a ring with unit, <math>C_*</math> is a chain complex of free modules over <math>R</math>, <math>G</math> is any <math>(R,S)</math>-bimodule for some ring with a unit <math>S</math>, and <math>\operatorname{Ext}</math> is the [[Ext functor|Ext group]]. The differential <math>d^r</math> has degree <math>(1-r,r)</math>.

Similarly for homology,
:<math>E_{p,q}^2=\operatorname{Tor}^{R}_q(H_p(C_*),G)\Rightarrow H_*(C_*;G),</math>
for <math>\operatorname{Tor}</math> the [[Tor functor|Tor group]] and the differential <math>d_r</math> having degree <math>(r-1,-r)</math>.