In homological algebra, Zeeman's comparison theorem, introduced by Christopher Zeeman, gives conditions for a morphism of spectral sequences to be an isomorphism.
As an illustration, we sketch the proof of Borel's theorem, which says the cohomology ring of a classifying space is a polynomial ring.
First of all, with G as a Lie group and with as coefficient ring, we have the Serre spectral sequence for the fibration . We have: since EG is contractible. We also have a theorem of Hopf stating that , an exterior algebra generated by finitely many homogeneous elements.
Next, we let be the spectral sequence whose second page is and whose nontrivial differentials on the r-th page are given by and the graded Leibniz rule. Let . Since the cohomology commutes with tensor products as we are working over a field, is again a spectral sequence such that . Then we let
Note, by definition, f gives the isomorphism A crucial point is that f is a "ring homomorphism"; this rests on the technical conditions that are "transgressive" (cf. Hatcher for detailed discussion on this matter.) After this technical point is taken care, we conclude: as ring by the comparison theorem; that is,