Zeeman's comparison theorem
In homological algebra, Zeeman's comparison theorem, introduced by, gives conditions for a morphism of spectral sequences to be an isomorphism.Statement
Illustrative example
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" After this technical point is taken care, we conclude: as ring by the comparison theorem; that is,
