In homological algebra, Zeeman's comparison theorem, introduced by Christopher Zeeman,[1] gives conditions for a morphism of spectral sequences to be an isomorphism.
Comparison theorem — Let
be first quadrant spectral sequences of flat modules over a commutative ring and
r
a morphism between them.
Then any two of the following statements implies the third: As an illustration, we sketch the proof of Borel's theorem, which says the cohomology ring of a classifying space is a polynomial ring.
[citation needed] First of all, with G as a Lie group and with
{\displaystyle \mathbb {Q} }
as coefficient ring, we have the Serre spectral sequence
for the fibration
{\displaystyle G\to EG\to BG}
since EG is contractible.
We also have a theorem of Hopf stating that
, an exterior algebra generated by finitely many homogeneous elements.
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.
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,
{\displaystyle E_{2}^{p,0}=H^{p}(BG;\mathbb {Q} )\simeq \mathbb {Q} [y_{1},\dots ,y_{n}].}
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