Zeeman's comparison theorem

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In homological algebra, Zeeman's comparison theorem, introduced by Christopher Zeeman,[1] gives conditions for a morphism of spectral sequences to be an isomorphism.

Statement

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Comparison theorem — Let   be first quadrant spectral sequences of flat modules over a commutative ring and   a morphism between them. Then any two of the following statements implies the third:

  1.   is an isomorphism for every p.
  2.   is an isomorphism for every q.
  3.   is an isomorphism for every p, q.

Illustrative example

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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   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,  

References

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Bibliography

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  • McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, vol. 58 (2nd ed.), Cambridge University Press, ISBN 978-0-521-56759-6, MR 1793722
  • Roitberg, Joseph; Hilton, Peter (1976), "On the Zeeman comparison theorem for the homology of quasi-nilpotent fibrations" (PDF), The Quarterly Journal of Mathematics, Second Series, 27 (108): 433–444, doi:10.1093/qmath/27.4.433, ISSN 0033-5606, MR 0431151
  • Zeeman, Erik Christopher (1957), "A proof of the comparison theorem for spectral sequences", Proc. Cambridge Philos. Soc., 53: 57–62, doi:10.1017/S0305004100031984, MR 0084769