Zeeman's comparison theorem
In homological algebra, Zeeman's comparison theorem, introduced by Zeeman (1957), gives conditions for a morphism of spectral sequences to be an isomorphism.
Statement
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:
- is an isomorphism for every p.
- is an isomorphism for every q.
- is an isomorphism for every p, q.
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.[1]
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
- McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, vol. 58 (2nd ed.), Cambridge University Press, doi:10.2277/0521567599, 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. Oxford. 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