Zeeman's comparison theorem

Short description: On when a morphism of spectral sequences in homological algebra is an isomorphism

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

Statement

Comparison theorem — Let [math]\displaystyle{ E^r_{p, q}, {}^{\prime}E^r_{p, q} }[/math] be first quadrant spectral sequences of flat modules over a commutative ring and [math]\displaystyle{ f: E^r \to {}^{\prime}E^r }[/math] a morphism between them. Then any two of the following statements implies the third:

1. [math]\displaystyle{ f: E_2^{p, 0} \to {}^{\prime} E_2^{p, 0} }[/math] is an isomorphism for every p.
2. [math]\displaystyle{ f: E_2^{0, q} \to {}^{\prime} E_2^{0, q} }[/math] is an isomorphism for every q.
3. [math]\displaystyle{ f: E_{\infty}^{p, q} \to {}^{\prime} E_{\infty}^{p, q} }[/math] 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.

First of all, with G as a Lie group and with [math]\displaystyle{ \mathbb{Q} }[/math] as coefficient ring, we have the Serre spectral sequence [math]\displaystyle{ E_2^{p,q} }[/math] for the fibration [math]\displaystyle{ G \to EG \to BG }[/math]. We have: [math]\displaystyle{ E_{\infty} \simeq \mathbb{Q} }[/math] since EG is contractible. We also have a theorem of Hopf stating that [math]\displaystyle{ H^*(G; \mathbb{Q}) \simeq \Lambda(u_1, \dots, u_n) }[/math], an exterior algebra generated by finitely many homogeneous elements.

Next, we let [math]\displaystyle{ E(i) }[/math] be the spectral sequence whose second page is [math]\displaystyle{ E(i)_2 = \Lambda(x_i) \otimes \mathbb{Q}[y_i] }[/math] and whose nontrivial differentials on the r-th page are given by [math]\displaystyle{ d(x_i) = y_i }[/math] and the graded Leibniz rule. Let [math]\displaystyle{ {}^{\prime} E_{r} = \otimes_i E_{r}(i) }[/math]. Since the cohomology commutes with tensor products as we are working over a field, [math]\displaystyle{ {}^{\prime} E_{r} }[/math] is again a spectral sequence such that [math]\displaystyle{ {}^{\prime} E_{\infty} \simeq \mathbb{Q} \otimes \dots \otimes \mathbb{Q} \simeq \mathbb{Q} }[/math]. Then we let

[math]\displaystyle{ f: {}^{\prime} E_r \to E_r, \, x_i \mapsto u_i. }[/math]

Note, by definition, f gives the isomorphism [math]\displaystyle{ {}^{\prime} E_r^{0, q} \simeq E_r^{0, q} = H^q(G; \mathbb{Q}). }[/math] A crucial point is that f is a "ring homomorphism"; this rests on the technical conditions that [math]\displaystyle{ u_i }[/math] are "transgressive" (cf. Hatcher for detailed discussion on this matter.) After this technical point is taken care, we conclude: [math]\displaystyle{ E_2^{p, 0} \simeq {}^{\prime} E_2^{p, 0} }[/math] as ring by the comparison theorem; that is, [math]\displaystyle{ E_2^{p, 0} = H^p(BG; \mathbb{Q}) \simeq \mathbb{Q}[y_1, \dots, y_n]. }[/math]

Bibliography

• McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, 58 (2nd ed.), Cambridge University Press, ISBN 978-0-521-56759-6 
• Roitberg, Joseph; Hilton, Peter (1976), "On the Zeeman comparison theorem for the homology of quasi-nilpotent fibrations", The Quarterly Journal of Mathematics, Second Series 27 (108): 433–444, doi:10.1093/qmath/27.4.433, ISSN 0033-5606, http://doc.rero.ch/record/300344/files/27-4-433.pdf 
• Zeeman, Erik Christopher (1957), "A proof of the comparison theorem for spectral sequences", Proc. Cambridge Philos. Soc. 53: 57–62, doi:10.1017/S0305004100031984