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 $E_{p,q}^{r},{}^{\prime }E_{p,q}^{r}$ be first quadrant spectral sequences of flat modules over a commutative ring and $f:E^{r}\to {}^{\prime }E^{r}$ a morphism between them. Then any two of the following statements implies the third:

$f:E_{2}^{p,0}\to {}^{\prime }E_{2}^{p,0}$ is an isomorphism for every p.
$f:E_{2}^{0,q}\to {}^{\prime }E_{2}^{0,q}$ is an isomorphism for every q.
$f:E_{\infty }^{p,q}\to {}^{\prime }E_{\infty }^{p,q}$ 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 $\mathbb {Q}$ as coefficient ring, we have the Serre spectral sequence $E_{2}^{p,q}$ for the fibration $G\to EG\to BG$ . We have: $E_{\infty }\simeq \mathbb {Q}$ since EG is contractible. We also have a theorem of Hopf stating that $H^{*}(G;\mathbb {Q} )\simeq \Lambda (u_{1},\dots ,u_{n})$ , an exterior algebra generated by finitely many homogeneous elements.

Next, we let $E(i)$ be the spectral sequence whose second page is $E(i)_{2}=\Lambda (x_{i})\otimes \mathbb {Q} [y_{i}]$ and whose nontrivial differentials on the r-th page are given by $d(x_{i})=y_{i}$ and the graded Leibniz rule. Let ${}^{\prime }E_{r}=\otimes _{i}E_{r}(i)$ . Since the cohomology commutes with tensor products as we are working over a field, ${}^{\prime }E_{r}$ is again a spectral sequence such that ${}^{\prime }E_{\infty }\simeq \mathbb {Q} \otimes \dots \otimes \mathbb {Q} \simeq \mathbb {Q}$ . Then we let

f:{}^{\prime }E_{r}\to E_{r},\,x_{i}\mapsto u_{i}.

Note, by definition, f gives the isomorphism ${}^{\prime }E_{r}^{0,q}\simeq E_{r}^{0,q}=H^{q}(G;\mathbb {Q} ).$ A crucial point is that f is a "ring homomorphism"; this rests on the technical conditions that $u_{i}$ are "transgressive" (cf. Hatcher for detailed discussion on this matter.) After this technical point is taken care, we conclude: $E_{2}^{p,0}\simeq {}^{\prime }E_{2}^{p,0}$ as ring by the comparison theorem; that is, $E_{2}^{p,0}=H^{p}(BG;\mathbb {Q} )\simeq \mathbb {Q} [y_{1},\dots ,y_{n}].$

References

Hatcher, Theorem 1.34

McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, 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

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[1] Hatcher, Theorem 1.34