A proof of the comparison theorem for spectral sequences

Extract

The comparison theorem is an algebraic theorem, which corresponds to the following topological situation. A map between two fibre spaces induces a homomorphism between the two corresponding spectral sequences relating the homologies of the base, fibre and fibre space; if the map induces an isomorphism on any two of these quantities, then it also does on the third. J. C. Moore (l) proved the theorem using a mapping cylinder, which requires that the spectral sequences arise from filtered differential groups. The present proof assumes only the existence of the spectral sequences from the E² terms onwards. Moreover, we generalize the theorem, assuming that the homomorphisms on the two given quantities are isomorphic only up to given dimensions, and deducing the same for the third quantity. For algebraic translucence we use a simpler hypothesis (property (iii)) than Moore's concerning the E² terms. However, if his hypothesis is assumed, together with the existence of E¹ terms, then his theorem can be deduced as a corollary. The analogous result for cohomology is similar, and is stated at the end.