Morphism of spectral sequences

A morphism of spectral sequences from page s is a family of bidegree (0,0) maps $f_r:E^r\to D^r$ for r≥s, where both sequences are defined, such that $d_D^r{f}_r=f_r{d}_E^r$ and $f_{r+1}{\alpha }_r^E={\alpha }_r^{D}H(f_r)\mathrm{.}$ Use Homological spectral sequence or Cohomological spectral sequence as appropriate; the map on homology is the unique cycle-quotient map of A chain map induces a well-defined map on homology. A map of abutments is additional filtered data, compatible with the stable graded maps; it is not inherent in f.

If $f_s$ is an isomorphism at every bidegree, all later $f_r$ are isomorphisms. In fact the inverse of $f_s$ commutes with differentials by multiplying the commutation equation on both sides by inverses. Thus $f_s$ and its inverse carry kernels and boundary images into each other and induce inverse quotient maps on homology. The displayed compatibility expresses $f_{s+1}$ as a composite of three isomorphisms. Repeating this argument a finite number of times proves the assertion at every r≥s.

Weibel, Chapter 5, §5.2 pp.122–125 and Definitions 5.4.2–4 pp.132–133; Sharifi, §4.1 pp.87–89 and Definition 4.2.1 p.90. Mapping Lemma 5.2.4 and Sharifi Lemma 4.1.4.