Derived exact couple

For a page-$r$ exact couple $(D,E,i,j,k)$, let $d=jk$. The square-zero identity allows the quotient $D_{p,q}^{\mathrm{\prime }}=\mathrm{im}(i_{p-1,q+1}:D_{p-1,q+1}\to D_{p,q}),E_{p,q}^{\mathrm{\prime }}=\ker (d_{p,q})\mathrm{/}\mathrm{im}(d_{p+r,q-r+1})\mathrm{.}$ Define the following maps by their local formulas: $i_{p,q}^{\mathrm{\prime }}:D_{p,q}^{\mathrm{\prime }}\to D_{p+1,q-1}^{\mathrm{\prime }},i^{\mathrm{\prime }}(a)=i(a),$ $j_{p,q}^{\mathrm{\prime }}:D_{p,q}^{\mathrm{\prime }}\to E_{p-r,q+r}^{\mathrm{\prime }},j^{\mathrm{\prime }}(ix)=[jx](x\text{ at }{D}_{p-1,q+1}),$ $k_{p,q}^{\mathrm{\prime }}:E_{p,q}^{\mathrm{\prime }}\to D_{p-1,q}^{\mathrm{\prime }},k^{\mathrm{\prime }}([e])=ke\mathrm{.}$ In the last formula $e$ is a $d$-cycle. The local preimage $x$ and local representative $e$ are understood after an epimorphism onto a test object's domain, as in Spectral sequence subquotient and local lifting calculus. They do not mean a chosen section of $i$ or of a quotient map.

The existence and independence of these maps are established in The derived couple maps are well defined ↗, and their exactness in The derived couple is exact ↗. These data form the derived exact couple, a page-$(r+1)$ couple: the degrees of $i^{\mathrm{\prime }}$ and $k^{\mathrm{\prime }}$ are unchanged, while $j^{\mathrm{\prime }}$ has degree $(-r,r)$. For an initial couple this changes the degree of $j$ from $(0,0)$ to $(-1,1)$. Zero kernels, images and quotients are permitted; every construction is componentwise, with no infinite sum.