Exact couple

Fix an integer $r\ge 1$ and an abelian category. A page-$r$ homological exact couple is a pair of indexed families $(D_{p,q}{)}_{p,q\in \mathbb{Z}}$ and $(E_{p,q}{)}_{p,q\in \mathbb{Z}}$ and maps $i_{p,q}:D_{p,q}\to D_{p+1,q-1},j_{p,q}:D_{p,q}\to E_{p+1-r,q+r-1},k_{p,q}:E_{p,q}\to D_{p-1,q}\mathrm{.}$ For every $(p,q)$ require the three subobject equalities $\mathrm{im}(i_{p-1,q+1})=\ker (j_{p,q})\text{ in }{D}_{p,q},$ $\mathrm{im}(j_{p+r-1,q-r+1})=\ker (k_{p,q})\text{ in }{E}_{p,q},$ $\mathrm{im}(k_{p+1,q})=\ker (i_{p,q})\text{ in }{D}_{p,q}\mathrm{.}$ These are exactness at the three positions, meaning the corresponding kernel modulo image is zero. In particular each consecutive composite $ji$, $kj$ or $ik$, with the displayed shifts understood, is zero.

An initial exact couple means $r=1$: then $j$ has bidegree $(0,0)$. Its first spectral page will be called $E^1$, not $E^0$. A morphism between two page-$r$ couples is a pair of bidegree-zero families $u_{p,q}:D_{p,q}\to {\tilde{D}}_{p,q}$ and $w_{p,q}:E_{p,q}\to {\tilde{E}}_{p,q}$ satisfying $u_{p+1,q-1}{i}_{p,q}={\tilde{i}}_{p,q}{u}_{p,q},w_{p+1-r,q+r-1}{j}_{p,q}={\tilde{j}}_{p,q}{u}_{p,q},u_{p-1,q}{k}_{p,q}={\tilde{k}}_{p,q}{w}_{p,q}\mathrm{.}$ All-zero families are allowed. Graded objects here mean indexed families: no infinite direct sum, convergence or abutment is part of the definition.