Remark 12.20.3 (Variant). It is often the case that the terms of a spectral sequence have additional structure, for example a grading or a bigrading. To accomodate this (and to get around certain technical issues) we introduce the following notion. Let $\mathcal{A}$ be an abelian category. Let $(T_ r)_{r \geq 1}$ be a sequence of translation or shift functors, i.e., $T_ r : \mathcal{A} \to \mathcal{A}$ is an isomorphism of categories. In this setting a spectral sequence is given by a system $(E_ r, d_ r)_{r \geq 1}$ where each $E_ r$ is an object of $\mathcal{A}$, each $d_ r : E_ r \to T_ rE_ r$ is a morphism such that $T_ rd_ r \circ d_ r = 0$ so that

$\xymatrix{ \ldots \ar[r] & T_ r^{-1}E_ r \ar[r]^-{T_ r^{-1}d_ r} & E_ r \ar[r]^-{d_ r} & T_ rE_ r \ar[r]^{T_ r d_ r} & T_ r^2E_ r \ar[r] & \ldots }$

is a complex and $E_{r + 1} = \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(T_ r^{-1}d_ r)$ for $r \geq 1$. It is clear what a morphism of spectral sequences means in this setting. In this setting we can still define

$0 = B_1 \subset B_2 \subset \ldots \subset B_ r \subset \ldots \subset Z_ r \subset \ldots \subset Z_2 \subset Z_1 = E_1$

and $Z_\infty$ and $B_\infty$ (if they exist) as above.

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