Lemma 12.24.2. Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet , F)$ be a filtered complex of $\mathcal{A}$. There is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ in the category of bigraded objects of $\mathcal{A}$ associated to $(K^\bullet , F)$ such that $d_ r$ has bidegree $(r, - r + 1)$ and such that $E_ r$ has bigraded pieces $E_ r^{p, q}$ and maps $d_ r^{p, q} : E_ r^{p, q} \to E_ r^{p + r, q - r + 1}$ as given above. Furthermore, we have $E_0^{p, q} = \text{gr}^ p(K^{p + q})$, $d_0^{p, q} = \text{gr}^ p(d^{p + q})$, and $E_1^{p, q} = H^{p + q}(\text{gr}^ p(K^\bullet ))$.

**Proof.**
If $\mathcal{A}$ has countable direct sums and if countable direct sums are exact, then this follows from the discussion above. In general, we proceed as follows; we strongly suggest the reader skip this proof. Consider the bigraded object $A = (F^{p + 1}K^{p + 1 + q})$ of $\mathcal{A}$, i.e., we put $F^{p + 1}K^{p + 1 + q}$ in degree $(p, q)$ (the funny shift in numbering to get numbering correct later on). We endow it with a differential $d : A \to A[0, 1]$ by using $d$ on each component. Then $(A, d)$ is a differential bigraded object. Consider the map

which is given in degree $(p, q)$ by the inclusion $F^{p + 1}K^{p + 1 + q} \to F^ pK^{p + 1 + q}$. This is an injective morphism of differential objects $\alpha : (A, d) \to (A, d)[-1, 1]$. Hence, we can apply Remark 12.22.6 with $S = [0, 1]$ and $T = [1, -1]$. The corresponding spectral sequence $(E_ r, d_ r)_{r \geq 0}$ of bigraded objects is the spectral sequence we are looking for. Let us unwind the definitions a bit. First of all we have $E_ r = (E_ r^{p, q})$. Then, since $T^ rS = [r, -r + 1]$ we have $d_ r : E_ r \to E_ r[r, -r + 1]$ which means that $d_ r^{p, q} : E_ r^{p, q} \to E_ r^{p + r, q - r + 1}$.

To see that the description of the graded pieces hold, we argue as above. Namely, first we have

and by our choice of numbering above this gives

The first differential is given by $d_0^{p, q} = \text{gr}^ pd^{p + q} : E_0^{p, q} \to E_0^{p, q + 1}$. Next, the description of the boundaries $B_ r$ and the cocycles $Z_ r$ in Remark 12.22.6 translates into a straightforward manner into the formulae for $Z_ r^{p, q}$ and $B_ r^{p, q}$ given above. $\square$

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