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
\alpha : A \to A[-1, 1]
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
E_0 = \mathop{\mathrm{Coker}}(\alpha : A \to A[-1, 1])[0, -1] = \mathop{\mathrm{Coker}}(\alpha [0, -1] : A[0, -1] \to A[-1, 0])
and by our choice of numbering above this gives
E_0^{p, q} = \mathop{\mathrm{Coker}}(F^{p + 1}K^{p + q} \to F^ pK^{p + q}) = \text{gr}^ pK^{p + q}
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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