The Stacks project

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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