Lemma 12.22.4. Let $\mathcal{A}$ be an abelian category. Let $K^{\bullet , \bullet }$ be a double complex. The spectral sequences associated to $K^{\bullet , \bullet }$ have the following terms:

1. ${}'E_0^{p, q} = K^{p, q}$ with ${}'d_0^{p, q} = (-1)^ p d_2^{p, q} : K^{p, q} \to K^{p, q + 1}$,

2. ${}''E_0^{p, q} = K^{q, p}$ with ${}''d_0^{p, q} = d_1^{q, p} : K^{q, p} \to K^{q + 1, p}$,

3. ${}'E_1^{p, q} = H^ q(K^{p, \bullet })$ with ${}'d_1^{p, q} = H^ q(d_1^{p, \bullet })$,

4. ${}''E_1^{p, q} = H^ q(K^{\bullet , p})$ with ${}''d_1^{p, q} = (-1)^ q H^ q(d_2^{\bullet , p})$,

5. ${}'E_2^{p, q} = H^ p_ I(H^ q_{II}(K^{\bullet , \bullet }))$,

6. ${}''E_2^{p, q} = H^ p_{II}(H^ q_ I(K^{\bullet , \bullet }))$.

Proof. Omitted. $\square$

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