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The Stacks project

Lemma 12.25.1. 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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