The Stacks project

Remark 84.8.4. Assumptions and notation as in Lemma 84.8.3 except we do not require $K$ in $D(\mathcal{C}_{total})$ to be bounded below. We claim there is a natural spectral sequence in this case also. Namely, suppose that $\mathcal{I}^\bullet $ is a K-injective complex of sheaves on $\mathcal{C}_{total}$ with injective terms representing $K$. We have

\begin{align*} R\Gamma (\mathcal{C}_{total}, K) & = R\mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, K) \\ & = R\mathop{\mathrm{Hom}}\nolimits ( \ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z}, K) \\ & = \Gamma (\mathcal{C}_{total}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet ( \ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z}, \mathcal{I}^\bullet )) \\ & = \text{Tot}_\pi (A^{\bullet , \bullet }) \end{align*}

where $A^{\bullet , \bullet }$ is the double complex with terms $A^{p, q} = \Gamma (\mathcal{C}_ p, \mathcal{I}^ q_ p)$ and $\text{Tot}_\pi $ denotes the product totalization of this double complex. Namely, the first equality holds in any site. The second equality holds by Lemma 84.8.1. The third equality holds because $\mathcal{I}^\bullet $ is K-injective, see Cohomology on Sites, Sections 21.34 and 21.35. The final equality holds by the construction of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet $ and the fact that $\mathop{\mathrm{Hom}}\nolimits (g_{p!}\mathbf{Z}, \mathcal{I}^ q) = \Gamma (\mathcal{C}_ p, \mathcal{I}^ q_ p)$. Then we get our spectral sequence by viewing $\text{Tot}_\pi (A^{\bullet , \bullet })$ as a filtered complex with $F^ i\text{Tot}^ n_\pi (A^{\bullet , \bullet }) = \prod _{p + q = n,\ p \geq i} A^{p, q}$. The spectral sequence we obtain behaves like the spectral sequence $({}'E_ r, {}'d_ r)_{r \geq 0}$ in Homology, Section 12.25 (where the case of the direct sum totalization is discussed) except for regularity, boundedness, convergence, and abutment issues. In particular we obtain $E_1^{p, q} = H^ q(\mathcal{C}_ p, K_ p)$ as in Lemma 84.8.3.

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