Remark 85.8.4. Assumptions and notation as in Lemma 85.8.3 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
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 85.8.1. 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 85.8.3.
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