The Stacks project

Lemma 85.14.7. In Situation 85.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. Let $(K_ n, K_\varphi )$ be a simplicial system of the derived category of modules. Assume

  1. $f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ is flat for $\varphi : [m] \to [n]$,

  2. $(K_ n, K_\varphi )$ is cartesian,

  3. $\mathop{\mathrm{Hom}}\nolimits (K_ i[t], K_ i) = 0$ for $i \geq 0$ and $t > 0$.

Then there exists a cartesian object $K$ of $D(\mathcal{O})$ whose associated simplicial system is isomorphic to $(K_ n, K_\varphi )$.

Proof. The proof is exactly the same as the proof of Lemma 85.13.7 with the following changes

  1. use $g_ n^* = Lg_ n^*$ everywhere instead of $g_ n^{-1}$,

  2. use $f_\varphi ^* = Lf_\varphi ^*$ everywhere instead of $f_\varphi ^{-1}$,

  3. refer to Lemma 85.10.1 instead of Lemma 85.8.1,

  4. in the construction of $Y'_{m, n}$ use $\mathcal{O}_ m$ instead of $\mathbf{Z}$,

  5. compare with the proof of Lemma 85.14.4 rather than the proof of Lemma 85.13.4.

This ends the proof. $\square$

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