The Stacks project

Lemma 83.14.4. In Situation 83.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $K, K' \in D(\mathcal{O})$. Assume

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

  2. $K$ is cartesian,

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

  4. $\mathop{\mathrm{Hom}}\nolimits (K_ i[i + 1], K'_ i) = 0$ for $i \geq 0$.

Then any map $K \to K'$ which induces the zero map $K_0 \to K'_0$ is zero.

Proof. The proof is exactly the same as the proof of Lemma 83.13.4 except using Lemma 83.14.3 instead of Lemma 83.13.3. $\square$


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