The Stacks project

Lemma 85.8.5. In Situation 85.3.3. Let $K$ be an object of $D(\mathcal{C}_{total})$.

  1. If $H^{-p}(\mathcal{C}_ p, K_ p) = 0$ for all $p \geq 0$, then $H^0(\mathcal{C}_{total}, K) = 0$.

  2. If $R\Gamma (\mathcal{C}_ p, K_ p) = 0$ for all $p \geq 0$, then $R\Gamma (\mathcal{C}_{total}, K) = 0$.

Proof. With notation as in Remark 85.8.4 we see that $R\Gamma (\mathcal{C}_{total}, K)$ is represented by $\text{Tot}_\pi (A^{\bullet , \bullet })$. The assumption in (1) tells us that $H^{-p}(A^{p, \bullet }) = 0$. Thus the vanishing in (1) follows from More on Algebra, Lemma 15.103.1. Part (2) follows from part (1) and taking shifts. $\square$

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