The Stacks project

Lemma 21.53.2. Let $\mathcal{C}$ be a site with final object $X$. Let $\Lambda $ be a ring. Let

1. $K$ a perfect object of $D(\Lambda )$,

2. a finite complex $K^\bullet $ of finite projective $\Lambda $-modules representing $K$,

3. $\mathcal{L}^\bullet $ a complex of sheaves of $\Lambda $-modules, and

4. $\varphi : \underline{K} \to \mathcal{L}^\bullet $ a map in $D(\mathcal{C}, \Lambda )$.

Then there exists a covering $\{ U_ i \to X\} $ and maps of complexes $\alpha _ i : \underline{K}^\bullet |_{U_ i} \to \mathcal{L}^\bullet |_{U_ i}$ representing $\varphi |_{U_ i}$.