The Stacks project

Lemma 21.52.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site with set of coverings $\text{Cov}_\mathcal {C}$. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $\text{Cov} \subset \text{Cov}_\mathcal {C}$ be subsets. Assume that

1. For every $\mathcal{U} \in \text{Cov}$ we have $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ with $I$ finite, $U, U_ i \in \mathcal{B}$ and every $U_{i_0} \times _ U \ldots \times _ U U_{i_ p} \in \mathcal{B}$.

2. For every $U \in \mathcal{B}$ the coverings of $U$ occurring in $\text{Cov}$ is a cofinal system of coverings of $U$.

Then for $U \in \mathcal{B}$ the object $j_{U!}\mathcal{O}_ U$ is a compact object of $D^+(\mathcal{O})$ in the following sense: if $M = \bigoplus _{i \in I} M_ i$ in $D(\mathcal{O})$ is bounded below, then $\mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathcal{O}_ U, M) = \bigoplus _{i \in I} \mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathcal{O}_ U, M_ i)$.