The Stacks project

Definition 35.12.1. Let $S$ be a scheme. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\}$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules on $(\mathit{Sch}/S)_\tau$.

1. $\mathcal{F}$ is called parasitic¹ if for every flat morphism $U \to S$ we have $\mathcal{F}(U) = 0$.

2. $\mathcal{F}$ is called parasitic for the $\tau$-topology if for every $\tau$-covering $\{ U_ i \to S\} _{i \in I}$ we have $\mathcal{F}(U_ i) = 0$ for all $i$.