The Stacks project

Definition 18.13.1. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi or ringed sites.

1. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal {C}$-modules. We define the pushforward of $\mathcal{F}$ as the sheaf of $\mathcal{O}_\mathcal {D}$-modules which as a sheaf of abelian groups equals $f_*\mathcal{F}$ and with module structure given by the restriction via $f^\sharp : \mathcal{O}_\mathcal {D} \to f_*\mathcal{O}_\mathcal {C}$ of the module structure

   \[ f_*\mathcal{O}_\mathcal {C} \times f_*\mathcal{F} \longrightarrow f_*\mathcal{F} \]

   from Lemma 18.12.1.

2. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_\mathcal {D}$-modules. We define the pullback $f^*\mathcal{G}$ to be the sheaf of $\mathcal{O}_\mathcal {C}$-modules defined by the formula

   \[ f^*\mathcal{G} = \mathcal{O}_\mathcal {C} \otimes _{f^{-1}\mathcal{O}_\mathcal {D}} f^{-1}\mathcal{G} \]

   where the ring map $f^{-1}\mathcal{O}_\mathcal {D} \to \mathcal{O}_\mathcal {C}$ is $f^\sharp $, and where the module structure is given by Lemma 18.12.2.