Definition 6.26.1. Let $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces.

Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. We define the

*pushforward*of $\mathcal{F}$ as the sheaf of $\mathcal{O}_ Y$-modules which as a sheaf of abelian groups equals $f_*\mathcal{F}$ and with module structure given by the restriction via $f^\sharp : \mathcal{O}_ Y \to f_*\mathcal{O}_ X$ of the module structure given in Lemma 6.24.5.Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ Y$-modules. We define the

*pullback*$f^*\mathcal{G}$ to be the sheaf of $\mathcal{O}_ X$-modules defined by the formula\[ f^*\mathcal{G} = \mathcal{O}_ X \otimes _{f^{-1}\mathcal{O}_ Y} f^{-1}\mathcal{G} \]where the ring map $f^{-1}\mathcal{O}_ Y \to \mathcal{O}_ X$ is the map corresponding to $f^\sharp $, and where the module structure is given by Lemma 6.24.6.

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