Pullback of a module along a morphism of ringed spaces

Let

be a morphism of ringed spaces, and let $\mathcal{G}$ be an ${\mathcal{O}}_Y$-module. The pullback of $\mathcal{G}$ along $f$ is the ${\mathcal{O}}_X$-module $f^{\ast }\mathcal{G}:={\mathcal{O}}_X{\otimes }_{f^{-1}{\mathcal{O}}_Y}{f}^{-1}\mathcal{G}\mathrm{.}$

Here the ring map

is the morphism corresponding to $f^{\mathrm{♯}}$ under the inverse/direct-image adjunction Inverse image is left adjoint to direct image on sheaves. The inverse-image construction is taken with its algebraic structure: applying the neighbourhood-colimit construction to the restriction-compatible ring operations of ${\mathcal{O}}_Y$, and then sheafifying, makes $f^{-1}{\mathcal{O}}_Y$ a sheaf of rings. The same construction makes $f^{-1}\mathcal{G}$ an $f^{-1}{\mathcal{O}}_Y$-module (Presheaves and sheaves of groups, rings, and modules).

Consequently ${\mathcal{O}}_X$ is an $f^{-1}{\mathcal{O}}_Y$-algebra, the displayed sheaf tensor product is well typed, and it carries the asserted ${\mathcal{O}}_X$-module structure.