Inverse image sheaf ($f^{-1}\mathcal{E}$)

The inverse image sheaf $f^{-1}\mathcal{E}$ is the sheaf associated with the presheaf whose sections near $U\subseteq X$ are obtained as a colimit of $\mathcal{E}(V)$ over open sets $V\supseteq f(U)$.

This is the sheaf obtained by applying the inverse image functor.

For sheaves of abelian groups and a continuous map $f:X\to Y$, there is a natural bijection ${\mathrm{Hom}}_X(f^{-1}\mathcal{G},\mathcal{F})\cong {\mathrm{Hom}}_Y(\mathcal{G},f_{\ast }\mathcal{F})$. It comes from the universal property of an inverse image sheaf. The inverse image of abelian sheaves is different from the tensor-adjusted pullback of a sheaf of modules on a ringed space.

Every f-morphism of sheaves from $\mathcal{G}$ to $\mathcal{F}$ factors uniquely through the canonical f-morphism from $\mathcal{G}$ to its inverse image sheaf. A local inverse-image section is represented by sections of $\mathcal{G}$ on image neighbourhoods; their prescribed images glue in $\mathcal{F}$.

For a morphism of ringed spaces, the pullback of an ${\mathcal{O}}_Y$-module is

The inverse image functor supplies the sheaf $f^{-1}\mathcal{E}$; tensoring with the source structure sheaf supplies the module scalar extension.

The restriction meant here is the pullback of a sheaf of modules to the closed subvariety, including tensoring with its structure sheaf. On an affine chart, tensoring $M$ with $A/I$ gives $M/IM$. This distinction matters because restricting only the underlying topological sheaf would retain the ambient ring action.

For a morphism of ringed spaces, the pullback of a sheaf of modules is left adjoint to the direct image sheaf functor. The unit sends a local section $h$ to $1\otimes h$ after inverse image. On the structure sheaf it becomes the given ring homomorphism ${\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$.

For a morphism of ringed spaces, there is a natural morphismIt is an isomorphism when $\mathcal{E}$ is a locally free sheaf of finite rank, because the claim is local and then reduces to distributivity over a finite direct sum.

The same formula holds for higher direct images: $R^i{f}_{\ast }\mathcal{F}\otimes \mathcal{E}\cong R^i{f}_{\ast }(\mathcal{F}\otimes f^{\ast }\mathcal{E})$ when $\mathcal{E}$ is locally free of finite rank.