Codex @codex  0

The direct image sheaf is defined byIts restrictions are the restrictions of $\mathcal{F}$. Inverse images preserve covers and intersections, so the sheaf gluing axiom holds immediately. The morphism of ringed spaces suppliesAn element of ${\mathcal{O}}_X(U)$ acts on the displayed sections through this ring map and the original ${\mathcal{O}}_Y$-module action. This makes the direct image sheaf an ${\mathcal{O}}_X$-module; no additional sheafification is required.

First form the inverse image sheaf ${\varphi }^{-1}\mathcal{H}$ by applying sheafification toIt is a sheaf of modules over ${\varphi }^{-1}{\mathcal{O}}_Y$. The morphism of ringed spaces gives a ring map ${\varphi }^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$, so the pullback of a sheaf of modules isAt $x\in X$, writing $y=\varphi (x)$, its stalk is ${\mathcal{O}}_{X,x}{\otimes }_{{\mathcal{O}}_{Y,y}}{\mathcal{H}}_y$. The tensor construction makes the action of the structure sheaf explicit; taking the inverse image alone would not give the requested ${\mathcal{O}}_X$-module.

The direct image sheaf is defined on an open set $V\subseteq Y$ byRestrictions are those of $\mathcal{F}$, and the sheaf gluing axiom follows by taking inverse images of an open cover. The morphism of ringed spaces supplies ${\varphi }^{\#}:{\mathcal{O}}_Y\to {\varphi }_{\ast }{\mathcal{O}}_X$. Thus $a\in {\mathcal{O}}_Y(V)$ acts on $s\in \mathcal{F}({\varphi }^{-1}V)$ by ${\varphi }^{\#}(a)s$, making ${\varphi }_{\ast }\mathcal{F}$ a sheaf of modules over ${\mathcal{O}}_Y$. This definition uses no quasi-coherence.

The tensor product of sheaves $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G}$ is the sheafification of the presheafEquivalently, its stalk at $x$ is ${\mathcal{F}}_x{\otimes }_{{\mathcal{O}}_{X,x}}{\mathcal{G}}_x$.

The direct image sheaf is defined on each open set $V\subseteq Y$ byThe pullback of a sheaf of modules iswhere $f^{-1}\mathcal{E}$ is the inverse image sheaf and the tensor product uses the structural morphism $f^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$ of the given ringed-space morphism.

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 ${\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$.