Sheaf of modules

Remarks

The structure sheaf $\mathcal{O}_X$ is a sheaf of $\mathcal{O}_X$-modules, with each $\mathcal{O}_X(U)$ acting on itself by multiplication. Condition 2 holds because restriction maps of $\mathcal{O}_X$ are ring homomorphisms. On $X = \operatorname{Spec} \Z$, the constant sheaf $\underline{\Z}$ admits no structure of $\mathcal{O}_X$-module. Its sections over $D(2)$ form the group $\Z$, and clause 1 would need a module structure over $\mathcal{O}_X(D(2)) = \Z[1/2]$. This is impossible, since multiplication by $2$ is not invertible on $\Z$. For each point $x \in X$, the stalk $\mathcal{F}_x$ is a module over the local ring $\mathcal{O}_{X,x}$. The $\mathcal{O}_X$-modules form an abelian category, with tensor products $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ and sheaves of homomorphisms.