Example of an $\mathcal{O}_{X,p}$- module Consider a point $p$ on a variety $X$. Let $\mathcal{O}_{X,p}$ be the local ring of polynomials at $p$. In Algebraic geometry, we often study $\mathcal{O}_{X,p}$- modules. Could I get an example of such a module? I used to mainly take $\mathcal{O}_{X,p}$ as a module over itself, but that seems to be a pretty unhelpful example. Thank you

The comment is pretty much the answer, but it's worth saying a few more things. For example, let $L$ be a line bundle/invertible sheaf on $X$. Then the stalk $L_p$ is isomorphic to $\mathcal O_{X,p}$ by definition, but is a module over it in a slightly more interesting way. More generally, for a locally free sheaf of rank $r$ the stalk at $p$ is an $\mathcal O_{X,p}$-module isomorphic to $\mathcal O_{X,p}^{\oplus r}$. Even more generally, for $\mathcal F$ a coherent sheaf, the stalk $\mathcal F_p$ is isomorphic to a finite direct sum power of $\mathcal O_{X,p}$, but the power can vary with $p$.
