Vector bundles on punctured disc around isolated surface singularity

Question

Let $X/k$ be a surface (over some field), smooth except for an isolated (closed) point $x$. One may look at the punctured local ring

$X:=\mathrm{Spec}(\mathcal{O}_{X,x}) - x$.

Are there non-trivial vector bundles on $X$? If $X$ is smooth at the puncture as well, every vector bundle on $X$ must be trivial, thanks to low-dimensionality (since it must be the pullback of a vector bundle on the entire local ring).

I suspect the answer is yes, but I would be interested in explicit examples, literature on this, etc. Any advice?

Answer 1

I am just posting my comment as an answer. For a Noetherian local ring $\mathcal{O}_{X,x}$ with maximal ideal $\mathfrak{m}_{X,x}$, the $\mathcal{O}_{X,x}$-module $\mathcal{O}_{X,x}/\mathfrak{m}_{X,x}$ has a finite free resolution if and only if the local ring is regular. Thus, for every surjection of $\mathcal{O}_{X,x}$-modules, $$\phi:\mathcal{O}_{X,x}^{\oplus (r+1)} \to \mathfrak{m}_{X,x},$$ the kernel of $\phi$ is a finitely generated $\mathcal{O}_{X,x}$ that is not free, yet the restriction of the kernel of $\phi$ to $\text{Spec}(\mathcal{O}_{X,x}) \setminus \{\mathfrak{m}_{X,x}\}$ is locally free.