Let $A = \mathbb{C}[x,y,z]/(x y - z^k)$. In fact $A$ is the ring of $\mu_k$ invariants: $A = \mathbb{C}[u,v]^{\mu_k}$ where $g \in \mu_k$ acts by $g(u,v) = (g u, g^{-1} v)$.

This allows one to understand vector bundles on the smooth surface $Spec\ A - (0,0,0)$ as $\mu_k$ equivariant bundles on $Spec\ \mathbb{C}[u,v]$.

However I would like to understand how many of these vector bundles extend to all of $Spec\ A$.

**QUESTION**

It's well known for $k = 2$ that $Pic(Spec\ A)$ is trivial. But what about higher $k$ and higher rank?

One approach is to consider the natural map $\mathbb{C}[x,y]\to A$ which is finite and flat and then (identifying vector bundles with projective modules) work with modules over $\mathbb{C}[x,y]$ equipped with an endomorphism $\phi$ such that $\phi^k = x y$. But I didn't get very far with this.