Let $A$ be a noetherian ring. Let us define $A$ to be $n$-contractible if

- All locally free sheaves of rank $\le n$ over $\text{Spec} A$ is trivial.
- There exist a non-trivial locally free sheaf of rank $n+1$.

Note that all $\ge 1$-contractible normal noetherian domains are UFDs and convesely. All the PIDs and local rings are infinitely contractible. By the extremely nontrivial **Quillen–Suslin theorem**, all the polynomial rings over a field are also infinitely contractible.

**Questions:**

- Example of a $1$-contractible UFD. Examples of $k$-contractible UFDs for small $k$ are all welcome.
Examples of infinitely contractible UFDs other than PIDs, local rings and polynomial rings.

Can we (expect to) categorize rings by their degree of contractiblity? Can we (expect to) find some criterion which will tell the order of the cobtractibility?