4
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Let $A$ be a noetherian ring. Let us define $A$ to be $n$-contractible if

  1. All locally free sheaves of rank $\le n$ over $\text{Spec} A$ is trivial.
  2. 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:

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

  3. 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?

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  • 1
    $\begingroup$ Not all local rings are UFDs! For one thing, every UFD is integrally closed. $\endgroup$ – Laurent Moret-Bailly Jun 3 '16 at 8:46
  • $\begingroup$ This could perhaps be a separate MO question but: is there a relation, for a $\mathbb{C}$-algebra $A$, between $A$ being contractible and $(\mathrm{Spec}(A))^\mathrm{an}$ being a contractible topological space? $\endgroup$ – Qfwfq Jun 3 '16 at 9:08
  • $\begingroup$ If you do not mind, I can put this up in my list of questions. Let me know if that's okay! $\endgroup$ – Shubhodip Mondal Jun 3 '16 at 9:11
  • $\begingroup$ Qfwfq: I think that question is open but the natural implication is expected. A negative result for quasi-affines can be found here: arxiv.org/abs/0710.3607 $\endgroup$ – Joe Berner Jun 3 '16 at 12:18
3
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Examples for 1 and 2 are well known. A general hypersurface in $\mathbb{C}^3$ is a UFD by Noether-Lefshcetz theorem. If the degree is sufficiently large, its second Chow group (codimension 2 cycles) is large by a theorem of Mumford. So, there are are rank 2 projective modules which are not free by Serre construction. Any affine open subset of $\mathbb{C}^2$ (or more generally any affine smooth rational surface which is a UFD over complex numbers) is infinitely contractible, since the second Chow group is zero.

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  • $\begingroup$ Thanks for your answer! I must confess that I am not familiar with the theorems you quote. Just out of curiosity, can you similarly construct $k$-contractible UFDs for any $k$? $\endgroup$ – Shubhodip Mondal Jun 3 '16 at 13:11
  • $\begingroup$ Cancellation theorems for projective modules in the middle range are hard to come by, so no general results are known to me for arbitrary $k$. $\endgroup$ – Mohan Jun 3 '16 at 13:22

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