Let $R=\mathbb{Z}[t^{\pm 1}]$ be the ring of Laurent polynomials, and let $S \subset R$ be the multiplicative subset generated by the polynomial $t-1$. I am interested in the ring $S^{-1}R=\mathbb{Z}[t^{\pm 1},(t-1)^{-1}]$ obtained by inverting $t-1$. More specifically, I know that finitely generated projective $R$-modules are free (e.g. by the Quillen-Suslin theorem) and I would like to know whether finitely generated projective $S^{-1}R$-modules are free?
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asked May 16, 2020 at 21:33
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$\begingroup$ Out of curiosity: is there a reason you don't ask the stronger question about $\mathbf Z[t]$? That would be an even closer analogue to Quillen–Suslin. $\endgroup$– R. van Dobben de BruynCommented May 16, 2020 at 22:24
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$\begingroup$ @R.vanDobbendeBruyn since there's a single question in the post, I'm not sure what you mean by "the stronger question". Do you mean, for every $S$ instead of this specific one? $\endgroup$– YCorCommented May 16, 2020 at 22:54
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$\begingroup$ @YCor: see Mohan's answer (vector bundles on regular schemes of dimension $\leq 2$ extend). $\endgroup$– R. van Dobben de BruynCommented May 17, 2020 at 1:03
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$\begingroup$ @R.vanDobbendeBruyn : no sensible reason, other than it is a vice of my trade, knot theorists often work with $\mathbb{Z}[t^{\pm 1}]$ instead of $\mathbb{Z}[t]$. $\endgroup$– Anthony ConwayCommented May 17, 2020 at 2:38
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1 Answer
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The answer is yes. Given any projective module $P$ over $S^{-1}A$, where $A=\mathbb{Z}[t]$ (and works for many other rings too), it is the localization $S^{-1}M$ of a projective module over $A$. The reason is, you can always find such a finitely generated module $M$ with $S^{-1}M=P$, but you may replace $M$ with its double dual without affecting the localization, but any reflexive module over $A$ is projective (and thus free, by Seshadri's theorem, which precedes Quillen-Suslin by many years).
To answer your questions in the comments below, double dual of any finitely generated module over $A$ is reflexive. Since $P$ is projective (and hence reflexive), it follows that if $S^{-1}M=P$,then so is $S^{-1}(M^{**})$. For your last question, for any Noetherian ring $A$ and $S\subset A$ a multiplicatively closed set, given any finitely generated module $P$ over $S^{-1}A$, there exists a finitely generated module $M$ over $A$ such that $S^{-1}M=P$. Further, if $P$ reflexive, then you may replace $M$ by $M^{**}$ and thus assume it is reflexive.
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2$\begingroup$ Actually the result that every projective module over $A[t]$ or $A[t,u]$ for $A$ PID was proved in the Vaserstein-Suslin paper (attributed to Suslin in Bass' Bourbaki seminar), which precedes Quillen-Suslin by... few years. $\endgroup$– YCorCommented May 16, 2020 at 22:58
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$\begingroup$ Thank you for your answer. Do you mind expanding a bit please? From what I read, you are saying that $S^{-1}M=S^{-1}M^{**}$ for $M$ f.g. which then implies that $M$ is reflexive? I fear I don't understand either of these steps. Where are you using that $P$ is projective? Did you use anything about my specific localisation or are you working with any $A$ of dimension $\leq 2$ and any multiplicative subset $S \subset A$? $\endgroup$ Commented May 17, 2020 at 2:34
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$\begingroup$ @AnthonyConway I have added a few more comments in my answer. $\endgroup$– MohanCommented May 17, 2020 at 3:00
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$\begingroup$ @AnthonyConway: more precisely, $S^{-1}(M^{**}) = (S^{-1}M)^{**}$. But $S^{-1}M$ is finite projective, so in particular reflexive. $\endgroup$ Commented May 17, 2020 at 3:15
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1$\begingroup$ @Mohan You really should explain what you use about the ring. For instance, it would not work when $A$ is a polynomial ring in three variables over the reals. $\endgroup$ Commented May 17, 2020 at 7:06