Consider the following ring: $\mathbb{Z}_p[[X,Y]]\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$ (this is the power series ring over $\mathbb{Z}_p$ of two indeterminates tensored with $\mathbb{Q}_p$). Is it the case that all finitely generated projective modules over it are free?


Quillen's question is whether every finite projective module over $R_f$ is free, when $R$ is a regular local ring and $f \in \mathfrak m \setminus \mathfrak m^2$. There are many results of this nature in the literature. In particular, this is proved by Gabber for all three dimensional regular local rings in "Some theorems on Azumaya algebras", see https://link.springer.com/chapter/10.1007/BFb0090480

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    $\begingroup$ Maybe it's worth saying explicitly that Gabber's result solves the question positively (with $R=\mathbb{Z}_p[[X,Y]]$, which is local regular of Krull dimension 3, and $f=p$). $\endgroup$
    – YCor
    Apr 16 '17 at 14:11

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