Quillen's conjecture $Q_n$: are all projective modules over the localization of a regular local ring at a regular parameter free?

Quillen's conjecture, posed by Daniel Quillen in 1976 as an avenue of attack on the Bass–Quillen conjecture, predicts that if $(R,\mathfrak m)$ is a Noetherian regular local ring of dimension $n$ and $r\in\mathfrak m\setminus\mathfrak m^2$ is a regular parameter, then every finitely generated projective module over the localization $R[1/r]$ is free. The conjecture has been confirmed when $R$ contains a field (Bhatwadekar–Rao, Rao, Popescu) and when $\dim R\le 3$ (Gabber, Swan), but it remains open for regular local rings of mixed characteristic, especially ramified ones, in dimension at least $4$.

Background

A finitely generated module $P$ over a commutative ring $A$ is called projective if it is a direct summand of a free module; geometrically, projective modules correspond to algebraic vector bundles on the affine scheme $\mathrm{Spec}\,A$. The classical Serre problem asked whether every finitely generated projective module over a polynomial ring $k[X_1,\dots,X_d]$ over a field $k$ is free; this was settled affirmatively and independently in 1976 by Quillen and Suslin (the Quillen–Suslin theorem). Its natural extension, the Bass–Quillen conjecture, predicts that for a regular ring $R$ (a Noetherian ring whose localization at every prime ideal has finite projective dimension over itself, the algebraic analogue of smoothness), every finitely generated projective module over the polynomial ring $R[t_1,\dots,t_d]$ is extended from $R$, i.e. of the form $P\otimes_R R[t_1,\dots,t_d]$; when $R$ is local this forces freeness. By a reduction of Quillen now called Quillen patching, the conjecture reduces to the case of a regular local ring $(R,\mathfrak m)$ and a single variable. A Noetherian local ring $(R,\mathfrak m)$ of dimension $n$ is a regular local ring if its maximal ideal can be generated by $n$ elements; such a minimal system of generators $r_1,\dots,r_n$ is called a regular system of parameters, and a single $r\in\mathfrak m\setminus\mathfrak m^2$ that belongs to one is called a regular parameter. To extend a vector bundle on the affine line $\mathbb A^1_R$ to the projective line $\mathbb P^1_R$ (the input needed for Horrocks' theorem, a classical criterion asserting that a vector bundle on the affine plane over a local ring that is trivial on a line is free, and which then delivers Bass–Quillen), one is led via formal gluing (a Beauville–Laszlo-type technique that compares modules over $R$, over its $r$-adic completion, and over the localization of that completion) to exactly the following question, recorded by Quillen as comment (2) on page 170 of his 1976 paper: is every finitely generated projective module over the localization $R[1/r]$ free? An affirmative answer to this question, conventionally called Quillen's conjecture or $Q_n$ where $n=\dim R$, implies the Bass–Quillen conjecture. The hypothesis that $r$ be a regular parameter is essential: Bhatwadekar and Rao showed that over the ring $R=\mathbb R[x,y,z]_{(x,y,z)}$, localized at $r=x^2+y^2+z^2\in\mathfrak m^2$, the kernel of the surjection $R[1/r]^3\to R[1/r]$, $(a,b,c)\mapsto ax+by+cz$, is a projective module of rank $2$ that is not free. A regular local ring is of equicharacteristic if it contains a field and of mixed characteristic otherwise; in mixed characteristic $0$ it has characteristic $0$ but residue characteristic $p$, and it is called unramified if $R/pR$ is regular. The conjecture is known for equicharacteristic rings (Bhatwadekar–Rao for localizations of affine algebras over a field; Popescu in general, via his general Néron desingularization, a smoothing theorem for regular homomorphisms) and for rings of dimension at most $3$ (Gabber, with a simpler proof by Swan). What remains open is the mixed-characteristic case, especially for ramified rings, in dimension at least $4$. Mathew and Keshari [2] prove freeness results for projective modules over monoid algebras $A[M]$ (the semigroup algebra of a commutative monoid $M$ with coefficients in $A$), where $M$ is cancellative ($a+c=b+c$ implies $a=b$ in $M$), torsion-free, and seminormal (a mild algebraicity condition on the monoid), and $A$ is an overring of a polynomial ring built from a regular local ring containing a field, and they recall Quillen's conjecture $Q_n$ as the open question behind this circle of ideas.

Problem Statement

Let $(R,\mathfrak m)$ be a Noetherian regular local ring and let $r\in\mathfrak m\setminus\mathfrak m^2$ be a regular parameter, i.e. an element of some regular system of parameters of $R$. Must every finitely generated projective module over the localization $R[1/r]$ be free? Equivalently, must every finitely generated projective $R[1/r]$-module extend to a finitely generated projective $R$-module? This is Quillen's conjecture $Q_n$ with $n=\dim R$, recorded as comment (2) on page 170 of Quillen's 1976 paper; a positive answer for all regular local rings would imply the Bass–Quillen conjecture. The conjecture is known when $R$ contains a field and when $\dim R\le 3$; the unresolved regime is regular local rings of mixed characteristic in dimension at least $4$, especially ramified ones.

References

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