Bass–Quillen conjecture for regular local rings of ramified mixed characteristic

The Bass–Quillen conjecture predicts that for a Noetherian regular ring $R$, every finitely generated projective module over a polynomial ring $R[T_1,\dots,T_n]$ is extended from $R$. All cases are settled except one: regular local rings of mixed characteristic $(0,p)$ in which $p$ lies in the square of the maximal ideal (the ramified case, e.g. $\mathbb{Z}_p[\zeta_p][[u_1,\dots,u_{d-1}]]$), in dimension at least 4, and in dimension 3 with residue characteristic 2 or 3. Dorin Popescu's positive resolution of Swan's approximation question (2018) does not close this gap, because it is unknown — and, by an example of Lam, possibly false — that the property 'all finitely generated projectives are extended' ascends along the ramified ring extensions $A[W]/(W^e-b)$ used in the inductive-limit construction. The ramified mixed-characteristic core remains open; the adjacent purity conjecture for flat cohomology stated in the same paper was resolved by Česnavičius–Scholze (Annals of Mathematics, 2024).

Background

A finitely generated projective module over a ring $S$ is a direct summand of a finite-rank free $S$-module. Serre's problem (1955–1976) asked whether finitely generated projective modules over $k[T_1,\dots,T_n]$, with $k$ a field, are free; Quillen and Suslin proved this in 1976. Bass and Quillen then posed the natural relative version (Bass, 'Some problems in classical algebraic K-theory', Problem IX, 1973): for a regular ring $R$ — here 'regular' means every finitely generated module has a finite resolution by projectives, the algebraic counterpart of smoothness — is every finitely generated projective $P$ over the polynomial algebra $R[T_1,\dots,T_n]$ extended from $R$, i.e. isomorphic to $R[T_1,\dots,T_n]\otimes_R(P/(T)P)$, the module obtained by base change from its fiber at the origin? By Quillen's patching theorem the conjecture for arbitrary regular rings reduces to the case where $R$ is a regular local ring, and for local $R$ the question is whether $P$ is free precisely when $P/(T)P$ is free.

The conjecture is known in the following cases: dimension $\le 1$ (Quillen 1976, Suslin 1976); dimension $2$ (Murthy 1966); dimension $3$ with residue characteristic $\ne 2,3$ (Rao 1988); and regular local rings whose residue characteristic $p$ is $0$ or does not lie in $\mathfrak{m}^2$ — the 'unramified' mixed-characteristic case — by Lindel (1981) and Swan for rings essentially of finite type over $\mathbb{Z}$, extended by Popescu (1989) to all such regular local rings. Here $\mathfrak{m}$ is the maximal ideal, a regular system of parameters is a minimal system of generators of $\mathfrak{m}$, and a mixed-characteristic ring $(R,\mathfrak{m},k)$ has characteristic $0$ while its residue field $k=R/\mathfrak{m}$ has characteristic $p>0$; the case $p\in\mathfrak{m}^2$ is called ramified, e.g. $\mathbb{Z}_p[\zeta_p]$ where $p$ is a power of $1-\zeta_p$ up to a unit.

The remaining case is exactly the ramified mixed-characteristic one. Swan noticed that a useful tool would be: every regular local ring is a filtered inductive limit (directed colimit) of regular local rings essentially of finite type over $\mathbb{Z}$ ('essentially' means a localization of a finite-type algebra). Popescu proved this positively in his paper ('On a question of Swan', 2018, Theorem 17 and Corollary 19: one can even take the approximants excellent). However, as the paper's Remark 18 records, this does not settle Bass–Quillen: to run the limit argument one would need that the conjecture ascends through the specific ramified extensions of the form $A[W]/(W^e-b)$, $b$ in a regular system of parameters, that appear in the inductive-limit construction — and an example recorded in Lam's 1978 lecture notes (over $B=(A[X_2]/(X_2^3-X_1^2))_{(X_1,X_2)}$ with $A=\mathbb{R}[X_1]_{(X_1)}$, $\mathbb{R}$ the real numbers, there are rank-one non-free projectives over $B[T]$) shows that extendedness of all projectives can genuinely fail along such ramified quotients of regular rings. Popescu's companion survey (2020) reduces the full conjecture to the derived question whether Bass–Quillen holds for $R/(p-b)$ where $(R,\mathfrak{m})$ is essentially smooth over $\mathbb{Z}_{(p)}$ and $b\in\mathfrak{m}^2$ (Question 6, via Theorem 12); a regular morphism here is a flat morphism with geometrically regular fibers, and essentially smooth means essentially of finite type and smooth. The paper's appendix also stated a folklore purity conjecture for flat cohomology — vanishing of fppf cohomology with supports $H^i_{\mathfrak{m}}(R,G)=0$ for $i<\dim R$ for every commutative finite flat group scheme $G$ over a regular local ring $R$, where fppf cohomology is sheaf cohomology for the topology of flat, finitely presented coverings — reduced there to complete rings; that secondary question has since been proven in a stronger form by Česnavičius–Scholze (2024), but it does not touch the projective-module core.

Problem Statement

Let $(R,\mathfrak{m},k)$ be a regular local ring of ramified mixed characteristic: $R$ has characteristic $0$ while its residue field $k=R/\mathfrak{m}$ has characteristic $p>0$, and $p\in\mathfrak{m}^2$ (for instance $R=\mathbb{Z}_p[\zeta_p][[u_1,\dots,u_{d-1}]]$ for a prime $p>3$). Is every finitely generated projective module over the polynomial algebra $R[T_1,\dots,T_n]$, for every $n\ge 1$, extended from $R$, i.e. isomorphic to $R[T_1,\dots,T_n]\otimes_R(P/(T)P)$? This is exactly the remaining open case of the Bass–Quillen conjecture for regular rings (which Quillen patching reduces to regular local rings): the cases of dimension $\le 1$, dimension $2$, dimension $3$ with residue characteristic $\ne 2,3$, and residue characteristic $0$ or $p\notin\mathfrak{m}^2$ being settled, the frontier is ramified mixed characteristic in dimension $\ge 4$ together with dimension $3$ and residue characteristic $2$ or $3$. A negative answer may be given as an explicit regular local ring $R$ of ramified mixed characteristic and a finitely generated projective $R[T_1,\dots,T_n]$-module that is not extended from $R$.

References

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