Bass–Quillen for ramified mixed-characteristic test rings $R/(p-b)$ (Popescu's Question 6)

Dorin Popescu proved that the Bass–Quillen Conjecture for all regular rings reduces to a single explicit family of test rings: he showed that a positive answer to his Question 6 — for every regular local ring $(R,\mathfrak m)$ essentially smooth over $\mathbf Z_{(p)}$ and every $b\in\mathfrak m^2$, does the Bass–Quillen Conjecture hold for the ring $R/(p-b)$, i.e. is every finitely generated projective $(R/(p-b))[T_1,\dots,T_n]$-module extended from $R/(p-b)$ — would imply the conjecture for every regular ring. Up to étale neighborhood, the rings $R/(p-b)$ with $b\notin(p)$ are exactly the ramified mixed-characteristic regular local rings essentially of finite type over $\mathbf Z$ (those of residue characteristic $p>0$ with $p$ lying in the square of the maximal ideal), the core case in which Bass–Quillen has remained open since the 1980s. No subsequent work resolving the question is known.

Background

A finitely generated projective module over a ring $A$ is a direct summand of a finite free $A$-module; it is extended from $A$ over a polynomial algebra $A[T_1,\dots,T_n]$ if it is isomorphic to $A[T_1,\dots,T_n]\otimes_A M$ for a projective $A$-module $M$ (equivalently, it is obtained by base change from the coefficient ring). The Bass–Quillen Conjecture (Bass, around 1973; Quillen, 1976) predicts that if $R$ is a regular ring (a noetherian ring whose localizations at prime ideals are regular local rings, i.e. their maximal ideals are generated by a regular sequence), then every finitely generated projective $R[T_1,\dots,T_n]$-module is extended from $R$. It grew out of Serre's problem (the case $R$ a field, settled affirmatively by Quillen and Suslin in 1976) and is one of the central open problems of commutative algebra and algebraic $K$-theory.

Known cases delimit the frontier. Quillen (1976) settled dimension $\le 1$, and Murthy (1966) dimension $\le 2$, in any characteristic. Lindel (1981) proved the conjecture for regular rings essentially of finite type over a field, and Popescu (1989), using his General Néron Desingularization (the theorem that a regular morphism of noetherian rings is a filtered colimit of smooth morphisms), extended this to all regular rings containing a field and, in mixed characteristic, to the unramified local case. Here a noetherian local ring $(A,\mathfrak n,k)$ of residue characteristic $p>0$ and characteristic $0$ (mixed characteristic $(0,p)$) is called ramified when $p\in\mathfrak n^2$ and unramified when $p\notin\mathfrak n^2$. The ramified mixed-characteristic case with $\dim\ge 3$ is exactly the case in which the conjecture remains open.

Swan asked whether every regular local ring is a filtered inductive limit (filtered colimit) of regular local rings essentially of finite type over $\mathbf Z$; Popescu answered this affirmatively (2019, with an appendix by K. Česnavičius), and in a 2018/2020 paper refined the structure theory: every regular local ring $(R,\mathfrak m)$ of residue characteristic $0\ne p=\operatorname{char}k\in\mathfrak m^2$ is a filtered inductive limit of regular local rings essentially smooth (essentially of finite type with geometrically regular fibers) over rings of the form $A/(p-b)$, where $(A,\mathfrak a)$ is a regular local ring essentially smooth over the localization $\mathbf Z_{(p)}$ of $\mathbf Z$ at $p$ and $b\in\mathfrak a^2$. Since $A$ essentially smooth over $\mathbf Z_{(p)}$ forces $p\notin\mathfrak a^2$, the element $p-b$ is part of a regular system of parameters, so $A/(p-b)$ is again a regular local ring, in which the image of $p$ equals the image of $b$ and lies in the square of the maximal ideal of $A/(p-b)$; whenever $b\notin(p)$ it is of mixed characteristic $(0,p)$, a ramified ring essentially of finite type over $\mathbf Z_{(p)}$; conversely, every regular local ring essentially of finite type over $\mathbf Z$ with $p\in\mathfrak m^2$ has this shape up to étale neighborhood. Combining this structure theory with Quillen's patching and limit arguments, Popescu proved (Theorem 12 of that paper) that a positive answer for this single family of test rings would imply the Bass–Quillen Conjecture for all regular rings. The question for these test rings — stated as Question 6 in that paper — is the subject of this problem record.

Problem Statement

Popescu's Question 6 (arXiv:1810.00617; published version DOI 10.1007/978-3-030-52111-0_9): Let $p$ be a prime, $(R,\mathfrak m)$ a regular local ring essentially smooth over $\mathbf Z_{(p)}$, and $b\in\mathfrak m^2$ an element of the square of the maximal ideal. The quotient $R/(p-b)$ is a regular local ring in which the image of $p$ lies in the square of the maximal ideal; whenever $b\notin(p)$ (equivalently, the image of $p$ in $R/(p-b)$ is nonzero) it is of mixed characteristic $(0,p)$, a ramified ring essentially of finite type over $\mathbf Z_{(p)}$; for $b\in(p)$, instead, $R/(p-b)=R/(p)$ is an equal-characteristic-$p$ regular local ring, a case in which Bass–Quillen is already known. Is every finitely generated projective module over the polynomial algebra $(R/(p-b))[T_1,\dots,T_n]$ extended from $R/(p-b)$ — that is, isomorphic to $(R/(p-b))[T_1,\dots,T_n]\otimes_{R/(p-b)}M$ for some finitely generated projective $R/(p-b)$-module $M$ — for every $n\ge 1$? Equivalently: does the Bass–Quillen Conjecture hold for all rings of the form $R/(p-b)$ with $R$ essentially smooth over $\mathbf Z_{(p)}$ and $b\in\mathfrak m^2$? By Theorem 12 of the source, an affirmative answer (uniformly in $R$, $b$, and $n$) would imply the Bass–Quillen Conjecture for all regular rings; a single counterexample within this family would show that the conjecture fails in general. The question is to be answered in its full stated generality, with no restriction on the regular local ring $R$, the element $b\in\mathfrak m^2$, the residue field, the dimension, or the number of polynomial variables.

References

1. Dorin Popescu, "On the Bass-Quillen Conjecture and Swan's question", arXiv:1810.00617 (2020), https://arxiv.org/abs/1810.00617; published as "The Bass-Quillen Conjecture and Swan's Question", in Combinatorial Structures in Algebra and Geometry, Springer Proceedings in Mathematics & Statistics, Springer (2020), pp. 115–121, DOI 10.1007/978-3-030-52111-0_9, https://doi.org/10.1007/978-3-030-52111-0_9
2. Dorin Popescu, "On a question of Swan. With an Appendix by Kęstutis Česnavičius", arXiv:1803.06956 (published in Algebraic Geometry 6(6) (2019), 716–729), https://arxiv.org/abs/1803.06956
3. Daniel Quillen, "Projective modules over polynomial rings", Inventiones Mathematicae 36 (1976), 167–171, DOI 10.1007/BF01390008, https://doi.org/10.1007/BF01390008
4. Hartmut Lindel, "On the Bass-Quillen conjecture concerning projective modules over polynomial rings", Inventiones Mathematicae 65 (1981), 319–323, DOI 10.1007/BF01389017, https://doi.org/10.1007/BF01389017
5. M. P. Murthy, "Projective A[X]-Modules", Journal of the London Mathematical Society s1-41 (1966), 453–456, DOI 10.1112/jlms/s1-41.1.453, https://doi.org/10.1112/jlms/s1-41.1.453
6. Dorin Popescu, "Polynomial rings and their projective modules", Nagoya Mathematical Journal 113 (1989), 121–128, DOI 10.1017/S0027763000001288, https://doi.org/10.1017/S0027763000001288
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