Constructive Hermite rings conjecture and constructive Bass–Quillen conjecture for projective modules over polynomial rings (Lombardi–Quitté closing problem list)

The constructive theory of finitely generated projective modules over polynomial rings, as developed by Lombardi and Quitté, closes with two named open conjectures. The first is the Hermite rings conjecture: if $A$ is a Hermite ring (every stably free finitely generated $A$ -module is free, equivalently every unimodular row over $A$ can be completed to an invertible matrix), then so is $A[X]$ ; in equivalent local form, $A[X]$ is Hermite for every residually discrete local ring $A$ . The second is the constructive Bass–Quillen conjecture: for every regular coherent Noetherian ring $A$ , every finitely generated projective $A[X_1,\dots,X_n]$ -module is extended from $A$ , with a constructive (algorithmic, choice-free) proof demanded. In the published record, the constructive Bass–Quillen conjecture is open even in the classically settled Krull dimension $\le 2$ case, and the Hermite rings conjecture remains open, although a 2023 unrefereed preprint of Schäppi claims an explicit counterexample via symplectic K-theory.

Background

A finitely generated projective module over a commutative ring $A$ is a direct summand of a finite free module; such modules are the algebraic counterpart of algebraic vector bundles. Serre observed in 1955 that it was unknown whether every finitely generated projective module over a polynomial ring $K[X_1,\dots,X_n]$ over a field $K$ is free; the affirmative answer proved independently by Quillen and Suslin in 1976 is the Quillen–Suslin theorem (Serre's problem). Two directions of generalization immediately arise.

First, one can weaken the coefficient ring. A module-theoretic property of $A$ -modules is studied for $A[X_1,\dots,X_n]$ -modules via extension: a projective $A[X_1,\dots,X_n]$ -module $P$ is extended from $A$ if $P \cong Q \otimes_A A[X_1,\dots,X_n]$ for a projective $A$ -module $Q$ . The Bass–Quillen conjecture predicts that if $A$ is a regular Noetherian ring — regular here means that every finitely presented module admits a finite resolution by finitely generated projective modules, and coherent means every finitely generated ideal is finitely presented, a condition automatic classically for Noetherian rings — then every finitely generated projective $A[X_1,\dots,X_n]$ -module is extended from $A$ . Classically this is known for regular rings of Krull dimension $\le 2$ and for regular rings containing a field, and it remains open in general (mixed characteristic, dimension $\ge 3$ ).

Second, one can weaken the conclusion from 'extended' to 'free'. A finitely generated projective module $P$ is stably free if $P \oplus A^r \cong A^m$ for some $r,m$ . A unimodular row is a row vector with entries in $A$ generating the unit ideal; completing such a row to an invertible matrix is equivalent to the associated stably free module being free. Following Lam, a ring $A$ is called Hermite if every stably free finitely generated $A$ -module is free, equivalently if every unimodular row over $A$ is completable. The Hermite rings conjecture, posed by Lam, states that $A$ Hermite implies $A[X]$ Hermite; via Quillen patching it is equivalent to its local form: $A[X]$ is Hermite whenever $A$ is a residually discrete local ring (a local ring whose residue field is discrete, i.e., equality of elements is decidable), and it is also equivalent to Swan's Question (B) of 1978, which asks whether every stably free $A[X]$ -module is extended from $A$ .

The problem belongs to the constructive mathematics tradition. In Bishop-style constructive algebra, existence statements must be witnessed by explicit finite data and proofs must avoid the law of excluded middle and arguments passing through prime or maximal ideals, which classical proofs of the surrounding literature use heavily. Lombardi and Quitté's monograph 'Commutative Algebra: Constructive Methods' rebuilds the theory of finitely generated projective modules constructively: Quillen's patching and local-global principles, Horrocks' theorem, and theorems of Bass–Simis–Vasconcelos, Maroscia–Brewer–Costa, and Lequain–Simis (finitely generated projective modules over $A[X_1,\dots,X_n]$ are extended from $A$ for every arithmetical ring $A$ — a ring whose localizations at prime ideals are valuation rings; when an arithmetical ring is a domain it is exactly a Prüfer domain, the setting of Lequain and Simis's original theorem) all receive constructive proofs, the last via an induction scheme due to Yengui combined with constructive Krull dimension bounds for finitely generated valuation rings. The chapter's closing section, 'Conclusion: a few conjectures', records exactly two conjectures left open by this constructive theory: the Hermite rings conjecture, in local form (H') and global form (H), with special cases known (rings of Krull dimension $\le 1$ in one variable, ranks above the dimension, valuation rings); and the Bass–Quillen conjecture, stated there for regular coherent Noetherian rings in global form (BQ) and local form (BQ'), with the explicit remark that even the classically settled Krull dimension $\le 2$ case has no constructive proof at the time of writing. A constructive proof of either conjecture would in particular settle the corresponding classical conjecture in full generality and would yield explicit algorithms, for instance for completing unimodular rows over polynomial rings.

Problem Statement

Settle, in the constructive sense demanded by Lombardi–Quitté's closing list of conjectures (Chapter XVI, 'Conclusion: a few conjectures'), the following two statements, each of which must be either proved constructively or refuted by an explicit counterexample:

(1) Hermite rings conjecture.

(H) If $A$ is a Hermite ring (every stably free finitely generated $A$ -module is free, equivalently every unimodular row over $A$ is completable to an invertible matrix), then so is $A[X]$ .

Equivalently, by Quillen patching: (H') if $A$ is a residually discrete local ring, then $A[X]$ is a Hermite ring. A constructive proof must be valid without the law of excluded middle and without prime or maximal ideal arguments, and should provide terminating algorithms that complete unimodular rows over $A[X]$ (equivalently trivialize stably free $A[X]$ -modules) from explicit finite input data. A refutation consists of an explicit Hermite ring $A$ (or a residually discrete local ring $A$ for (H')) together with an explicit stably free $A[X]$ -module, given by presentation matrices, certified non-free.

(2) Constructive Bass–Quillen conjecture.

(BQ) If $A$ is a regular coherent Noetherian ring (regular: every finitely presented $A$ -module admits a finite projective resolution), then every finitely generated projective $A[X_1,\dots,X_n]$ -module is extended from $A$ .

Equivalently, in local form: (BQ') if $A$ is a regular coherent Noetherian residually discrete local ring, every finitely generated projective $A[X_1,\dots,X_n]$ -module is free. Since $A$ regular Noetherian implies $A[X]$ regular Noetherian, it suffices to treat $n=1$ . The constructive demand is an algorithmic, choice-free proof, i.e., an explicit terminating procedure that, given a presentation matrix of a finitely generated projective $A[X]$ -module over such a ring $A$ (given by suitable finite data), produces matrices realizing the isomorphism with a module extended from $A$ . A sub-goal explicitly flagged by the source and still open is the constructive proof of the classically settled case of Krull dimension $\le 2$ (for arbitrary $n$ ). A refutation consists of an explicit regular coherent Noetherian ring $A$ and an explicit finitely generated projective $A[X_1,\dots,X_n]$ -module certified not extended from $A$ .

The two items are jointly the closing open-problems list of the constructive theory of projective modules over polynomial rings: the common objective is constructive (algorithmic, choice-free) control of stably free and extended projective modules under polynomial extension. The problem is completely settled when each item is settled (constructive proof, or explicit counterexample); note that the constructive Lequain–Simis statement without Krull-dimension hypotheses, sometimes mentioned alongside these, is already proved in the source book and is not part of this problem.

References

Henri Lombardi and Claude Quitté, Commutative Algebra: Constructive Methods (Finitely generated projective modules), Algebra and Applications 20, Springer, 2015. DOI: 10.1007/978-94-017-9944-7; revised version: arXiv:1605.04832, https://arxiv.org/abs/1605.04832 (Chapter XVI, closing list of conjectures).
Sami Barhoumi, Henri Lombardi, and Ihsen Yengui, Projective modules over polynomial rings: a constructive approach, Mathematische Nachrichten 282 (2009), 792–799. DOI: 10.1002/mana.200610772, https://doi.org/10.1002/mana.200610772.
Daniel Schäppi, A counterexample to the Hermite ring conjecture, preprint, arXiv:2311.04104 (2023), https://arxiv.org/abs/2311.04104.
Daniel Schäppi, Formal Laurent series rings and the Hermite ring conjecture, preprint, arXiv:2204.06303 (2022), https://arxiv.org/abs/2204.06303.
Daniel Schäppi, Symplectic K-theory and a problem of Murthy, preprint, arXiv:2311.03077 (2023), https://arxiv.org/abs/2311.03077.
Dorin Popescu, On the Bass–Quillen Conjecture and Swan's question, preprint, arXiv:1810.00617 (2018), https://arxiv.org/abs/1810.00617.
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Yves Lequain and Aron Simis, Projective modules over R[X1,…,Xn], R a Prüfer domain, Journal of Pure and Applied Algebra 18 (1980), 165–171. DOI: 10.1016/0022-4049(80)90127-9, https://doi.org/10.1016/0022-4049(80)90127-9.
Thierry Coquand, Henri Lombardi, and Claude Quitté, Heitmann dimension of distributive lattices and commutative rings, preprint, arXiv:2312.00684 (2023), https://arxiv.org/abs/2312.00684.