Over a commutative noetherian local ring, finitely generated projective modules are free (Kaplansky), and over two-dimensional regular local rings even every finitely generated reflexive module is free (Serre). Ramras asked in 1971 for a general answer: exactly which local rings have the property that all finitely generated reflexive modules are free? The problem remains open in this generality. It is completely settled only for restricted classes — quasi-reduced rings (where the property forces regularity of dimension at most two), BNSI rings (rings whose non-free modules have strictly increasing Betti numbers), and certain artinian rings such as truncated polynomial rings k[X1,…,Xm]/(X1,…,Xm)nk[X_1,\dots,X_m]/(X_1,\dots,X_m)^n — while an active program maps the boundary in dimension one through Arf rings, almost Gorenstein rings, and trace ideals.

Background

Throughout, (R,m,k)(R,\mathfrak{m},k) is a commutative noetherian local ring: RR has a unique maximal ideal m\mathfrak{m} with residue field k=R/mk=R/\mathfrak{m}, and every ideal is finitely generated. For a finitely generated RR-module MM, its dual is M∗=Hom⁡R(M,R)M^{*}=\operatorname{Hom}_R(M,R), and MM is called reflexive if the natural map M→M∗∗M\to M^{**}, m↦(f↦f(m))m\mapsto(f\mapsto f(m)), is an isomorphism. Reflexivity sits between projectivity and arbitrary finite generation: every finitely generated projective module is reflexive, and over an artinian ring (a ring in which descending chains of ideals stabilize) every reflexive module is automatically finitely generated. Kaplansky proved that projective modules over local rings are free, so the class of reflexive modules is the first natural enlargement of the free modules, and one asks when this enlargement adds nothing new.

Two classical anchors shape the question. First, as a generalization of Seshadri's result, Serre observed in 1958 that over a regular local ring of dimension two — regular meaning that the minimal number of generators of m\mathfrak{m} equals the Krull dimension of RR (the supremum of the lengths of chains of prime ideals), i.e. a local ring with the homological properties of a localized polynomial ring — every finitely generated reflexive module is free; over such rings the finitely generated reflexive modules are exactly the second syzygies, i.e. the kernels of the map F1→F0F_1\to F_0 in a minimal free resolution ⋯→F2→F1→F0→M→0\cdots\to F_2\to F_1\to F_0\to M\to 0 of a finitely generated module. Second, Ramras introduced homological growth conditions: the Betti numbers βi(M)\beta_i(M) are the ranks of the free modules FiF_i in the minimal free resolution of MM, and a local ring is BNSI (Betti numbers strictly increasing) if βi(M)>βi−1(M)\beta_i(M)>\beta_{i-1}(M) for all i>1i>1 for every non-free finitely generated module MM. Ramras proved that every finitely generated reflexive module over a BNSI ring is free, and that over any local ring of depth zero (depth being the length of a maximal regular sequence in m\mathfrak{m}, a regular sequence being one whose entries are successively non-zerodivisors modulo the preceding ones), freeness of the dual M∗M^{*} already forces freeness of MM.

That freeness of reflexives is genuinely subtle even over simple rings was noted by Lam: over R=k[X,Y]/(X,Y)2R=k[X,Y]/(X,Y)^2, the only obvious reflexive modules are the free powers RnR^n, and confirming that these are the only ones required real work (via the classical Kronecker classification of the indecomposable modules over the square-zero algebra k⊕mk\oplus\mathfrak{m}). In the problem list of the 1971 Park City conference proceedings, Ramras posed the general question, restated as Problem 1.1 of Asgharzadeh's Reflexivity revisited: when are finitely generated reflexive modules free?

The settled fragments delimit what is known. A local ring is quasi-reduced if it satisfies Serre's condition (S1)(\mathrm{S}_1) (for every prime p\mathfrak{p}, depth⁡Rp≥min⁡{1,dim⁡Rp}\operatorname{depth}R_{\mathfrak{p}}\ge\min\{1,\dim R_{\mathfrak{p}}\}) and is generically Gorenstein (RpR_{\mathfrak{p}} is a Gorenstein, i.e. finite-injective-dimension, artinian local ring for every minimal prime p\mathfrak{p}). Over quasi-reduced rings the problem has a complete answer: all reflexive modules are free exactly when RR is regular of dimension at most two. On the artinian side, freeness holds for the truncated polynomial rings k[X1,…,Xm]/(X1,…,Xm)nk[X_1,\dots,X_m]/(X_1,\dots,X_m)^n with m>1m>1 and for rings with m2=0\mathfrak{m}^2=0 and μ(m)>1\mu(\mathfrak{m})>1 (μ\mu denoting the minimal number of generators of m\mathfrak{m}), both via the Betti-growth criterion; Asgharzadeh further introduced the class of eventually BNSI rings (strict growth of Betti numbers only from some point on) and established freeness results there. In dimension one, a separate program studies reflexive modules through ideal theory: over a one-dimensional Arf local ring (a ring of the class introduced through Arf's and Lipman's study of curve singularities, in which every integrally closed ideal containing a non-zerodivisor is the integral closure of a principal ideal, together with a companion closure condition on quotients, so that the ideal theory mimics that of a discrete valuation ring), Isobe and Kumashiro proved that reflexive modules admit an explicit direct-sum decomposition, that integrally closed ideals, trace ideals (the trace ideal of a module MM being the image of the natural evaluation map M⊗RM∗→RM\otimes_RM^{*}\to R), and divisorial ideals (reflexive ideals of rank one) coincide up to isomorphism, and that analytically irreducible Arf rings (rings whose m\mathfrak{m}-adic completion is a domain) have only finitely many indecomposable reflexive modules.

Problem Statement

Ramras' problem (1971 Park City conference problem list, p. 380; restated as Problem 1.1 of Asgharzadeh, Reflexivity revisited): give necessary and sufficient conditions on a commutative noetherian local ring RR under which every finitely generated reflexive RR-module is free. The requested answer is a characterization in intrinsic ring-theoretic terms of the class

{ (R,m):R commutative noetherian local, and every finitely generated reflexive R-module is free },\{\,(R,\mathfrak{m}) : R \text{ commutative noetherian local, and every finitely generated reflexive } R\text{-module is free}\,\},

with no restriction on dimension, singularity type, or residue characteristic. Equivalently, the characterization is an equivalence: every ring satisfying the stated conditions has no non-free finitely generated reflexive module, and every ring failing them admits a non-free finitely generated reflexive module.

References

  1. M. Ramras, "Betti numbers and reflexive modules", in R. Gordon (ed.), Ring theory (Proc. Conf., Park City, Utah, 1971), Academic Press, New York, 1972, pp. 297–308; the problem discussed here appears at p. 380 of the proceedings volume.
  2. Mohsen Asgharzadeh, "Reflexivity revisited", arXiv preprint arXiv:1812.00830 (v1 December 2018, v7 revised 3 June 2025), DOI: 10.48550/arXiv.1812.00830, https://arxiv.org/abs/1812.00830
  3. Ryotaro Isobe and Shinya Kumashiro, "Reflexive modules over Arf local rings", Taiwanese Journal of Mathematics 28 (2024), no. 5, DOI: 10.11650/tjm/240605; preprint arXiv:2105.07184, https://arxiv.org/abs/2105.07184
  4. Victor D. Mendoza-Rubio and Victor H. Jorge-Pérez, "On modules whose dual is of finite Gorenstein dimension", Collectanea Mathematica 77 (2025), no. 1, pp. 233–255, DOI: 10.1007/s13348-025-00466-y; preprint arXiv:2312.06124, https://arxiv.org/abs/2312.06124
  5. Toshinori Kobayashi and Shinya Kumashiro, "The set of trace ideals of curve singularities", Israel Journal of Mathematics 265 (2024), no. 1, pp. 153–184, DOI: 10.1007/s11856-024-2661-6, https://doi.org/10.1007/s11856-024-2661-6
  6. Pietro Campochiaro, Marco D'Anna, and Francesco Strazzanti, "Some classes of one-dimensional rings characterized by their reflexive ideals", Ricerche di Matematica 75 (2026), no. 2, pp. 941–954 (published online 12 February 2026), DOI: 10.1007/s11587-026-01060-5, https://doi.org/10.1007/s11587-026-01060-5
  7. Mohsen Asgharzadeh and Elham Mahdavi, "Remarks on modules of finite projective dimension", arXiv preprint arXiv:2602.09899 (v1 10 February 2026, v2 revised 30 April 2026), DOI: 10.48550/arXiv.2602.09899, https://arxiv.org/abs/2602.09899