Ramras' 1972 problem asks for necessary and sufficient conditions on a commutative noetherian local ring RR under which every finitely generated reflexive RR-module (a module MM with M≅M∗∗M\cong M^{**}, where M∗=Hom⁡R(M,R)M^{*}=\operatorname{Hom}_R(M,R)) is free. Asgharzadeh's Reflexivity revisited' (2018, v7 2025) records that the problem is settled only for quasi-reduced rings, where the answer is regular of dimension at most two', and for rings with strictly increasing Betti numbers, which can be singular artinian rings on which all reflexives are nevertheless free. The full-generality problem remains open.

Background

For a local ring (R,m)(R,\mathfrak{m}), the dual of an RR-module MM 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^{**} is an isomorphism. Free modules are always reflexive, so the question is when the converse holds for finitely generated modules. The question is subtle even over very small rings: Lam's textbook `Lectures on Modules and Rings' remarks that over R=k[X,Y]/(X,Y)2R=k[X,Y]/(X,Y)^2 the only obvious reflexive modules are the free ones, while over the artinian Gorenstein ring k[x]/(x2)k[x]/(x^2) the residue field kk is already a nonfree reflexive module, since k∗=Hom⁡R(k,R)k^{*}=\operatorname{Hom}_R(k,R) is the socle (the annihilator of the maximal ideal) (x)/(x2)≅k(x)/(x^2)\cong k and k∗∗≅kk^{**}\cong k.

Three classical anchors delimit the problem. First, over a regular local ring (a local ring whose maximal ideal is generated by dim⁡R\operatorname{dim}R elements, where dim⁡R\operatorname{dim}R is the Krull dimension of RR; equivalently a ring of finite global dimension equal to its dimension) of dimension at most two, every finitely generated reflexive module is a second syzygy (the kernel appearing two steps into a free resolution of a module) and hence free; Vasconcelos' 1968 study of reflexive modules over Gorenstein rings (a Gorenstein local ring is one of finite injective dimension as a module over itself) placed the question inside homological module theory. Second, Ramras posed the general problem in the problem list of the 1971 Park City ring theory conference (published 1972, page 380) and proved that over a BNSI ring — one for which the Betti numbers βi(M)=dim⁡kTor⁡iR(k,M)\beta_i(M)=\dim_k\operatorname{Tor}_i^R(k,M) of every non-free module MM satisfy βi(M)>βi−1(M)\beta_i(M)>\beta_{i-1}(M) for all i>1i>1 — every finitely generated reflexive module is free; this applies to k[X,Y]/(X,Y)2k[X,Y]/(X,Y)^2 and answers Lam's remark nontrivially. Third, Treger's 1978 work on reflexive modules showed, among other results, that over complete normal local rings (normal rings complete with respect to the topology defined by powers of the maximal ideal) the freeness of all reflexive modules of rank at most two already forces regularity, so any positive class of rings on which all reflexives are free is constrained.

Asgharzadeh's `Reflexivity revisited' (arXiv:1812.00830, latest version 2025) surveys and extends this line. Its Proposition 4.22 answers Ramras' problem for local quasi-reduced rings — rings satisfying Serre's condition (S1)(S_1) (every associated prime of RR is minimal; equivalently depth Rp≥min⁡(1,dim⁡Rp)R_{\mathfrak{p}}\geq\min(1,\dim R_{\mathfrak{p}}) for all primes p\mathfrak{p}) that are generically Gorenstein (RpR_{\mathfrak{p}} is Gorenstein for each minimal prime p\mathfrak{p}; this includes all reduced rings with Gorenstein total ring of fractions and all artinian Gorenstein rings): over such a ring, every finitely generated reflexive module is free if and only if RR is regular of dimension at most two. The paper also introduces eventually BNSI rings (those for which the Betti numbers of every non-free module are strictly increasing from some point on) and proves that every finitely generated totally reflexive module over such a ring is free (a totally reflexive module is one with Ext⁡Ri(M,R)=Ext⁡Ri(M∗,R)=0\operatorname{Ext}_R^i(M,R)=\operatorname{Ext}_R^i(M^{*},R)=0 for all i>0i>0); its Example 4.10 shows that total reflexivity cannot be replaced by reflexivity here, since the eventually BNSI ring A/fnA/f\mathfrak{n} (for (A,n)(A,\mathfrak{n}) a regular local ring of dimension greater than one and f∈nf\in\mathfrak{n} nonzero) carries the nonfree reflexive module fRfR. The two known families of rings on which all reflexives are free genuinely diverge: the quasi-reduced answer forces regularity and dimension at most two, while singular artinian non-Gorenstein BNSI rings such as k[X,Y]/(X,Y)2k[X,Y]/(X,Y)^2 have all reflexives free. Hence no naive guess (regular of dimension at most two, artinian, Gorenstein, etc.) gives the general criterion; a full answer must blend dimension, singularity, and Betti-number growth. For general artinian non-Gorenstein rings the picture includes a further wrinkle recorded in the same paper: the Cohen–Macaulay type type⁡(R)=dim⁡kExt⁡Rdim⁡R(k,R)\operatorname{type}(R)=\dim_k\operatorname{Ext}_R^{\dim R}(k,R) controls the growth of iterated duals ℓ(Mn∗)\ell(M^{n*}) (here ℓ\ell is length), and iterated duals of nonfree modules need not be reflexive at all.

Problem Statement

Give necessary and sufficient conditions, in intrinsic ring-theoretic or homological terms, for a commutative noetherian local ring RR to have the property that every finitely generated reflexive RR-module is free (Ramras' Problem 1.1, posed in 1972). The classification is sought for all commutative noetherian local rings, with no restriction to the quasi-reduced, Gorenstein, artinian, or Betti-number-monotone classes; the two known extremes are that over local quasi-reduced rings the property holds exactly for regular rings of dimension at most two, while over singular artinian BNSI rings such as k[X,Y]/(X,Y)2k[X,Y]/(X,Y)^2 (which are not quasi-reduced) it nevertheless holds. Equivalently: determine exactly which local rings satisfy the property.

References

  1. Mohsen Asgharzadeh, Reflexivity revisited, arXiv:1812.00830 (2018; latest version v7, 2025; preprint). zbMATH 900500997. https://arxiv.org/abs/1812.00830
  2. Mark Ramras, Betti numbers and reflexive modules, in: Ring theory (Proc. Conf., Park City, Utah, 1971), R. Gordon (ed.), Academic Press, New York, 1972, pp. 297–308 (Ramras' problem is posed at p. 380 of the volume). zbMATH 3373890. https://zbmath.org/3373890
  3. W. V. Vasconcelos, Reflexive modules over Gorenstein rings, Proceedings of the American Mathematical Society 19 (1968), 1349–1355. DOI 10.1090/s0002-9939-1968-0237480-2. https://doi.org/10.1090/s0002-9939-1968-0237480-2
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