Gorenstein-projective versus semi-Gorenstein-projective modules: the open questions left by Ringel and Zhang (2020)

Ringel and Zhang (Algebra & Number Theory 14, 2020) constructed, over a 6-dimensional noncommutative local algebra $\Lambda(q)$, a module that is semi-Gorenstein-projective with semi-Gorenstein-projective dual but not reflexive, thereby proving that Auslander's three classical conditions for Gorenstein dimension zero are independent over artin algebras. Their paper closes with six open questions that jointly probe one objective: determine exactly when semi-Gorenstein-projectivity fails to force Gorenstein-projectivity, and how such failures distribute between the commutative and noncommutative worlds and between a ring and its opposite. The six questions ask: (1) whether a commutative ring can carry a module satisfying the Ext-vanishing conditions (G1) and (G2) but not reflexivity (G3); (2) whether finiteness of the class of Gorenstein-projective modules, or the condition that all Gorenstein-projective modules are projective, forces the algebra to be left weakly Gorenstein; (3) whether left weak Gorensteinness is left–right symmetric, a converse of which implies the Gorenstein symmetry conjecture; (4) whether the existence of a nonreflexive module whose module and dual are both semi-Gorenstein-projective is symmetric under passing to the opposite algebra, already for $\Lambda(q)$ itself; (5) whether a nonzero semi-Gorenstein-projective module with zero dual (a Nunke module) exists, equivalently the strong Nakayama conjecture for artin algebras; and (6) whether, over commutative artinian rings, the Jorgensen–Şega total-reflexivity conditions $(\mathrm{TR}_i)$ holding for all but finitely many $i$ force a module to be Gorenstein-projective. All six questions remain open in general, with substantial partial results recorded below.

Background

An artin algebra is a ring that is finitely generated as a module over a commutative artinian center; the standard example is a finite-dimensional algebra over a field. For such a ring $A$ one studies the category of finitely generated left $A$-modules, in which every module has finite length and a minimal projective resolution. A complete projective resolution of a module $M$ is a doubly infinite exact sequence $\cdots\to P_{-1}\to P_0\to P_1\to\cdots$ of projective modules that stays exact after applying $\operatorname{Hom}_A(-,A)$, with $M$ the image of $P_0\to P_1$; modules admitting one are called Gorenstein-projective (also known as totally reflexive modules or maximal Cohen–Macaulay modules), and they form the class $\mathrm{gp}(A)$. A weaker condition is semi-Gorenstein-projectivity: $\operatorname{Ext}^i_A(M,A)=0$ for all $i\ge 1$. Writing $M^*=\operatorname{Hom}_A(M,{}_AA)$ for the $A$-dual and $\varphi_M\colon M\to M^{**}$ for the canonical evaluation map, $M$ is torsionless if $\varphi_M$ is injective (equivalently, $M$ embeds in a projective module) and reflexive if $\varphi_M$ is an isomorphism. Auslander's modules of Gorenstein dimension zero are characterized by three classical conditions: (G1) $M$ is semi-Gorenstein-projective; (G2) $M^*$ is semi-Gorenstein-projective; (G3) $M$ is reflexive. Gorenstein-projective modules satisfy all three, and the conjunction (G1)+(G2)+(G3) is equivalent to Gorenstein-projectivity. An artin algebra $A$ is called left weakly Gorenstein if every semi-Gorenstein-projective left module is Gorenstein-projective, that is, the class $\perp A$ of semi-Gorenstein-projective modules coincides with $\mathrm{gp}(A)$; right weak Gorensteinness is the same property for the opposite algebra $A^{\mathrm{op}}$ (the ring with the multiplication reversed). The noncommutative left/right distinction is genuine, which is why the two one-sided notions are studied separately.

History of the independence problem. Whether conditions (G1)–(G3) are logically independent was asked explicitly by Avramov and Martsinkovsky. Jorgensen and Şega (2006) gave the first examples showing partial independence, over a commutative artinian algebra of dimension 8: modules satisfying (G1) and (G3) but not (G2), and modules satisfying (G2) and (G3) but not (G1); their work also produced the first semi-Gorenstein-projective module that is not Gorenstein-projective. The remaining combination, (G1)+(G2) without (G3), resisted commutative methods. Ringel and Zhang (2020) settled the general independence problem: they introduced the 6-dimensional local algebra $\Lambda(q)$ over a field $k$, generated by $x,y,z$ with relations $x^2$, $y^2$, $z^2$, $yz$, $xy+qyx$, $xz-zx$, $zy-zx$ (where $q\in k^*$), and proved that when the multiplicative order $o(q)$ of $q$ is infinite, the 3-dimensional local module $M(q)=\Lambda/\Lambda(x-qy)+\operatorname{soc}\Lambda$ satisfies (G1) and (G2) but is not torsionless, hence not reflexive. Their technical engine is the $\mho$-operator, which sends a module $M$ to the cokernel of a minimal left $\operatorname{add}(A)$-approximation (a map from $M$ into a finite direct sum of copies of $A$ through which every homomorphism $M\to A$ factors) and acts as a kind of inverse to the syzygy operator $\Omega$ (which sends a module to the kernel of a projective cover of it); the $\mho$-quiver, whose vertices are indecomposable non-projective modules, has connected components only of shapes $A_n$, $\tilde A_n$, $\mathbb{N}$, $-\mathbb{N}$, $\mathbb{Z}$, and the shape of a component detects homological behavior: modules of $\mho$-type $-\mathbb{N}$ are exactly the semi-Gorenstein-projective modules that are not Gorenstein-projective, so $A$ is left (respectively right) weakly Gorenstein if and only if it has no $\mho$-components of type $-\mathbb{N}$ (respectively $\mathbb{N}$).

Neighboring conditions and conjectures. For a minimal projective presentation $P_1\to P_0\to M\to 0$, the Auslander transpose $\operatorname{Tr}\,M$ is the cokernel of the dual map $P_0^*\to P_1^*$; $M$ is called $t$-torsionfree if $\operatorname{Ext}^i_A(\operatorname{Tr}\,M, A_A)=0$ for $1\le i\le t$ and $\infty$-torsionfree if this holds for all $i\ge 1$. Jorgensen and Şega formulated, for each integer $i\ne 0$, the condition $(\mathrm{TR}_i)$: $\operatorname{Ext}^i_A(M,A)=0$ for $i\ge 1$, and $\operatorname{Ext}^{-i}_A(\operatorname{Tr}\,M, A_A)=0$ for $i\le -1$. A module is totally (that is, Gorenstein-) projective exactly when it satisfies all conditions $(\mathrm{TR}_i)$. A Nunke module is a nonzero semi-Gorenstein-projective module with zero dual; the assertion that no Nunke module exists is the Nunke condition, which Colby and Fuller conjectured holds for every artin algebra under the name strong Nakayama conjecture. The strong Nakayama conjecture implies the generalized Nakayama conjecture (every simple module has some nonvanishing $\operatorname{Ext}^i(-,A)$) and is implied by the finitistic dimension conjecture, all classical open homological conjectures. Finally, the Gorenstein symmetry conjecture asserts that for an artin algebra, finite injective dimension of the left regular module forces finite (and then equal) injective dimension of the right regular module. Ringel and Zhang observed that if $_AA$ had injective dimension $\le d$ while $A_A$ had injective dimension $>d$, then $A$ would be right weakly Gorenstein but not left weakly Gorenstein; consequently, the statement that every right weakly Gorenstein artin algebra is left weakly Gorenstein implies the Gorenstein symmetry conjecture. Known positive results confine weak Gorensteinness to special classes: Yoshino (certain commutative noetherian rings, including all commutative artinian rings) and, in a generalization to arbitrary artin algebras, Beligiannis proved that finiteness of the number of isomorphism classes of indecomposable semi-Gorenstein-projective modules forces left weak Gorensteinness; Marczinzik proved the same for torsionless-finite algebras (those with only finitely many indecomposable torsionless modules up to isomorphism) and related representation-finite conditions (representation-finite means having only finitely many indecomposable modules at all). A modern offshoot is the class of virtually Gorenstein artin algebras, defined by the identity $(\mathrm{GP})^\perp={}^\perp(\mathrm{GI})$: the right orthogonal of the class of Gorenstein-projective modules equals the left orthogonal of the class of Gorenstein-injective modules, the injective-side analogues of the Gorenstein-projectives.

Problem Statement

The common objective of the six questions below (Ringel–Zhang, Section 9) is to determine, without restriction to special classes of algebras, exactly when semi-Gorenstein-projectivity fails to imply Gorenstein-projectivity, and how the resulting failures are distributed between the commutative and noncommutative settings and between an artin algebra and its opposite. Throughout, $A$ denotes an artin algebra or a commutative ring, all modules are finitely generated (left modules by default, right modules where a right module or the opposite algebra $A^{\mathrm{op}}$ is indicated), $M^*=\operatorname{Hom}_A(M,A)$ is the $A$-dual, and the following definitions apply: (G1) $M$ is semi-Gorenstein-projective, that is, $\operatorname{Ext}^i_A(M,A)=0$ for all $i\ge 1$; (G2) $M^*$ is semi-Gorenstein-projective; (G3) $M$ is reflexive, that is, the canonical evaluation map $M\to M^{**}$ is an isomorphism; a module is Gorenstein-projective exactly when it satisfies (G1), (G2), and (G3); $A$ is left (respectively right) weakly Gorenstein if every semi-Gorenstein-projective left (respectively right) $A$-module is Gorenstein-projective; $\Lambda(q)$ is the 6-dimensional algebra defined in the Background and $o(q)$ is the multiplicative order of $q$; for $i\ge 1$ the condition $(\mathrm{TR}_i)$ requires $\operatorname{Ext}^i_A(M,A)=0$, and for $i\le -1$ it requires $\operatorname{Ext}^{-i}_A(\operatorname{Tr}\,M,A_A)=0$, where $\operatorname{Tr}\,M$ is the Auslander transpose of $M$ (the cokernel of the dual of a minimal projective presentation of $M$); and a Nunke module is a nonzero semi-Gorenstein-projective module $M$ with $M^*=0$.

(1) [9.1] Does there exist a commutative ring $A$ with a module $M$ satisfying (G1) and (G2) but not (G3) — that is, a commutative example of a semi-Gorenstein-projective module with semi-Gorenstein-projective dual that is not reflexive? The known examples, including $M(q)$ over the noncommutative algebra $\Lambda(q)$ with $o(q)=\infty$, are all noncommutative.

(2) [9.2] (a) If the indecomposable Gorenstein-projective $A$-modules form only finitely many isomorphism classes, must $A$ be left weakly Gorenstein? (b) The weaker question: if all Gorenstein-projective $A$-modules are projective, must $A$ be left weakly Gorenstein?

(3) [9.3] Is every left weakly Gorenstein artin algebra also right weakly Gorenstein (equivalently, in terms of the $\mho$-quiver defined in the Background: can an algebra have $\mho$-components of type $\mathbb{N}$ without having any of type $-\mathbb{N}$)? The source further discusses the converse direction: it proves that the statement “every right weakly Gorenstein artin algebra is left weakly Gorenstein” would imply the Gorenstein symmetry conjecture, since $\operatorname{inj.dim}{}_AA\le d<\operatorname{inj.dim}A_A$ would make $A$ right weakly Gorenstein but not left weakly Gorenstein; this converse statement remains open.

(4) [9.4] If an artin algebra $A$ has a nonreflexive module $M$ such that both $M$ and $M^*$ are semi-Gorenstein-projective, must the same hold for $A^{\mathrm{op}}$? Even for $A=\Lambda(q)$ with $o(q)=\infty$ the answer is unknown: no right module of the form $m_\alpha\Lambda$ works (the modules $m_\alpha\Lambda$, introduced in the source paper, are reflexive as soon as they and their duals are semi-Gorenstein-projective), and by the Part II classification no right module of dimension at most 3 works, but right modules of arbitrary dimension remain unexcluded.

(5) [9.5] Does there exist a Nunke module, that is, a nonzero semi-Gorenstein-projective module $M$ with $M^*=0$? A negative answer for all artin algebras is precisely the strong Nakayama conjecture; such a module would be an extreme example satisfying (G1) and (G2) but not (G3).

(6) [9.6] Over a commutative artinian ring (the setting of Jorgensen and Şega), if a module satisfies the conditions $(\mathrm{TR}_i)$ for all but finitely many integers $i\ne 0$, must it be Gorenstein-projective? Over general (noncommutative) artin algebras the answer is no — any module with $M$ and $M^*$ semi-Gorenstein-projective satisfies $(\mathrm{TR}_i)$ for all $i\notin\{-1,-2\}$, so the $\Lambda(q)$-modules give counterexamples — but the commutative case is open.

The six questions are linked by the shared objective above, but each is posed as an independently refutable open problem: a proof or disproof of any one of (1)–(6) is an independently sufficient complete solution to this task, while the shared objective is the umbrella program that the six questions jointly probe. Item (2) lists a stronger question (2)(a) together with a weaker variant (2)(b) that (2)(a) implies, so item (2) is resolved by a proof or disproof of (2)(a), whereas a proof or disproof of the weaker variant (2)(b) alone does not resolve item (2) and counts only as partial progress.

References

1. Claus Michael Ringel and Pu Zhang, “Gorenstein-projective and semi-Gorenstein-projective modules”, Algebra & Number Theory 14 (2020), no. 1, 1–36. DOI: 10.2140/ant.2020.14.1, https://doi.org/10.2140/ant.2020.14.1 (arXiv:1808.01809). Primary source; Section 9 contains the open questions.
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