Coincidence of injective dimension, eventual Ext-vanishing against $D(\Lambda^{op})$, and projective dimensions of the minimal injective resolution for an arbitrary Artin algebra (Li's Conjecture 3.16)

For a virtually Gorenstein Artin algebra $\Lambda$, Li (2026) proved that three invariants coincide: the injective dimension $\mathrm{id}_\Lambda(\Lambda)$, the eventual vanishing degree of $\mathrm{Ext}_\Lambda^{n+l}(D(\Lambda^{op}),\Lambda)$, and the supremum of the projective dimensions of the terms $E_i$ in the minimal injective resolution of $\Lambda$. Li's Conjecture 3.16 asserts the same identity for an arbitrary Artin algebra. The conjecture remains open, and it implies the Gorenstein Symmetry Conjecture, the Nakayama conjecture, Tachikawa's First Conjecture, and (as a generalized extension) the Auslander–Gorenstein Conjecture — all longstanding open problems in the representation theory of algebras.

Background

An Artin algebra is a ring $\Lambda$ that is finitely generated as a module over a commutative artinian center ring; equivalently for most purposes, a finite-dimensional algebra over a field, with finitely many isomorphism classes of simple modules. One studies its right modules, their projective dimension $\mathrm{pd}$ and injective dimension $\mathrm{id}$, and $\mathrm{Ext}$ groups. The right regular module $\Lambda_\Lambda$ embeds in its injective envelope, and iterating gives the minimal injective resolution $0 \to \Lambda \xrightarrow{h_0} E_0 \xrightarrow{h_1} E_1 \xrightarrow{h_2} \cdots$ whose terms $E_i$ are finite direct sums of indecomposable injectives (there are only finitely many indecomposable injectives, one per simple module). The standard duality $D=\mathrm{Hom}_R(-,E)$ between right and left finite-length modules sends $\Lambda^{op}$ to $D(\Lambda^{op})$, the direct sum of all indecomposable injective right modules, i.e. the minimal injective cogenerator. An Artin algebra is self-injective (quasi-Frobenius) when $\Lambda$ is injective as a module over itself, and Gorenstein (in Iwanaga's sense) when both $\mathrm{id}_\Lambda(\Lambda)$ and $\mathrm{id}_{\Lambda^{op}}(\Lambda^{op})$ are finite. A web of homological conjectures about Artin algebras, open since the 1950s–1990s, can be phrased through this resolution. The Nakayama conjecture (Nakayama 1958, in the equivalent form due to Müller) asserts that if every $E_i$ is projective then $\Lambda$ is self-injective. The generalized Nakayama conjecture (Auslander–Reiten 1975) asserts that every indecomposable injective occurs as a direct summand of some $E_i$. The Auslander–Gorenstein Conjecture (Auslander–Reiten 1994) asserts that $\mathrm{pd}(E_i)\le i$ for all $i$ forces $\Lambda$ Gorenstein. Tachikawa's First Conjecture (Tachikawa 1973) asserts that $\mathrm{Ext}^i_\Lambda(D(\Lambda^{op}),\Lambda)=0$ for all $i\ge 1$ forces $\Lambda$ self-injective. The Gorenstein Symmetry Conjecture (Auslander–Reiten 1991) asserts that $\mathrm{id}_\Lambda(\Lambda)<\infty$ implies $\mathrm{id}_{\Lambda^{op}}(\Lambda^{op})<\infty$. The Finitistic Dimension Conjecture asserts that the supremum of the projective dimensions of all finitely generated modules is finite; the generalized Nakayama conjecture, and with it the Nakayama conjecture, is a formal consequence of it. All of these conjectures remain open in general. Gorenstein homological algebra refines this picture. A module is Gorenstein projective when it is the syzygy of a complete projective resolution — an exact complex of projectives that stays exact after applying $\mathrm{Hom}(-,Q)$ for every projective $Q$ — and semi-Gorenstein-projective when merely $\mathrm{Ext}^i_\Lambda(M,\Lambda)=0$ for all $i\ge 1$; the second class can be strictly larger. Ringel and Zhang (2020) called an Artin algebra (left) weakly Gorenstein when every finitely generated semi-Gorenstein-projective module is Gorenstein projective, and asked (Question 9.2) whether this holds when all finitely generated Gorenstein projective modules are projective. Beligiannis and Reiten (2007) called $\Lambda$ virtually Gorenstein when the classes $(\mathcal{GP})^{\perp}$ and ${}^{\perp}\mathcal{GI}$ coincide (modules orthogonal to all Gorenstein projectives, respectively to all Gorenstein injectives, in Ext-degree one); this is a common enlargement of the class of Gorenstein algebras, closed under passing to the opposite algebra. Li (2026) proved that a virtually Gorenstein Artin algebra is weakly Gorenstein in the strong sense that $\mathrm{Ext}^i_\Lambda(M,\Lambda)=0$ for all $i\ge 1$ already forces $M$ Gorenstein projective (answering the Ringel–Zhang question for this class), and established (Theorem 3.14) that for virtually Gorenstein $\Lambda$ the five quantities $\mathrm{id}_\Lambda(\Lambda)$, $\mathrm{id}_{\Lambda^{op}}(\Lambda^{op})$, the eventual Ext-vanishing bound $\inf\{n\mid \mathrm{Ext}^{n+l}_\Lambda(D(\Lambda^{op}),\Lambda)=0\ \forall l\ge 1\}$, its opposite-side analogue, and $\sup\{\mathrm{pd}_\Lambda(E_i)\mid i\ge 0\}$ (over the terms of the minimal injective resolution) are identical. The proof uses Holm's result that finite Gorenstein injective dimension of $\Lambda$ agrees with the ordinary one, and the Strong Nakayama property for virtually Gorenstein algebras. Notably, the equality of the two Ext-vanishing bounds with each other relies only on the standard duality $D$ and therefore holds for every Artin algebra — this is why the conjecture below subsumes the Gorenstein Symmetry Conjecture. The proof of the remaining identifications, however, genuinely uses the virtually Gorenstein hypothesis, and Li closes the paper by posing the conjecture for arbitrary Artin algebras.

Problem Statement

Let $\Lambda$ be an arbitrary Artin algebra (no virtual-Gorenstein or Gorenstein hypothesis), let $0 \to \Lambda \xrightarrow{h_0} E_0 \xrightarrow{h_1} E_1 \xrightarrow{h_2} \cdots$ be a minimal injective resolution of $\Lambda$ as a right module over itself, let $D$ be the standard duality between right and left finite-length $\Lambda$-modules, and let $D(\Lambda^{op})$ denote the minimal injective cogenerator of the module category. Prove, or refute by an explicit counterexample, that the following three quantities (each valued in $\mathbb{N} \cup \{\infty\}$, with the conventions that the infimum in quantity 2 is $0$ when all the relevant Ext groups vanish and $\infty$ when no such $n$ exists, and that the supremum in quantity 3 is $\infty$ when the projective dimensions are unbounded or some $\mathrm{pd}_\Lambda(E_i)$ is infinite) are identical:

1. $\mathrm{id}_\Lambda(\Lambda)$, the injective dimension of $\Lambda$ as a right $\Lambda$-module;
2. $\inf\{n \in \mathbb{N} \mid \mathrm{Ext}^{n+l}_\Lambda(D(\Lambda^{op}),\Lambda) = 0 \text{ for all } l \ge 1\}$;
3. $\sup\{\mathrm{pd}_\Lambda(E_i) \mid i \ge 0\}$, where the index runs over the terms $E_0, E_1, E_2, \ldots$ of the resolution above.

This is Conjecture 3.16 of Li (arXiv:2608.10049, 2026), where the identity is proved for virtually Gorenstein Artin algebras and posed for arbitrary Artin algebras.

References

1. Weiqing Li, Virtually Gorenstein Artin algebras are weakly Gorenstein, preprint (2026), arXiv:2608.10049, DOI 10.48550/arXiv.2608.10049, https://arxiv.org/abs/2608.10049
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