Virtually Gorenstein Algebras Overview

• Virtually Gorenstein algebras are associative structures where the categories of Gorenstein-projective and Gorenstein-injective modules exhibit a balanced duality, extending classical Gorenstein properties.
• They are characterized by cotorsion pairs, stable equivalences, and invariance under recollement, which facilitates the classification of homological phenomena in Artin and Noetherian settings.
• Applications include the study of cluster-tilting theory, finite Cohen–Macaulay type, and the validation of the Nakayama conjecture through relationships with dominant dimension and self-injectivity.

A virtually Gorenstein algebra is an associative algebra whose module category exhibits a generalized symmetry between Gorenstein-projective and Gorenstein-injective modules, extending many of the favorable properties of classical Gorenstein algebras. The concept appears in both the commutative and noncommutative settings—typically for Artin algebras, finite-dimensional algebras, and Noetherian rings—and serves as a robust framework for systematizing homological phenomena that fall short of full Gorensteinness yet retain significant duality properties relevant to dominant dimension, self-injectivity, and relative homological algebra. Virtual Gorensteinness plays pivotal roles in the structure theory of algebras, the representation theory of self-orthogonal modules, the behavior of recollements in triangulated categories, and as a unifying principle connecting cluster-tilting theory, Auslander correspondence, and canonical/defect categories.

1. Definitions and Characterizations

The definition of a virtually Gorenstein algebra varies slightly depending on context.

• For Artin algebras: An Artin algebra $A$ is virtually Gorenstein if

$$(\mathcal{G}\text{proj}\ A)^\perp = {}^\perp(\mathcal{G}\text{inj}\ A)$$

where $(\mathcal{G}\text{proj}\ A)^\perp$ is the class of $A$-modules $M$ for which $\operatorname{Ext}^i_A(G, M) = 0$ for all $i > 0$ and for all Gorenstein-projective $G$, and ${}^\perp(\mathcal{G}\text{inj}\ A)$ is defined dually with respect to Gorenstein-injective modules (Zareh-Khoshchehreh et al., 2014, Chen, 2017).

• For commutative Noetherian rings of finite Krull dimension: A ring $R$ is virtually Gorenstein if the two orthogonal classes defined by Gorenstein-projective (GP) and Gorenstein-injective (GI) modules coincide:

$\mathrm{GP}^- = {}^-\mathrm{GI}$

with \begin{align*} \mathrm{GP}^- & = { M \mid \operatorname{Ext}^i_R(Q, M) = 0,\ \forall i > 0,\ Q \ \mathrm{Gorenstein\text{-}proj} } \ {}^-\mathrm{GI} & = { M \mid \operatorname{Ext}^i_R(M, E) = 0,\ \forall i > 0,\ E \ \mathrm{Gorenstein\text{-}inj} } \end{align*} (Zareh-Khoshchehreh et al., 2014).

• For stable categories: A finite-dimensional algebra $A$ is virtually Gorenstein if the subcategory of compact objects in the stable category of Gorenstein-projective modules coincides with the finitely generated Gorenstein-projective modules, i.e.,

$(\mathrm{GP}(A))^c = \mathsf{Gp}(A)$

(Shen et al., 2023).

• For weakly Gorenstein (or nearly Gorenstein) algebras: The terms are sometimes used synonymously for algebras such that the class of modules with vanishing $\operatorname{Ext}_A^i(M, A)$ for all $i \geq 1$ coincides with the Gorenstein-projective modules (Marczinzik, 2016).

The following table summarizes the principal definitions:

Context	Virtually Gorenstein definition	Reference
Artin algebra	$$(\mathcal{G}\mathrm{proj})^\perp = {}^\perp(\mathcal{G}\mathrm{inj})$$	(Zareh-Khoshchehreh et al., 2014, Chen, 2017)
Noetherian ring	$\mathrm{GP}^- = {}^-\mathrm{GI}$	(Zareh-Khoshchehreh et al., 2014)
Stable category (triangulated)	$(\mathrm{GP}(A))^c = \mathsf{Gp}(A)$	(Shen et al., 2023)
Weakly Gorenstein	Semi-Gorenstein-projective = Gorenstein-projective	(Marczinzik, 2016)

Key equivalent characterizations include: the functor $\operatorname{Hom}_R(-,~)$ being right balanced by the pair $(\mathrm{GP}, \mathrm{GI})$; the coincidence of GP-pure and GI-copure exact sequences; and invariance under derived and stable equivalences (Zareh-Khoshchehreh et al., 2014, Shen et al., 2023). Every finite Krull-dimensional Gorenstein ring is virtually Gorenstein (Zareh-Khoshchehreh et al., 2014).

2. Homological Invariants and Structural Properties

Virtually Gorenstein algebras exhibit a balance between projective and injective behavior that extends many results valid for Gorenstein algebras. Homological invariants central to their classification and paper include:

• Decomposition Property: For an Artin algebra $A$, the condition that every Gorenstein-projective $A$-module is a direct sum of finitely generated ones is equivalent to $A$ being virtually Gorenstein of finite Cohen–Macaulay (CM) type (Beligiannis, 2013). The principal equivalence, established in Theorem 4.10, is:

$$\text{Every GP module is a sum of finitely generated ones} \iff A\ \text{is virtually Gorenstein and of finite CM type}$$

• Cotorsion Pair: The pair $$(\mathcal{G}\text{proj}, (\mathcal{G}\text{proj})^\perp)$$ is a cotorsion pair, and virtually Gorenstein algebras are precisely those for which this pair is complete and "balanced", meaning every module has both a Gorenstein-projective envelope and preenvelope (Chen, 2017).
• GP-pure Homology: Over a commutative Noetherian ring $R$ of finite Krull dimension, Gorenstein homology is GP-pure if and only if $R$ is virtually Gorenstein; that is, relative Ext functors computed via GP-resolutions and GI-resolutions agree (Zareh-Khoshchehreh et al., 2014).
• Stability under Recollement and Equivalence: Virtual Gorensteinness is invariant under derived and stable equivalences, and is preserved—under suitable conditions—by recollement structures of triangulated categories (Shen et al., 2023, Asadollahi et al., 2014). In the language of $n$-recollements, the property is transferred along the ladder: if the stable category of Gorenstein-projective $A$-modules admits an $n$-recollement relative to those of $B$ and $C$ (with $n\geq 2$), then $A$ is virtually Gorenstein if and only if $B$ and $C$ are virtually Gorenstein.

3. Interplay with Dominant Dimension, Self-Injectivity, and the Nakayama Conjecture

The connection between virtual Gorensteinness, dominant dimension, and self-injectivity is exemplified by results addressing the Nakayama conjecture. The conjecture asserts that an Artin algebra with infinite dominant dimension must be self-injective.

• If $A$ is an algebra of infinite dominant dimension with vanishing $\operatorname{Ext}_A^n(D(A),A) = 0$ for all $n \geq 1$, and $A$ is virtually Gorenstein, then $A$ is self-injective [(Chen et al., 5 Sep 2025), Theorem 3.1]. That is, for virtually Gorenstein $A$,

$$(\mathrm{domdim}\ A = \infty)\ \wedge\ \big(\forall n\geq 1,\ \operatorname{Ext}_A^n(D(A),A) = 0\big) \implies A\ \text{is self-injective}$$

• In module-theoretic terms, for a generator-cogenerator $M$ over $A$ with

$$\operatorname{Ext}_A^n(M \oplus \nu_A(M), M) = 0\quad \forall n \ge 1$$

and End$_A(M)$ virtually Gorenstein, one must have $M$ projective [(Chen et al., 5 Sep 2025), Theorem 3.2]. This verifies a strong form of the Nakayama conjecture in the context of virtually Gorenstein algebras and ties the property to the absence of "hidden" non-projective extensions under strong orthogonality.

• For strongly Morita algebras (endomorphism algebras of generators $M$ over self-injective $A$ with $(M) = (\nu_A M)$) that are virtually Gorenstein and of infinite dominant dimension, self-injectivity follows (Chen et al., 5 Sep 2025).

These results indicate that virtual Gorensteinness serves as a structural witness for robust dualities forced by high dominant dimension, blocking "unexpected" non-injectivity under natural self-orthogonality hypotheses.

4. Construction and Inheritance via Frobenius Extensions, Gluings, and Recollements

Virtual Gorensteinness is preserved and transferred under various algebraic operations:

• Frobenius Extensions: If $B \subseteq A$ is a Frobenius extension and $B$ is semisimple and virtually Gorenstein, so is $A$. In the split case, if $A$ is virtually Gorenstein, then so is $B$ [(Chen et al., 5 Sep 2025), Prop. 3.8]. This provides systematic methods for constructing new virtually Gorenstein algebras from known ones via extension theory.
• Gluing and Triangular Matrix Extensions: The property is preserved under simple gluing of algebras: if $A$, $B$ are Gorenstein (or virtually Gorenstein), their "simple gluing" algebra is Gorenstein (and thus virtually Gorenstein). The Gorenstein projective modules, singularity categories, and defect categories of the glued algebra can be constructed from those of the components (Lu, 2017, Wang, 2014). Similarly, for certain triangular matrix rings, the virtually Gorenstein property can be inherited under projectivity and homological constraints on bimodule structure (Wang, 2014).
• Derived and Stable Recollements: As discussed above, recollements and $n$-recollement ladders allow transfer and detection of virtual Gorensteinness via "local" properties and enable the reduction of the paper of this notion to block analysis (Shen et al., 2023, Asadollahi et al., 2014).

5. Relationships with Decomposition Properties and Finite Cohen-Macaulay Type

Virtually Gorenstein algebras of finite (Cohen-Macaulay) type are closely characterized by the decomposition property for Gorenstein-projective modules. This extends and generalizes classical results due to Auslander, Ringel-Tachikawa, Chen, and Yoshino (Beligiannis, 2013):

• For an Artin algebra $A$:

$$\text{Every GP module is a direct sum of finitely generated GPs} \iff A\ \text{is virtually Gorenstein and of finite CM-type}$$

and, in this setting, the stable category of Gorenstein-projective modules is "phantomless" and the associated functor category is Frobenius.

• Cluster tilting: If a Gorenstein-projective module $T$ is cluster-tilting, the stable category of Gorenstein-projective modules modulo projectives is equivalent to the module category of the endomorphism algebra $\mathrm{End}_A(T)^{\mathrm{op}}$, adding a strong combinatorial structure in terms of cluster categories (Beligiannis, 2013).

This establishes virtually Gorenstein algebras as a natural generalization of classical finite CM-type theory, supporting categorical and homological rigidity analogous to algebras of finite representation type.

6. Examples, Counterexamples, and Applications

Several classes of algebras serve as canonical examples or counterexamples:

• Gorenstein and Quasi-Gorenstein Rings: Every Gorenstein algebra is virtually Gorenstein (Zareh-Khoshchehreh et al., 2014). Quasi-Gorenstein rings also frequently display this property in applications to Rees algebras and blowup algebras (Iai, 29 May 2024).
• Radical Square Zero Rings and Group Algebras: Many such algebras are virtually Gorenstein; for example, radical square zero algebras and certain finite group rings.
• Mackey Algebras: For the integral Mackey algebra $\mu_\mathbb{Z}(G)$, the Gorenstein (and hence virtually Gorenstein) property is equivalent to the group order $|G|$ being square-free. When $|G|$ is not square-free, $\mu_\mathbb{Z}(G)$ is not virtually Gorenstein (Dell'Ambrogio et al., 2017).
• Almost Gorenstein and Rees Algebras: While almost Gorenstein rings (in the sense of embeddings into canonical modules with Ulrich cokernels) are not always virtually Gorenstein, in two-dimensional regular local rings with integrally closed ideals, Rees algebras of such ideals can be almost Gorenstein, and in certain circumstances this aligns with virtually Gorenstein behavior (Goto et al., 2015, Goto et al., 2015).
• Non-examples: Certain corner cases (e.g., $R = k[x,y,z]/(x^2, yz, y^2 - xz, z^2 - yx)$) are explicitly non-virtually Gorenstein, as witnessed by imbalances in the Ext vanishing conditions (Zareh-Khoshchehreh et al., 2014).

Applications encompass the classification of singularities, the paper of cluster categories, verification of the Nakayama conjecture in special classes, analysis of syzygies in gendo-symmetric algebras (Marczinzik, 2016), and the transfer of duality properties in commutative and noncommutative settings.

7. Open Problems, Generalizations, and Future Research

The following open directions and problems remain major themes:

• CM-finiteness vs. Virtual Gorensteinness: Does finite Cohen-Macaulay type always imply an algebra is virtually Gorenstein? Conversely, does virtual Gorensteinness, together with various finiteness criteria, imply CM-finiteness? [(Chen, 2017), App. C].
• Classification of Non-virtually Gorenstein Algebras: Systematic construction and classification of non-virtually Gorenstein algebras, e.g., via gendo-symmetric constructions or quantum exterior algebras, remains incomplete (Marczinzik, 2016).
• Generalization to Triangulated Categories and Higher Structures: The role of virtual Gorensteinness in the broader setting of relative Auslander algebras, Calabi–Yau categories, and higher homological algebra (e.g., in $n$-recollement theory) is an active area (Shen et al., 2023).
• Dominant Dimension and Homological Invariants: The precise relationship between dominant (and codominant) dimension, virtual Gorensteinness, and other homological invariants (finitistic dimension, singularity, and defect categories) is the subject of ongoing research (Marczinzik, 2016, Cruz et al., 2021).
• Extension to Non-associative and Non-Noetherian Settings: Extending these frameworks to a broader class of rings and algebras, and analyzing the corresponding implications for pure and relative homology, cluster-tilting, and derived equivalences (Khovanskii et al., 2021).

The continued interplay between categorical, homological, and structural aspects of virtually Gorenstein algebras ensures their centrality in both representation theory and commutative algebra.