Gorenstein Projective DG Modules

• Gorenstein projective DG modules are differential graded modules defined via totally acyclic DG projective resolutions, extending classical Gorenstein projectivity to derived settings.
• They facilitate the study of G-dimension, singularity categories, and duality theories in non-regular rings through categorical embeddings and model structures.
• Their framework underpins applications in Cohen–Macaulay representation theory and noncommutative geometry while addressing open problems in homological algebra.

Gorenstein projective DG modules are a differential graded (DG) generalization of the classical notion of Gorenstein projective modules, tightly linking triangulated and relative homological algebra to the structure and duality theory of non-regular rings, particularly in DG and singular contexts. The paper of these modules is motivated by the need to understand and classify objects in derived and stable categories that retain good projectivity properties, especially when projective and injective replacement functors are inadequate or unavailable. The DG (or differential) enhancement brings a stratified perspective and allows for the transfer of deep module-theoretic invariants into flexible derived settings.

1. Core Definitions and Foundational Properties

A Gorenstein projective DG module over a DG ring $A$ is typically defined as a DG module that admits a totally acyclic DG projective resolution. Specifically, an $A$-DG-module $M$ is Gorenstein projective if there exists a totally acyclic complex of DG projective modules

$\cdots \to P_1 \to P_0 \to P_{-1} \to \cdots$

such that $M \cong \operatorname{Ker}(P_0 \to P_{-1})$ and the total complex remains acyclic under application of the $\operatorname{Hom}_A(-, Q)$ functor for every DG projective $Q$.

For ordinary differential modules—that is, modules over $R[\varepsilon]$ where $\varepsilon^2=0$—a differential module $(M, d)$ is Gorenstein projective if and only if the underlying $R$-module $M$ is Gorenstein projective. Thus, the DG generalization conceptually extends the class of Gorenstein projective modules rigidly along the grading and differential, while echoing classical projectivity in the underlying algebra (Wei, 2012).

In the derived category of a commutative noetherian DG ring, reflexivity is formulated via the canonical map

$$X \longrightarrow \mathbb{R}\operatorname{Hom}_A(\mathbb{R}\operatorname{Hom}_A(X, A), A)$$

with DG Gorenstein projectives characterized as reflexive modules for which the supremum of $\mathbb{R}\operatorname{Hom}_A(X, A)$ equals $-\sup X$, with further specialization to modules $G$ for which this value is $0$ (Hu et al., 2023).

2. Categorical Embeddings, Functors, and Defect Categories

A central role is played by the relationship between totally acyclic complexes and singularity categories. Over a left noetherian ring $A$, the category of totally acyclic complexes of projective modules $K_{\operatorname{tac}}(\operatorname{proj}A)$ embeds fully faithfully (via brutal truncation at degree zero) into the singularity category

$$D_{sg}(A) = K^{-,b}(\operatorname{proj}A) / K^b(\operatorname{proj}A)$$

where the image corresponds to objects arising via Gorenstein projective resolutions. The triangle functor is dense if and only if $A$ is Gorenstein, leading to the Gorenstein defect category

$D^b(A) = D_{sg}(A) / \operatorname{Im}B_{proj}A,$

which measures the “gap” between all singularities and those captured by totally acyclic complexes (Bergh et al., 2012). For DG rings, these relationships generalize, providing a context for measuring how far a DG ring or DG module is from being Gorenstein in the triangulated sense.

3. Dimension Theory and Homological Invariants

The notion of G-dimension for DG-modules generalizes the Auslander–Bridger invariant to DG settings. For a reflexive DG-module $X\in D^{\mathrm{f}}(A)$,

$$\operatorname{G\text{-}dim}_A X = \sup\mathbb{R}\operatorname{Hom}_A(X, A)$$

and the global G-dimension of $A$ is

$$\operatorname{G\text{-}gldim}A = \sup\{\operatorname{G\text{-}dim}_A X + \inf X \mid X\in D^{\mathrm{f}}(A)\}.$$

The critical result is that finiteness of the G-dimension for objects such as the residue field $k$ characterizes the local Gorenstein property of $A$, paralleling the classical situation for local rings (Hu et al., 2023). For modules in the G-class $\mathcal{G}_0$ (those with G-dimension zero up to shift), these coincide with the DG-analogues of Gorenstein projective modules, and their stable categories yield singularity categories in the sense of the Buchweitz–Happel theorem generalized to the DG setting.

Furthermore, the “level” of a DG module (the minimal number of mapping cones over DG Gorenstein projectives needed to build it) satisfies

$$\operatorname{level}_{\mathcal{G}}(M) = \operatorname{Gpd}_A M + 1,$$

for modules $M$ of finite DG Gorenstein projective dimension, giving a direct quantitative link between generation time in the derived category and homological dimension (Awadalla et al., 2021).

4. Structural Models: Cotorsion Pairs and Recollement

The framework for Gorenstein projective DG modules naturally admits abelian model structures and cotorsion pairs. For a DG ring $A$ with appropriate finiteness assumptions (e.g., $R$ noetherian and with finite global dimension), the class of DG-modules of bounded Gorenstein projective dimension arises as the left half of a complete cotorsion pair, mirroring the module category structure. This arrangement is compatible with Hovey’s correspondence, resulting in model structures where cofibrant objects are DG Gorenstein projectives and the cotorsion pair (GP, W) is cogenerated by a set (Pérez, 2012, Ren, 2017).

In more elaborate settings, recollement structures and TTF triples involving categories of Gorenstein-projective DG modules can be “lifted” from the bounded and finitely generated contexts to larger ambient DG module categories via homological ring epimorphisms, as detailed for upper triangular matrix algebras and their derived categories (Gao et al., 2022).

5. Base Change, Extensions, and Transfer

The transferability of Gorenstein projectivity and other homological properties across extensions is a prominent theme. For a Frobenius extension $R\subset A$, a module is Gorenstein projective over $A$ if and only if its underlying $R$-module is Gorenstein projective, provided the extension is left-Gorenstein or separable; this framework directly encompasses DG modules viewed as complexes (Ren, 2017). For tensor rings $T_R(M)$, Gorenstein projective $T_R(M)$-modules are characterized categorically: $$\mathcal{GP}(T_R(M)) = \{ (X,u) \mid u: M\otimes_R X \to X\ \text{is monomorphic},\ \operatorname{coker}(u)\in\mathcal{GP}(R)\},$$ indicating that the projectivity property “lifts” provided an appropriate monomorphism structure and the cokernel carries the original property (Di et al., 30 Apr 2025). This applies to various settings, such as trivial extensions and Morita context rings.

6. Duality, Rigidity, and Silting Theory

Gorenstein projective DG modules inhabit a landscape intertwined with rigidity, duality, and silting theory. In the DG context, new characterizations arise via Auslander categories and duality: for a DG module $M$, DG Gorenstein projectivity is reflected by the conditions that $\operatorname{Hom}_A(M, E)\in\mathcal{B}_{DG}(A)$ (where $\mathcal{B}_{DG}(A)$ is the DG-Auslander category for a suitable dualizing complex) and that certain derived Ext vanishing properties hold for all DG projectives (Wu et al., 2015). There is a direct parallel to silting complexes, particularly 2-term Gorenstein silting complexes which are in bijection with module-theoretic Gorenstein silting modules, providing explicit generators for t-structures and torsion pairs in the derived module categories (Gao et al., 2021).

In stable categories, total reflexivity is characterized by duality: over generically Gorenstein rings, a complex is acyclic if and only if its dual is acyclic, and DG modules satisfying this reflexivity condition can be recovered as totally reflexive objects in the stable category (Yoshino, 2018).

7. Applications and Open Problems

The development of Gorenstein projective DG modules facilitates the paper of singularity categories, Cohen–Macaulay representation theory, and the structure of derived and stable categories for non-Gorenstein and singular rings. The Gorenstein defect category serves as a new invariant for quantifying non-Gorenstein behavior (Bergh et al., 2012), and approaches using eventually homological isomorphisms enable the transfer of categorical and conjectural properties between module categories, with implications for classical problems such as the Auslander–Reiten and Gorenstein symmetry conjectures (Qin, 2022).

Current open directions include the precise classification of DG rings or DG algebras whose associated defect categories are “small” or zero-dimensional, the model-theoretic behavior of Gorenstein projective (and flat/injective) DG modules under more general extension and base change, the explicit realization of cotorsion pairs and stable categories in broader DG settings, and the categorical role of these objects in noncommutative geometry and higher representation theory.

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Gorenstein projective DG modules thus unify and refine classical and derived approaches to projectivity, acyclicity, and duality, providing robust categorical and homological tools across a wide spectrum of modern algebra and geometry (Bergh et al., 2012, Wei, 2012, Pérez, 2012, Ren, 2017, Awadalla et al., 2021, Gao et al., 2021, Hu et al., 2023, Levins et al., 21 Feb 2025, Di et al., 30 Apr 2025).