Projectively Coresolved Gorenstein Flat Modules

• Projectively coresolved Gorenstein flat modules are defined as syzygies in totally acyclic complexes of projectives that remain acyclic upon tensoring with any injective module.
• They bridge the gap between Gorenstein flat and Gorenstein projective modules and play a crucial role in relative homological algebra, cotorsion theory, and model structures.
• Their strong closure and stability properties enable precise computation of homological dimensions and the construction of robust Frobenius and triangulated category frameworks.

A projectively coresolved Gorenstein flat module (PGF module) is a module-theoretic invariant central to relative and Gorenstein homological algebra. These modules were introduced by Šaroch and Šťovíček to interpolate between the notions of Gorenstein flat and Gorenstein projective modules and to provide a stable, resolving class for applications in cotorsion theory, model structures, and in various relative contexts including tensor rings, functor categories, and FP-n-generalizations. The defining feature of a PGF module is its occurrence as a syzygy (cycle) in a totally acyclic complex of projective modules that remains acyclic after tensoring with any injective module. This imposes both a projective and a flat-detectable acyclicity, leading to robust closure and stability properties and making the class an essential building block for modern homological constructions (Kaperonis et al., 2023, Stergiopoulou, 2022, Dalezios et al., 2022, Maaouy, 2023).

1. Definitions and Characterizations

For a ring $R$, a left $R$-module $M$ is projectively coresolved Gorenstein flat (PGF) if there exists an exact complex of projectives

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

with $M \cong \ker(P_0 \to P_{-1})$, such that for every injective right $R$-module $I$, the complex $I \otimes_R P_\bullet$ remains exact (Kaperonis et al., 2023, Stergiopoulou, 2022, Dalezios et al., 2022).

Equivalent characterizations include:

• Existence of a right-bounded projective resolution $0 \to M \to P^0 \to P^1 \to \cdots$ remaining exact under $I \otimes_R -$ for every injective $I$.
• For all injective $I$ and $i > 0$, $\mathrm{Tor}_i^R(I, M) = 0$.
• For any such $P_\bullet$, the syzygies (all cycles) are again PGF modules—an important closure property (Stergiopoulou, 2022).
• In functor categories: an additive functor $F: \mathcal{Q} \rightarrow R\text{-Mod}$ is PGF if and only if for all $q \in \mathcal{Q}$, $F(q)$ is PGF as an $R$-module (Di et al., 2022).

PGF modules are strictly contained in both the Gorenstein flat and Gorenstein projective classes: $$\operatorname{PGF}(R) \subseteq \operatorname{GFlat}(R) \cap \operatorname{GProj}(R)$$ (Kaperonis et al., 2023).

2. Homological Dimensions and Cotorsion Theory

The PGF-dimension of an $R$-module $M$ is the infimum $n \geq 0$ such that there exists an exact sequence

$$0 \to G_n \to G_{n-1} \to \cdots \to G_0 \to M \to 0,$$

with all $G_i \in \operatorname{PGF}(R)$. For modules with finite projective dimension, PGF-dimension coincides with projective dimension; for those with finite Gorenstein flat dimension, there are sharp comparison criteria (see below) (Maaouy, 2023, Dalezios et al., 2022, Stergiopoulou, 2022).

The left global PGF-dimension of $R$ is given by:

$\operatorname{PGF\mbox{-}gl.dim}(R) = \sup\{\operatorname{PGF\mbox{-}dim}_R(M) : M \in R\mbox{-}\mathrm{Mod}\}.$

A central theorem states $\operatorname{PGF\mbox{-}gl.dim}(R) = \operatorname{Ggl.dim}(R)$ (the global Gorenstein projective dimension) for arbitrary rings (Maaouy, 2023, Dalezios et al., 2022).

The pair $$(\operatorname{PGF}(R), \operatorname{PGF}(R)^\perp)$$ forms a complete hereditary cotorsion pair, and, under finiteness or perfection conditions, PGF modules are special precovering, which ensures the existence of PGF-covers for all modules (Becerril, 15 Mar 2024, Maaouy, 2023, Stergiopoulou, 2022).

3. Stability, Closure Properties, and Model Structures

PGF modules possess strong stability and closure properties:

• Closed under direct sums, direct summands, extensions, and kernels of epimorphisms (Kaperonis et al., 2023, Stergiopoulou, 2022, Dalezios et al., 2022).
• Stable under iteration in the “very Gorenstein process”: any syzygy in an exact complex of PGF modules, with each such complex remaining acyclic under tensoring with injectives, is itself PGF (Stergiopoulou, 2022, Kaperonis et al., 2023).
• The PGF dimension can be characterized using “strongly $n$-PGF” modules, i.e., as direct summands in short exact self-extensions by modules of bounded projective dimension and controlled $\operatorname{Tor}$-vanishing (Stergiopoulou, 2022).

In abelian model category contexts, PGF modules constitute cofibrant objects in Hovey triples, and the associated homotopy categories are triangulated equivalent to stable categories of modules modulo projectives (Maaouy, 2023, Dalezios et al., 2022).

4. Relations to Other Gorenstein-type Modules and Ring Conditions

For general rings,

• $\operatorname{PGF}(R) = \operatorname{DP}(R)$ (Ding projective) if and only if every Ding projective module is Gorenstein flat, which is always fulfilled when $R$ is coherent or of finite weak Gorenstein global dimension (Iacob, 2020).
• $\operatorname{PGF}(R) = \operatorname{GProj}(R)$ if and only if every Gorenstein projective module is Gorenstein flat, which is the case for coherent, left $n$-perfect rings (including Iwanaga–Gorenstein, Artin, or Ding–Chen rings) (Iacob, 2020, Kaperonis et al., 2023).

For rings that are left $n$-perfect ($\operatorname{fd}(F) \leq n$ for all flat $F$), every Gorenstein flat module has PGF-dimension at most $n$, and the cotorsion pair $$(\operatorname{PGF}(R), \operatorname{PGF}(R)^\perp)$$ is complete and hereditary (Kaperonis et al., 2023, Estrada et al., 2020, Becerril, 15 Mar 2024). Over Gorenstein, Ding–Chen, or commutative Noetherian rings of finite Krull dimension, the classes $\operatorname{GFlat}(R)$, $\operatorname{GProj}(R)$, and $\operatorname{PGF}(R)$ coincide (Kaperonis et al., 2023, Iacob, 2020).

5. Applications and Generalizations

Tensor rings and extensions: For a tensor ring $T_R(M)$ associated to an $N$-nilpotent bimodule $M$, $(X, u) \in \mathrm{Mod}(T_R(M))$ is PGF if and only if $u: M \otimes_R X \rightarrow X$ is injective and $\mathrm{coker}(u)$ is a PGF $R$-module, under explicit homological hypotheses. This allows description of PGF modules over trivial extensions, Morita context rings, and lower-triangular matrix rings in terms of PGF modules over the ground ring (Tang et al., 18 Nov 2025).

Relative versions (PGcF modules): For a general module $C$, the notion of projectively coresolved $C$-Gorenstein flat modules (PGcF) unifies standard PGF modules by specializing to $C=R$, and allows development of model structures, cotorsion pairs, and Bass classes for arbitrary $(R, C)$ (Maaouy, 13 Aug 2024).

Functor categories: In additive functor categories ${}_\mathcal{Q}\mathrm{Mod}_R$, PGF objects are pointwise PGF, and the major closure and model category results persist (Di et al., 2022).

FP$_n$-flat and relative dimensions: Projectively coresolved Gorenstein FP$_n$-flat modules generalize PGF modules, with associated global dimensions characterized in terms of the flat dimension of all right FP$_n$-injective modules, and underlying homological symmetry encoded in balanced cotorsion triples (Becerril, 2023).

Groups and group rings: For $RG$, the projectively coresolved Gorenstein flat dimension of a group $G$ over $R$ ($\operatorname{PGF\mbox{-}dim}_{RG} R$) refines both classical and Gorenstein cohomological dimensions, and for large geometric classes (e.g., LH$_3$, PR-groups), all Gorenstein projective modules are PGF, so the two classes collapse (Stergiopoulou, 2023).

6. Model Structures, Stable Categories, and Frobenius Theory

The existence of a Frobenius exact structure on the category of (finite) PGF-dimension modules enables the construction of stable categories $\operatorname{PGF}(R)/\mathrm{Proj}(R)$, which are triangulated and equivalent to homotopy categories of corresponding model structures. These categorical equivalences are crucial for applications in representation theory, approximation theory (precovers, preenvelopes), and the paper of finitistic dimension conjectures in various algebraic contexts (Dalezios et al., 2022, Di et al., 2022, Maaouy, 2023, Becerril, 15 Mar 2024).

7. Special Examples and Further Developments

PGF modules are special precovering in left $n$-perfect rings and provide left approximations for arbitrary modules. They admit self-dual extension-theoretic characterizations—for instance, $M$ is PGF if and only if it is a direct summand of a module $N$ fitting into a short exact sequence $0 \to N \to F \to N \to 0$ with $F$ projective and appropriate Tor-vanishing conditions (Stergiopoulou, 2022).

In summary, the theory of projectively coresolved Gorenstein flat modules provides a highly structured, robust framework that is compatible with model category theory, covers both absolute and relative settings, and unifies various lines of development in Gorenstein, Ding, and Cohen–Macaulay approximation theory. It does so by providing a class with strong closure, stability, and resolving properties, and by enabling precise calculations of homological dimensions and deep structural results across module, ring, group, and categorical contexts (Kaperonis et al., 2023, Tang et al., 18 Nov 2025, Maaouy, 2023, Di et al., 2022, Dalezios et al., 2022, Maaouy, 13 Aug 2024, Becerril, 2023, Becerril, 15 Mar 2024).