2. Preliminaries

Resolutions. Given a class $\mathcal{X}$ of R-modules and a class $\mathcal{Y}$ of right R-modules, an $\mathcal{X}$-resolution of an R-module M is an exact complex $\cdots\to X_1\to X_0\to M\to 0$ where $X_i\in\mathcal{X}$.

A sequence X of R-modules is called $\left(\mathcal{Y}\otimes_R-\right)$-exact (respectively, ${\rm Hom}_R(\mathcal{X},-)$-exact, $ {\rm Hom}_R(-,\mathcal{X})$-exact) if $Y\otimes_R\textbf{X}$ (respectively, ${\rm Hom}_R(X,\textbf{X})$, ${\rm Hom}_R(\textbf{X},X)$) is an exact complex for every $Y\in\mathcal{Y}$ (respectively, $X\in\mathcal{X}$).

An R-module M is said to have $\mathcal{X}$-resolution dimension at most an integer $n\geq 0$, ${\rm resdim}_\mathcal{X}(M)\leq n$, if M has a finite $\mathcal{X}$-resolution: $0\to X_n\to\cdots\to X_1\to X_0\to M\to 0.$ If n is the least non-negative integer for which such a sequence exists then its $\mathcal{X}$-resolution dimension is precisely $n,$ and if there is no such n then we define its $\mathcal{X}$-resolution dimension as $\infty$.

Given a class of R-modules $\mathcal{Z}$, the $\mathcal{X}$-dimension of $\mathcal{Z}$ is defined as:

\begin{equation*}{\rm resdim}_\mathcal{X}(\mathcal{Z})={\rm sup}\{{\rm resdim}_\mathcal{X}(Z)|Z\in\mathcal{Z}\}.\end{equation*}

$\mathcal{X}$-coresolutions and $\mathcal{X}$-coresolution dimensions are defined dually.

Gorenstein modules and dimensions. An R-module M is called Gorenstein flat if it is a syzygy of an $(\mathcal{I}(R^\mathrm{op})\otimes_R-)$-exact exact sequence of flat R-modules. Replacing flat with projective, we get the definition of projectively coresolved Gorenstein flat (PGF for short) modules [Reference Saroch and Stovicek32]. An R-module M is called Gorenstein projective if it is a syzygy of a ${\rm Hom}_R(-,\mathcal{P}(R))$-exact exact sequence of projective R-modules. Gorenstein injective modules are defined dually. We let $\mathcal{GF}(R)$ (respectively, $\mathcal{PGF}(R)$, $\mathcal{GP}(R)$ and $\mathcal{GI}(R)$) denote the subcategory of Gorenstein flat (respectively, PGF, Gorenstein projective and Gorenstein injective) R-modules.

The Gorenstein flat (respectively, PGF, Gorenstein projective, Gorenstein injective) dimension of an R-module M is defined as ${\rm Gfd}_R(M):={\rm resdim}_{\mathcal{GF}(R)}(M)$ (respectively, ${\rm {PGF}\mbox{-}pd}_R(M):={\rm resdim}_{\mathcal{PGF}(R)}(M)$, ${\rm Gpd}_R(M):={\rm resdim}_{\mathcal{GP}(R)}(M)$ and ${\rm Gid}_R(M):={\rm coresdim}_{\mathcal{GI}(R)}(M)$).

Let ${\rm GPD}(R)$ (respectively, ${\rm GID}(R)$) denote the global Gorenstein projective (respectively, global Gorenstein injective) dimension of R, i.e.

\begin{equation*}{\rm GPD}(R):={\rm Gpd}_R(R\text{-Mod})={\rm sup}\{{\rm Gpd}_R(M)|M\in R\text{-Mod}\}\end{equation*}

(respectively, ${\rm GID}(R):={\rm Gid}_R(R\text{-Mod})={\rm sup}\{{\rm Gid}_R(M)|M\in R\text{-Mod}\}$).

Note that 0-flat (respectively, Gorenstein 0-flat, 0-cotorsion, 0-Gorenstein cotorsion) modules coincide with the flat (respectively, Gorenstein flat, cotorsion, Gorenstein cotorsion) modules.

Cotorsion pairs. Given an abelian category $\mathcal{A}$, and a class of objects $\mathcal{X}$ in $\mathcal{A}$, we use the following standard notation: $\mathcal{X}^\perp=\{A\in\mathcal{A}|{\rm Ext}^1_\mathcal{A}(X,A)=0,\forall X\in\mathcal{X}\}$ and $^\perp\mathcal{X}=\{A\in\mathcal{A}|{\rm Ext}^1_\mathcal{A}(A,X)=0,\forall X\in\mathcal{X}\}.$

An $\mathcal{X}$-precover of an object $A\in\mathcal{A}$ is a morphism $f:X\to A$ with $X\in \mathcal{X}$, in such a way that $f_*:{\rm Hom}_\mathcal{A}(X',X)\to {\rm Hom}_\mathcal{A}(X',M)$ is surjective for every $X'\in \mathcal{X}$. An $\mathcal{X}$-precover is called an $\mathcal{X}$-cover if every endomorphism $g: X\to X$ such that fg = f is an automorphism of X. If every object has an $\mathcal{X}$-(pre)cover then the class $\mathcal{X}$ is said to be (pre)covering. An $\mathcal{X}$-precover is called special if it is epimorphism and ${\rm Ker} f\in\mathcal{X}^\perp$. $\mathcal{X}$-(special pre)envelopes can be defined dually.

A pair $\left(\mathcal{X},\mathcal{Y}\right)$ of classes of objects in $\mathcal{A}$ is called a cotorsion pair if $\mathcal{X}^\perp=\mathcal{Y}$ and $\mathcal{X}=^\perp\mathcal{Y}$. A cotorsion pair $\left(\mathcal{X},\mathcal{Y}\right)$ is said to be:

• • Hereditary if ${\rm Ext}^i_\mathcal{A}(X,Y)=0$ for every $X\in\mathcal{X}$, $Y\in\mathcal{Y}$ and $i\geq 1$.

• • Complete if any object in $\mathcal{A}$ has a special $\mathcal{X}$-precover and a special $\mathcal{Y}$-preenvelope.

• • Perfect if every module has an $\mathcal{X}$-cover and a $\mathcal{Y}$-envelope.

• • Cogenerated by a set if there is a set $\mathcal{S}$ such that $\mathcal{X}=\mathcal{S}^\perp$.

It is well-known that a perfect cotorsion pair $(\mathcal{X},\mathcal{Y})$ in R-Mod is complete. The converse holds when $\mathcal{X}$ is closed under direct limits [Reference Göbel and Trlifaj19, Corollary 2.3.7]. A well-known method for constructing complete cotorsion pairs in R-Mod is to cogenerate one from a set (see [Reference Göbel and Trlifaj19, Theorem 3.2.1], for instance).

Abelian model structures. Given a bicomplete category $\mathcal{A}$, a model structure on $\mathcal{A}$ is given by three classes of morphisms of $\mathcal{A}$, called cofibrations, fibrations and weak equivalences that satisfy a set of axioms [Reference Hovey23, Definition 1.1.3].

Over a (bicomplete) abelian category $\mathcal{A}$, Hovey defined abelian model structures and showed that we can shift all our attention from morphisms to objects [Reference Hovey24, Theorem 2.2]. Namely, an abelian model structure on $\mathcal{A}$, is equivalent to a triple (known as Hovey triple) $\mathcal{M}=(\mathcal{Q},\mathcal{W},\mathcal{R})$ of classes of objects in $\mathcal{A}$ such that $(\mathcal{Q}, \mathcal{W}\cap\mathcal{R})$ and $(\mathcal{Q}\cap\mathcal{W},\mathcal{R})$ are complete cotorsion pairs and $\mathcal{W}$ is thick. In this case, Q (respectively, $\mathcal{R}$, $\mathcal{W}$) is precisely the class of cofibrant (respectively, fibrant, trivial) objects and an abelian model structure (in the sense of [Reference Hovey23, Definition 1.1.3]) is one which the following two conditions are satisfied:

(i) A morphism is a (trivial) cofibration if and only if it is a monomorphism with (trivially) cofibrant cokernel.

(ii) A morphism is a (trivial) fibration if and only if it is an epimorphism with (trivially) fibrant kernel.

An abelian model structure is called hereditary if both of the associated cotorsion pairs are hereditary (see Gillespie [Reference Gillespie22] for a nice survey on hereditary abelian model categories). In this paper, we often identify abelian model structures with Hovey triples.

Given a model structure $\mathcal{M}$, the homotopy category, denoted as ${\rm Ho}_\mathcal{A}(\mathcal{M})$, is defined by formally inverting the weak equivalences of $\mathcal{M}$. More precisely, ${\rm Ho}_\mathcal{A}(\mathcal{M})$ is obtained after localizing $\mathcal{A}$ at the class of weak equivalences. If moreover, $\mathcal{M}$ is hereditary, the homotopy category is known to be an algebraic triangulated category and it encodes a variety of (relative) homological algebra on $\mathcal{A}$.

3. n-Gorenstein flat modules and their model structure

In this section, we aim at proving Theorems A and B in the introduction. First, we need some basic properties of n-Gorenstein flat modules.

Proof. It is clear that any projective module is Gorenstein flat, and then n-Gorenstein flat. On the other hand, taking W = R in [Reference Bennis, El Maaouy, García Rozas and Oyonarte2], we get that $\mathcal{GF}(R)={\rm G_WF}(R)$ and this class is closed under extensions by [Reference Saroch and Stovicek32, Theorem 4.11]. Therefore, the class $\mathcal{GF}_n(R)$ is closed under extensions and kernels of epimorphisms by [Reference Bennis, El Maaouy, García Rozas and Oyonarte2, Proposition 7.9] and direct sums and summands by [Reference Bennis, El Maaouy, García Rozas and Oyonarte2, Corollary 7.8(2)]. It remains to show that $\mathcal{GF}_n(R)$ is closed under direct limits.

Following [Reference Adámek and Rosický1, Corollary 1.7 and the Remark that follows it] it suffices to show that the class $\mathcal{GF}_n(R)$ is closed under well-ordered continuous direct limits of n-Gorenstein flat R-modules:

\begin{equation*}(M_\alpha)_{\alpha \lt \lambda}:= M_0\to M_1\to \cdots\to M_\omega\to M_{\omega+1}\to \cdots\end{equation*}

Here continuous (or smooth in the sense of [Reference Adámek and Rosický1]) means that for each limit ordinal $\beta \lt \lambda$, $M_\beta=\varinjlim_{\alpha \lt \beta} M_\alpha$.

• • If $\lambda \lt \omega$, then $\lim_{\alpha \lt \lambda} M_\alpha=M_{\lambda-1}\in \mathcal{GF}_n(R)$.

• • Assume now that $\lambda=\omega$. By [Reference Bennis, El Maaouy, García Rozas and Oyonarte2, Proposition 7.11], there exists an exact sequence $0\to M_0\to F_0\to G_0\to 0$ with $F_0\in\mathcal{F}_n(R)$ and G ₀ Gorenstein flat. Consider the push out

  

Since $M_1\in\mathcal{GF}_n(R)$ and G ₀ is Gorenstein flat, $V\in\mathcal{GF}_n(R)$. Using again [Reference Bennis, El Maaouy, García Rozas and Oyonarte2, Proposition 7.11], we get an exact sequence $0\to V\to F_1\to G\to 0$ with $F_1\in\mathcal{F}_n(R)$ and G Gorenstein flat.

Consider another push out

Since G ₀ and G are Gorenstein flat, so is G ₁. Therefore, we get the following morphism of exact sequences induced by the morphism $M_0\to M_1$:

Repeating this process, we get a commutative diagram with exact rows

where $F_i\in\mathcal{F}_n$ and G_i is Gorenstein flat for every $i\geq 0$. Applying the exact functor $\varinjlim$ to this commutative diagram, we get the induced exact sequence

\begin{equation*}\varinjlim A_m=: 0\to \varinjlim M_m\to \varinjlim F_m\to \varinjlim G_m\to 0\end{equation*}

with $\varinjlim G_m\in \mathcal{GF}(R)\subseteq \mathcal{GF}_n(R)$ by [Reference Yang and Liu36, Lemma 3.1] and $\varinjlim F_m\in \mathcal{F}_n(R)$ since the functor ${\rm Tor}^R_{n+1}(X,-)$ commutes with direct limits for any right R-module X. Hence, $M_\omega=\varinjlim M_m\in\mathcal{GF}_n(R)$.

Finally, using transfinite induction, it is clear that the above argument generalizes and we get that $\varinjlim_{\alpha \lt \lambda} M_\alpha\in\mathcal{GF}_n(R)$ as desired.

The following two lemmas will be used in several places in this paper.

The following characterization of n-Gorenstein flat modules is a key result to show many results in this section. The case n = 0 is due to Šaroch and [Reference Saroch and Stovicek32, Theorem 4.11].

Proof. $4.\Rightarrow 1.$ It follows by Proposition 3.1 since $F\in\mathcal{F}_n(R)\subseteq \mathcal{GF}_n(R)$ and $N\in \mathcal{PGF}(R)\subseteq \mathcal{GF}_n(R)$.

$1.\Rightarrow 4.$ Assume that ${\rm Gfd}_R(M)\leq n$ and proceed by induction on n.

The case n = 0 follows from [Reference Saroch and Stovicek32, Theorem 4.11(4)].

Assume that $n\geq 1$. There exists an exact sequence $0\rightarrow G_n\rightarrow\cdots\rightarrow G_0\rightarrow M\rightarrow 0$, where $G_i\in \mathcal{GF}(R)$ for every $i=0,\ldots,n.$ Let $K={\rm Ker}(G_0\to M)$. Clearly, ${\rm Gfd}_R(K)\leq n-1$. So, by induction there exists an exact sequence $0\rightarrow K\rightarrow F'\rightarrow N'\rightarrow 0,$ where $N'\in \mathcal{PGF}(R)$ and ${\rm fd}_R(F')\leq n-1$. Consider the pushout diagram

By the middle column D is Gorenstein flat. Using induction again, we get a short exact sequence of R-modules $0\to D\to F_0\to N\to 0$ where F ₀ is flat and $N\in\mathcal{PGF}(R)$.

Consider now another pushout diagram

It remains to see that ${\rm fd}_R(F)\leq n$. But this is true by the middle row since F ₀ is flat and ${\rm fd}_R(F')\leq n-1$.

$1.\Rightarrow 2.$ The case n = 0 holds by [Reference Saroch and Stovicek32, Theorem 4.11(2)]. So, we may assume that $n\geq 1$. By [Reference Saroch and Stovicek32, Theorem 4.9], there exists a special PGF precover $0\to N\to G\to M\to 0.$ By [Reference Bennis, El Maaouy, García Rozas and Oyonarte2, Proposition 7.9], ${\rm Gfd}_R(N)\leq {\rm sup}\{{\rm Gfd}_R(G),{\rm Gfd}_R(M)-1\}\leq n-1$ which implies by (4) that there exists an exact sequence of R-modules $0\to N\to K\to G'\to 0$ where ${\rm fd}_R(K)\leq n-1$ and $G'\in\mathcal{PGF}(R)$. Consider the pushout

Since $\mathcal{PGF}(R)$ is closed under extensions, $L\in\mathcal{PGF}(R)$. Then, there exits (by definition) an exact sequence $0\to L\to P\to L'\to 0$, with P projective and $L'\in \mathcal{PGF}(R)$. Consider now another pushout

Since ${\rm fd}_R(K)\leq n-1$ and P is projective (then flat), $H\in\mathcal{F}_n(R)$ by the middle row. It follows from the following commutative diagram with exact rows

that $0\to L\to K\to M\to 0$ is ${\rm Hom}_R(-,X)$-exact for any module $X\in\mathcal{F}_n(R)\cap\mathcal{C}_n(R)$.

$ 2.\Rightarrow 3.$ Similar to [Reference Saroch and Stovicek32, Theorem 4.11(2)$\Rightarrow$(3)], using the fact that any module $C\in \mathcal{PGF}(R)^\perp\cap \mathcal{C}_n(R)$ has a special n-flat precover which exists by Lemma 3.3.

$ 3.\Rightarrow 4.$ Similar to [Reference Saroch and Stovicek32, Theorem 4.11(3)$\Rightarrow$(4)] using the fact that every module $F\in \mathcal{PGF}(R)^\perp$ has a special n-flat precover which exists by Lemma 3.3 and that $\mathcal{F}_n(R)\subseteq \mathcal{PGF}(R)^\perp$ by Lemma 3.2.

Finally, it remains to show the equality. The inclusion $\mathcal{F}_n(R)\subseteq \mathcal{GF}_n(R)$ is clear and by Lemma 3.2 we also have $\mathcal{F}_n(R)\subseteq \mathcal{PGF}(R)^\perp$. Then, $\mathcal{F}_n(R)\subseteq \mathcal{GF}_n(R)\cap \mathcal{PGF}(R)^{\perp}$. Conversely, if $M\in\mathcal{GF}_n(R)\cap \mathcal{PGF}(R)^{\perp}$, then there exists by (4) a split exact sequence of R-modules $0\to M\to F\to N\to 0$, where $N\in\mathcal{PGF}(R)$ and ${\rm fd}_R(X)\leq n$. Then, $M\in \mathcal{F}_n(R)$.

The following result, which is Theorem A from the introduction, improves and generalizes [Reference Pérez30, Proposition 14.3.5] and [Reference Xu35, Lemma 4.4]. Here we have a more explicit description of the right hand of the n-Gorenstein flat cotorsion pair without any assumptions on the ring.

Now we are ready to prove our second main result in this section. The following result improves [Reference Pérez30, Theorem 14.3.1] and [Reference Xu35, Theorem 4.5] in the sense that we need no extra assumptions on the ring R with an obvious description of the trivial objects.

Theorem 3.6. For any ring R, there exists a hereditary abelian model structure on R-Mod

\begin{equation*}\mathcal{M}_n:=\left(\mathcal{GF}_n(R),\mathcal{PGF}(R)^\perp, \mathcal{C}_n(R)\right).\end{equation*}

Equivalently, there exists a hereditary abelian model structure on R-Mod, in which the cofibrations coincide with the monomorphisms with n-Gorenstein flat cokernel, the fibrations coincide with the epimorphisms with n-cotorsion kernel, and $\mathcal{PGF}(R)^\perp$ is the class of trivial objects.

Proof. By Theorem 3.4, $\mathcal{F}_n(R)=\mathcal{GF}_n(R)\cap \mathcal{PGF}(R)^\perp$. Then, by Lemma 3.3 and Corollary 3.5, there are two complete and hereditary cotorsion pairs:

\begin{equation*}(\mathcal{GF}_n(R),\mathcal{PGF}(R)^\perp\cap\mathcal{C}_n(R)) \text{ and } (\mathcal{GF}_n(R)\cap \mathcal{PGF}(R)^\perp,\mathcal{C}_n(R)).\end{equation*}

Finally, the class $\mathcal{PGF}(R)^\perp$ is thick by [Reference Saroch and Stovicek32, Theorem 4.9] which completes the proof.

Recall that a Frobenius category $\mathcal{A}$ is an exact category with enough injectives and projectives such that the projective objects coincide with the injective objects. In this case, we can form the stable category $\underline{\mathcal{A}}:=\mathcal{A}/\sim$, which has the same objects as $\mathcal{A}$ and ${\rm Hom}_{\mathcal{A}/\sim}(X,Y):={\rm Hom}_{\mathcal{A}}(X,Y)/\sim$, where $f\sim g$ if and only if f − g factors through a projective-injective object.

The following result can be proved directly as in the case n = 0 (see [Reference Bravo, Gillespie and Hovey8, Theorem 4.5]). However, we follow an alternative approach due to Gillespie [Reference Gillespie21]. Recall [Reference Gillespie21, Sections 4 and 5] that for any hereditary Hovey triple $\mathcal{M}=(\mathcal{Q},\mathcal{W},\mathcal{R})$, the subcategory $\mathcal{Q}\cap \mathcal{R}$ is a Frobenius exact category (with the exact structure given by the short exact sequences with terms in $\mathcal{Q}\cap \mathcal{R}$) whose projective-injective objects are precisely those in $\mathcal{Q}\cap\mathcal{W}\cap\mathcal{R}$. Moreover, the stable category $\underline{\mathcal{Q}\cap\mathcal{R}}$ is triangulated equivalent to ${\rm Ho}(\mathcal{M})$ (see also [Reference Gillespie22, Proposition 4.2 and Theorem 4.3]). Consequently, we have the following corollary.

Let $(\mathcal{T},\Sigma)$ be a triangulated category with coproducts and an autofunctor Σ. Recall that an object C of $\mathcal{T}$ is compact if for each family $\{Y_i|i\in I\}$ of objects of $\mathcal{T}$, the canonical morphism

\begin{equation*}\bigoplus_{i\in I} {\rm Hom}_\mathcal{T}(C,Y_i)\to {\rm Hom}_\mathcal{T}(C,\bigoplus_{i\in I}Y_i)\end{equation*}

is an isomorphism. The category $\mathcal{T}$ is called compactly generated if there exists a set $\mathcal{S}\subseteq \mathcal{T}$ of compact objects such that for each $0\neq Y\in\mathcal{Y}$ there is a morphism $0\neq f:\Sigma^m S\to Y$ for some $S\in\mathcal{S}$ and $m\in\mathbb{Z}$.

It follows by Liang and Wang [Reference Liang and Wang29, Theorem 1(a)] that the stable category $\mathcal{E}_0=\underline{\mathcal{GF}(R)\cap\mathcal{C}(R)}$ is compactly generated when R is right coherent with finite left global Gorenstein projective dimension. Based on their work, this property is inherited by the stable categories $\underline{\mathcal{PGF}(R)}$ and $\underline{\mathcal{GF}_n(R)\cap\mathcal{C}_n(R)}$ for every integer $n\geq 0$ under the weaker assumption that R has finite global Gorenstein AC-projective dimension. That is, the supremum of the $\mathcal{GP}_{AC}(R)$-resolution dimension of all R-modules,

\begin{equation*} {\rm GPD}_{AC}(R):={\rm sup}\{{\rm resdim}_{\mathcal{GP}_{AC}(R)}(M)| M \text{is an }R\text{-module} \}, \end{equation*}

is finite, where $\mathcal{GP}_{AC}(R)$ denotes the class of Gorenstein AC-projective modules in the sense of [Reference Bravo, Gillespie and Hovey7].

Corollary 3.8. There exist triangulated equivalences

\begin{equation*} \underline{\mathcal{GF}_n(R)\cap \mathcal{C}_n(R)} \backsimeq\cdots \backsimeq \underline{\mathcal{GF}_1(R)\cap \mathcal{C}_1(R)} \backsimeq \underline{\mathcal{GF}(R)\cap \mathcal{C}(R)}\simeq \underline{\mathcal{PGF}(R)}. \end{equation*}

Furthermore, if ${\rm GPD}_{AC}(R) \lt \infty$, then these triangulated categories are compactly generated.

Proof. By Theorem 3.6 and [Reference Saroch and Stovicek32, Theorem 4.9], we have the model structures

\begin{equation*} \mathcal{M}_n=(\mathcal{GF}_n(R),\mathcal{PGF}(R)^\perp,\mathcal{C}_n(R)) \text{ and } \mathcal{N}=(\mathcal{PGF}(R), \mathcal{PGF}(R)^\perp, R\text{-Mod}) \end{equation*}

with the same trivial objects and the following inclusions

\begin{equation*} \mathcal{PGF}\subseteq \mathcal{GF}(R)\subseteq \mathcal{GF}_1(R)\subseteq \cdots \subseteq \mathcal{GF}_n(R). \end{equation*}

Applying [Reference Estrada and Gillespie17, Lemma 5.4], we get the following triangulated equivalences

\begin{equation*} {\rm Ho}(\mathcal{M}_n) \backsimeq\cdots \backsimeq {\rm Ho}(\mathcal{M}_1) \backsimeq {\rm Ho}(\mathcal{M}_0)\simeq {\rm Ho}(\mathcal{N}). \end{equation*}

On the other hand, following Corollary 3.7 and [Reference Saroch and Stovicek32, p. 27], we have

\begin{equation*} {\rm Ho}(\mathcal{M}_n)\backsimeq \underline{\mathcal{GF}_n(R)\cap \mathcal{C}_n(R)} \text{ and }{\rm Ho}(\mathcal{N})\backsimeq \underline{\mathcal{PGF}(R)} \end{equation*}

as triangulated categories. Hence, the first statement follows.

Finally, if ${\rm GPD}_{AC}(R) \lt \infty$, then our last claim follows by [Reference Liang and Wang29, Theorem 24, Proposition 21 and Corollary 33] and the above triangulated equivalences.

It is well-known that the stable categories $\underline{\mathcal{GP}(R)}$ and $\underline{\mathcal{GI}(R)}$ are triangulated categories. Li and Wang showed in [Reference Liang and Wang29, Theorem 1] that these triangulated categories are compactly generated whenever ${\rm GPD}(R) \lt \infty$ with R right or left coherent. Their proof is mainly based on [Reference Liang and Wang29, Theorem 25]. However, the coherence assumption in [Reference Liang and Wang29, Theorem 25] is not needed if we replace ${\rm GPD}(R) \lt \infty$ with ${\rm GPD}_{AC}(R) \lt \infty$. Therefore, following the same argument in [Reference Liang and Wang29, Corollary 34], we have the improved result:

Corollary 3.9. Let R be a ring with ${\rm GPD}_{AC}(R) \lt \infty$. Then,

\begin{equation*} \underline{\mathcal{GP}(R)}\simeq \underline{\mathcal{GI}(R)}\simeq \underline{\mathcal{GF}(R)\cap\mathcal{C}(R)} \end{equation*}

are compactly generated.

4. PGF dimension of modules and rings

In this section, we are interested in the PGF dimension of modules and rings and its connection with other homological dimensions.

The following result is a key result to prove many of the results that will follow. It is a combination of [Reference Stergiopoulou33, Theorem 3.20] and [Reference Dalezios and Emmanouil11, Theorem 3.4$(i)\Leftrightarrow (ii)$], except for the last statement.

As a consequence, we get new characterizations of left (n-)perfect rings. Recall that [Reference Enochs, Jenda and López-Ramos16, Definition 2.1] a ring R is left n-perfect if every flat R-module has projective dimension at most n. In particular, left perfect rings are exactly left 0-perfect rings.

Theorem 4.2 can be interpreted from the point of view of global dimensions. For this purpose (and for later use), we introduce the following homological invariant.

Definition 4.3. The global PGF dimension of R with respect to a class $\mathcal{X}\subseteq R$-Mod is defined as: ${\rm PGFD}_{\mathcal{X}}(R):={\rm sup}\{{\rm {PGF}\mbox{-}pd}_R(X)|X\in\mathcal{X}\}.$

In particular, we set

\begin{equation*} {\rm PGFD}(R):={\rm PGFD}_{R\text{-Mod}}(R)={\rm sup}\{{\rm {PGF}\mbox{-}pd}_R(M)| M\in R\text{-Mod}\} \end{equation*}

and we simply call it the global PGF dimension of R.

We note that the global PGF dimension has been recently studied by Dalezios and Emmanouil [Reference Dalezios and Emmanouil11].

We now turn our attention to the questions raised in § 1.

Instead of asking whether any Gorenstein projective module is PGF or not, one could ask the following question:

(Q4): How far is a Gorenstein projective module from being PGF?

The following result reduces this problem to two cases.

In order to continue our investigations, we need the notion of definable classes. Recall from [Reference Prest31, Theorem 3.4.7] that a class of modules $\mathcal{D}$ is called definable if it is closed under direct products, direct limits and pure submodules. Such a class is, in particular, closed under direct sums and direct summands. We denote by $\langle\mathcal{X}\rangle$ the definable class generated by a class of modules $\mathcal{X}$, that is, the smallest definable class containing $\mathcal{X}$. The definable closure $\langle \mathcal{X}\rangle$ can be constructed, for instance, by closing $\mathcal{X}$ under direct products, then under pure submodules and, finally, under pure quotients. In case $\mathcal{X}=\{X\}$, we simply write $\langle\mathcal{X}\rangle=\langle X\rangle$.

It has been shown in many recent works [Reference Cortés-Izurdiaga and Saroch10, Reference Estrada, Iacob and Pérez18, Reference Saroch and Stovicek32] that the definable classes $\langle R\rangle$ and $\langle R^+\rangle$ are of great importance. They will also be of interest to us. Thus, we propose the following definition, giving names to the modules belonging to these classes.

Clearly, $\langle R\rangle=\langle \mathcal{P}(R)\rangle=\langle \mathcal{F}(R)\rangle$ and $\langle (_RR)^+\rangle=\langle\mathcal{I}(R^\mathrm{op})\rangle =\langle\mathcal{FI}(R^\mathrm{op})\rangle $. Moreover, R is right coherent if and only if definable flat R-modules coincide with flats if and only if definable injective right R-modules coincide withFP-injectives (see [Reference Prest31, Theorem 3.4.24]). Recall that a module M is called FP-injective if $Ext^1(F,M)_0$ for every finitely presented module F.

As we will see next, the class of definable flat modules is a key point for extending many results from the class of coherent rings to a larger class of rings.

Izurdiaga [Reference Cortés-Izurdiaga9] investigated rings for which the direct product of any family of flat R-modules has finite flat dimension. He calls them [Reference Cortés-Izurdiaga9, Section 4] right weak coherent rings. Moreover, if n is the maximum of the set consisting of all flat dimensions of all direct products of flat R-modules, then R is called right weak n-coherent. In particular, right weak 0-coherent rings are nothing but right coherent rings. It follows by Lemma 4.8 that R is right weak coherent if and only if and only if any definable flat module has finite flat dimension. In this case, R is right weak n-coherent with $n={\rm fd}_R(\langle R\rangle)$.

We also recall from [Reference Gillespie20] the notion of Ding projective modules (which are introduced for the first time in [Reference Ding, Li and Mao12] under a different name). An R-module M is called Ding projective if it is a syzygy of a ${\rm Hom}_R(-,\mathcal{F}(R))$-exact exact sequence of projective R-modules. The Ding projective dimension of an R-module M is defined as ${\rm Dpd}_R(M)={\rm resdim}_{\mathcal{DP}(R)}(M)$ where $\mathcal{DP}(R)$ denotes the class of Ding projective R-modules.

As a consequence of the above lemma, the assumption that R is right coherent in [Reference Iacob28, Theorem 2] and [Reference Cortés-Izurdiaga and Saroch10, Corollary 5.10] can be replaced by that R is right weak coherent.

Proof. We only need to show the equality $\mathcal{PGF}(R)=\mathcal{DP}(R)$. The inclusion $\mathcal{PGF}(R)\subseteq \mathcal{DP}(R)$ holds for any ring R. For the other inequality, let $M\in\mathcal{DP}(R)$, then M is a syzygy of a ${\rm Hom}_R(-,\mathcal{F}(R))-$exact exact sequence P of projective R-modules. Let X be a definable flat R-module. By Lemma 4.8, $m={\rm fd}_R(X) \lt \infty$. Then, X has a finite flat resolution

\begin{equation*} 0\to F_m\to \cdots \to F_0\to X\to 0. \end{equation*}

Applying the functor ${\rm Hom}_R(\textbf{P},-)$ to this exact sequence, we get an exact sequence of complexes

\begin{equation*}0\to {\rm Hom}_R(\textbf{P},F_m)\to \cdots \to{\rm Hom}_R(\textbf{P},F_0) \to {\rm Hom}_R(\textbf{P},X)\to 0.\end{equation*}

As the class of exact complexes is thick, it follows that the complex ${\rm Hom}_R(\textbf{P},X)$ is exact and hence $M\in\mathcal{PGF}(R)$.

The last claim follows by [Reference Saroch and Stovicek32, Theorem 4.9].

One could, as in Lemma 4.8, consider rings for which every definable flat has finite projective dimension. The following lemma, which we will use later, says that such rings are nothing new.

Proof. $(\Rightarrow)$ Assume $n={\rm pd}_R(\langle R\rangle) \lt \infty$. Then, ${\rm pd}_R(X)\leq n$ for all $X\in \langle R\rangle$. In particular, ${\rm fd}_R(X)\leq {\rm pd}_R(X)\leq n$ for all $X\in \langle R\rangle$ and ${\rm pd}_R(X)\leq n$ for all $X\in\mathcal{F}(R)\subseteq \langle R\rangle$. $(\Leftarrow)$ Assume that $m={\rm fd}_R(\langle R\rangle) \lt \infty$. Given a partial projective resolution of an R-module $M\in \langle R\rangle$

\begin{equation*} 0\to K_m\to P_{m-1}\to \cdots \to P_0\to M\to 0, \end{equation*}

we get that K_m is flat and hence ${\rm pd}_R(K_m)\leq n$ as $K_m\in \langle R\rangle$. Therefore, ${\rm pd}_R(M)\leq n+m$. Thus, ${\rm pd}_R(\langle R\rangle)\leq n+m \lt \infty$.

Now we are ready to give a partial answer to Question (Q3) raised in § 1. Recall that a ${\rm Hom}_R(-,\mathcal{P}(R))$-exact exact complex of projective modules is called totally acyclic complex of projectives.

Consequently, partial answers to questions (Q1) and (Q2) are obtained.

We now focus our attention on the global PGF dimension of R. As a first result, we provide simple ways to compute it (compare with [Reference Dalezios and Emmanouil11, Theorem 5.1]).

Theorem 4.14. The following assertions are equivalent:

1. (1) ${\rm PGFD}(R)\leq n$.

2. (2) The following two assertions hold.

   1. (i) ${\rm id}_R(M)\leq n$ for every definable flat R-module M.

   2. (ii) ${\rm pd}_R(M)\leq n$ for every injective R-module M.

3. (3) The following two assertions hold.

   1. (i) ${\rm fd}_{R^{op}}(M)\leq n$ for every definable injective right R-module M.

   2. (ii) ${\rm pd}_R(M)\leq n$ for every injective R-module M.

4. (4) The following two assertions hold.

   1. (i) ${\rm fd}_{R^{op}}(M)\leq n$ for every (FP-)injective right R-module M.

   2. (ii) ${\rm pd}_R(M)\leq n$ for every injective R-module M.

Consequently, the global PGF dimension of R can be computed by the following formulas:

\begin{align*} {\rm PGFD}(R)&= max\{{\rm pd}_R(\mathcal{I}(R)),{\rm id}_R(\langle R\rangle)\}\\ &= max\{{\rm pd}_R(\mathcal{I}(R)),{\rm fd}_{R^{op}}(\langle R^+\rangle)\}\\ &= max\{{\rm pd}_R(\mathcal{I}(R),{\rm fd}_{R^{op}}(\mathcal{I}(R^{op}))\}. \end{align*}

Proof. $1.\Rightarrow 2.$ (a) Consider an R-module N and a finite PGF resolution:

\begin{equation*}0\to G_n\to \cdots \to G_0\to N\to 0.\end{equation*}

By [Reference Saroch and Stovicek32, Corollary 4.5], ${\rm Ext}^{k\geq 1}_R(G_i,M)=0$ for any $M\in\langle R\rangle$ and i. Hence, ${\rm Ext}^{n+1}_R(N,M)\cong {\rm Ext}_R^1(G_n,M)=0$. Then, ${\rm id}_R(M)\leq n$.

(b) Let M be an injective R-module. Since ${\rm {PGF}\mbox{-}pd}_R(M)\leq n$, there exists a split short exact sequence $0\to M\to X\to G\to 0$ with ${\rm pd}_R(X)\leq n$ by Proposition 4.1(3). Hence, ${\rm pd}_R(M)\leq n$.

$2.\Rightarrow 3.$ We only prove (a) as (b) is clear. For any definable injective right R-module M, we have ${\rm fd}_{R^\mathrm{op}}(M)\leq n$ if and only if ${\rm Tor}^R_{n+1}(M,-)=0$ if and only if $ {\rm Ext}_R^{n+1}(-,M^+)\cong{\rm Tor}^R_{n+1}(M,-)^+=0$ if and only ${\rm id}_R(M^+)\leq n$. This later holds by 2(a) and [Reference Estrada, Iacob and Pérez18, Remark 2.10] as $M^+\in\langle R\rangle$. Hence, ${\rm fd}_{R^\mathrm{op}}(M)\leq n$.

$3.\Rightarrow 4.$ (b) is clear and (a) follows by $\mathcal{I}({R^\mathrm{op}})\subseteq \mathcal{FI}({R^\mathrm{op}})\subseteq \langle \mathcal{FI}({R^\mathrm{op}})\rangle= \langle (R_R)^+\rangle$.

$4.\Rightarrow 1.$ Let M be an R-module. Consider a projective and an injective resolution of M:

\begin{equation*}\cdots \to P_1\to P_0\to M\to 0 \text{ and } 0\to M\to I_0\to I_1\to\cdots,\end{equation*}

respectively. Decomposing these exact sequences into short exact ones we get, for every integer $i\in {\mathbb{N}}$,

\begin{equation*}0 \to L_{i+1}\to P_i\to L_i\to 0 \ \ \text{and} \ \ 0\to K_i\to I_i\to K_{i+1}\to 0\end{equation*}

where $L_i={\rm Coker}(P_{i+1}\to P_{i})$ and $K_i={\rm Ker}(I_i\to I_{i+1})$. Note that $M=L_0=K_0$. Adding the direct sum of the first sequences,

\begin{equation*}0 \to \bigoplus_{i\in{\mathbb{N}}}L_{i+1}\to \bigoplus_{i\in{\mathbb{N}}}P_i\to M\oplus( \bigoplus_{i\in{\mathbb{N}}}L_{i+1})\to 0\end{equation*}

to the direct product of the second ones,

\begin{equation*}0\to M\oplus( \prod_{i\in{\mathbb{N}}}K_{i+1})\to \prod_{i\in{\mathbb{N}}}I_i\to \prod_{i\in{\mathbb{N}}}K_{i+1}\to 0\end{equation*}

we get an exact sequence of the form $0\to N\to X\to N\to 0$, where

\begin{equation*}X=(\bigoplus_{i\in N} P_i)\oplus (\prod_{i\in N} I_i)\text{ and } N=M\oplus \left((\bigoplus_{i\in N} L_{i+1}) \oplus (\prod_{i\in N} K_{i+1})\right).\end{equation*}

By (b), ${\rm pd}_R(X)={\rm pd}_R((\prod_{i\in N} I_i))\leq n$ and by (a), ${\rm Tor}^R_{n+1}(E,N)=0$ for any injective right R-module E. And since M is a direct summand in N, it follows from Proposition 4.1 that ${\rm {PGF}\mbox{-}pd}_R(M)\leq n$ as desired.

Global dimensions of rings are known to characterize classical rings. And the global PGF dimension is no exception. Recall that a ring R is called quasi-Frobenius if injective left (respectively, right) R-modules coincide with projective left (respectively, right) modules.

At this point, it is natural to ask what the relationship between the global PGF dimension and the global Gorenstein projective dimension could be. Clearly, ${\rm GPD}(R)\leq {\rm PGFD}(R)$ as $\mathcal{PGF}(R)\subseteq \mathcal{GP}(R)$. Over an (Iwanaga-)Gorenstein ring R, it is an immediate consequence of Huang [Reference Huang25, Theorems 4.9 and 4.13] that we have an equality ${\rm GPD}(R)={\rm PGFD}(R)$. Dalezios and Emmanouil, on the other hand, have shown in [Reference Dalezios and Emmanouil11, Theorem 5.1] this equality when R has finite global PGF dimension.

Our last main result gives a positive answer to this question for any ring R. It is worth noting that the following result, as pointed out to me by Sergio Estrada, is contained in the proof of a recent result by Wang, Yang, Shao and Zhang [Reference Wang, Yang, Shao and Zhang34, Theorem 3.7]. However, our proof is different.

We end the paper with some consequences of Theorem 4.16.

Gorenstein injective modules are seen as dual to Gorenstein projective modules. Based on this, one could introduce a dual notion of a PGF module. Let us call an R-module Gorenstein definable injective if it is a syzygy of a ${\rm Hom}_R(\langle R^+\rangle,-)$-exact exact sequence of injective R-modules. Define its Gorenstein definable injective dimension as ${\rm Gdid}(M)={\rm resdim}_{\mathcal{GDI}(R)}(M)$ where $\mathcal{GDI}(R)$ denotes the class of all Gorenstein definable injective R-modules.

It is known that the global dimension of R can be computed either by projective or injective dimensions, that is, the following equality holds:

\begin{equation*} {\rm sup}\{{\rm pd}_R(M)|M\in R\text{-Mod}\}={\rm sup}\{{\rm id}_R(M)|M\in R\text{-Mod}\}. \end{equation*}

This fact was extended to the Gorenstein setting by Enochs and Jenda [Reference Enochs and Jenda15, Section 12.3] for (Iwanaga-)Gorenstein rings and later by Bennis and Mahdou [Reference Bennis and Mahdou6, Theorem 1.1] for any ring.

Inspired by this, one could ask whether the equality

\begin{equation*} {\rm sup}\{{\rm {PGF}\mbox{-}pd}_R(M)|M\in R\text{-Mod}\}={\rm sup}\{\mathrm{Gdid}_R(M)| M\in R\text{-Mod}\} \end{equation*}

holds true as well?

Using again Theorem 4.16, we give a positive answer to this question.

Corollary 4.17. For any ring R, we have the following equalities:

\begin{equation*} {\rm PGFD}(R)={\rm GID}(R)={\rm GDID}(R) \end{equation*}

with ${\rm GDID}(R):={\rm sup}\{\rm {Gdid}_R(M)|M \text{is an }R\text{-module}\}.$

Remark 4.18. It clear by the definitions that we have the inequalities

\begin{align*} {\rm GPD}(R) & \leq {\rm DPD}(R) \leq{\rm PGFD}(R)\leq {\rm GPD}_{AC}(R),\\ {\rm GID}(R) & \leq {\rm DID}(R) \leq{\rm GDID}(R)\leq {\rm GID}_{AC}(R) \end{align*}

with

\begin{align*} {\rm DPD}(R) & :={\rm sup}\{{\rm Dpd}_R(M)| M \text{is an }R\text{-module} \},\\ {\rm DID}(R) & :={\rm sup}\{{\rm Did}_R(M)| M \text{is an }R\text{-module} \},\\ {\rm GPD}_{AC}(R) & :={\rm sup}\{{\rm resdim}_{\mathcal{GI}_{AC}(R)}(M)| M \text{is an }R\text{-module} \}, \end{align*}

where $\mathcal{GI}_{AC}(R)$ denotes the class of all Gorenstein AC-injective modules in the sense of [Reference Bravo, Gillespie and Hovey7].

As stated above, Bennis and Mahdou showed in [Reference Bennis and Mahdou6, Theorem 1.1] the equality ${\rm GPD}(R)={\rm GID}(R)$ for any ring R. Generally, Huerta, Mendoza and Pérez have recently shown in [Reference Huerta, Mendoza and Pérez26, Corollary 8.8] the following equalities:

\begin{equation*} {\rm GPD}_{AC}(R)= {\rm DPD}(R)={\rm GPD}(R)={\rm GID}(R) ={\rm DID}(R)={\rm GID}_{AC}(R). \end{equation*}

The result [Reference Huerta, Mendoza and Pérez26, Corollary 8.8] is based on [Reference Huerta, Mendoza and Pérez26, Lemma 8.1], which is stated incorrectly as indicated in the corrigendum [Reference Huerta, Mendoza and Pérez27]. So, the result [Reference Huerta, Mendoza and Pérez26, Corollary 8.8] is no longer available and the question of whether the equalities ${\rm DPD}(R)={\rm DID}(R)$ and ${\rm GPD}_{AC}(R)={\rm GID}_{AC}(R)$ hold for any ring R is still open. However, following Corollary 4.17, we have a positive answer for the second equality. Generally, we have the following equalities for any ring R:

\begin{equation*} {\rm PGFD}(R)={\rm DPD}(R)={\rm GPD}(R)={\rm GID}(R)={\rm DID}(R)={\rm GDID}(R). \end{equation*}

Auslander’s theorem on the global dimension states that we can compute the global dimension of R by just computing the projective dimension of cyclic R-modules. That is, the formula ${\rm gldim}(R)={\rm sup}\{{\rm pd}_R(R/I)|I \text{ is a left idea}\}$ holds true. Bennis, Hu and Wang [Reference Bennis, Hu and Wang4, Theorem 1.1] extended this formula to the Gorenstein setting for commutative rings. However, one can see that the same proof also holds for non-commutative rings. Taking advantage of this fact, together with Theorem 4.16, we get a PGF version of Auslander’s theorem.

Corollary 4.19. (Auslander’s theorem on the global PGF dimension)

For any ring R, we have the following formula:

\begin{equation*}{\rm PGFD}(R)={\rm sup}\{{\rm {PGF}\mbox{-}pd}_R(R/I)|I \text{ is a left ideal}\}.\end{equation*}