1.1 (Cotorsion pairs)

Let $\mathsf {A}$ be an abelian category with enough projectives and injectives. A pair $(\mathsf {C},\, \mathsf {W})$ of subcategories of $\mathsf {A}$ is called a cotorsion pair if $\mathsf {C}^{\bot }=\mathsf {W}$ and $^{\bot }\mathsf {W}=\mathsf {C}$. Here

\begin{align*} {\mathsf{C}}^{\bot}& =\{N\in {\sf A}\ |\ {\rm Ext}_{\mathsf{A}}^{1}(C,N)=0 {\rm\ for\ all}\ C\in\mathsf{C}\},\ \text{and}\\ ^{\bot}{\mathsf{W}}& =\{M\in {\sf A}\ |\ {\rm Ext}_{\mathsf{A}}^{1}(M,W)=0 {\rm\ for\ all}\ W\in\mathsf{W}\}. \end{align*}

It is clear that $({^\perp (\mathsf {C}^\perp )},\,\mathsf {C}^\perp )$ is a cotorsion pair, which is called a cotorsion pair generated by $\mathsf {C}$. Recall from e.g. Enochs and Jenda [Reference Enochs and Jenda7] that a cotorsion pair $(\mathsf {C},\,\mathsf {W})$ is complete if for each object $M$ in $\mathsf {A}$ there is an exact sequence $0 \to M \to W \to C \to 0$ with $W\in \mathsf {W}$ and $C\in \mathsf {C}$ (or there is an exact sequence $0 \to W' \to C' \to M \to 0$ with $W'\in \mathsf {W}$ and $C'\in \mathsf {C}$). A cotorsion pair $(\mathsf {C},\,\mathsf {W})$ is called hereditary if $\mathsf {C}$ is closed under kernels of epimorphisms or $\mathsf {W}$ is closed under cokernels of monomorphisms.

1.4 (Flat objects)

Let $\mathsf {G}$ be a locally finitely presentable (in the sense of Crawley–Boevey [Reference Crawley-Boevey3, §1]) Grothendieck category admitting enough projectives. A sequence

\[ 0 \to K\to M\to N\to 0 \]

in $\mathsf {G}$ is said to be pure if for each finitely presentable object $T$, the sequence

\[ 0\to \operatorname{Hom}_{\mathsf{G}}\left(T,K\right) \to \operatorname{Hom}_{\mathsf{G}}\left(T,M\right) \to \operatorname{Hom}_{\mathsf{G}}\left(T,N\right) \to 0 \]

is exact (in this case, $M\to N\to 0$ is called a pure epimorphism). An object $F$ in $\mathsf {G}$ is called flat if every epimorphism $M\to F$ is a pure epimorphism. It is not difficult to check that $F$ is flat if and only if it is a filtered colimit of finitely generated projective objects. Let ${\sf Flat(\mathsf {G})}$ be the subcategory of flat objects in $\mathsf {G}$. Then ${\sf Flat(\mathsf {G})}$ is closed under extensions, filtered colimits and pure epimorphisms by Rump [Reference Rump21, prop. 7], and so the pair $({\sf Flat(\mathsf {G})},\, {\sf Flat(\mathsf {G})}^\perp )$ is a complete and hereditary cotorsion pair generated by a set by the proof of [Reference Holm and Jørgensen14, prop. A.3], that is, there is a set $\mathsf {S}\subseteq {\sf Flat(\mathsf {G})}$ such that $\mathsf {S}^\perp = {\sf Flat(\mathsf {G})}^\perp$.

1.5 It follows from [Reference Holm and Jørgensen14, prop. 3.12] that ${_{\mathcal {C},\,A}\mathsf {Mod}}$ is a locally finitely presentable Grothendieck category admitting enough projectives. Let $\mathsf {Flat}(\mathcal {C},\,A)$ be the subcategory of flat objects in ${_{\mathcal {C},\,A}\mathsf {Mod}}$. An object $X$ in ${_{\mathcal {C},\,A}\mathsf {Mod}}$ is called cotorsion if it is in $\mathsf {Flat}(\mathcal {C},\,A)^\perp$. Denote the subcategory of cotorsion objects by $\mathsf {Cot}(\mathcal {C},\,A)$. Then by 1.4, the pair $(\mathsf {Flat}(\mathcal {C},\,A),\, \mathsf {Cot}(\mathcal {C},\,A))$ is a complete and hereditary cotorsion pair generated by a set.

1.6 (Tensor products in ${_{\mathcal {C},\,A}\mathsf {Mod}}$)

Following Oberst and Röhrl [Reference Oberst and Röhrl19], we define a tensor product functor $-\otimes _{\mathcal {C},A}-: \mathsf {Mod}_{\mathcal {C},\,A}\times {_{\mathcal {C},\,A}\mathsf {Mod}}\to {_{\mathbb {k}}\mathsf {Mod}}$; it is right exact in each variable.

Let $X\in {_{\mathcal {C},\,A}\mathsf {Mod}}$. We construct for any $\mathbb {k}$-module $G$, an object $\operatorname {Hom}_{\mathbb {k}}\left (X,G\right )\in \mathsf {Mod}_{\mathcal {C},\,A}$ as follows:

• • For each $q\in \mathcal {C}^{\sf op}$ let $\operatorname {Hom}_{\mathbb {k}}\left (X,G\right )(q)=\operatorname {Hom}_{\mathbb {k}}\left (X(q),G\right )\in \mathsf {Mod}_{A}$.

• • For each morphism $f: p\to q$ in $\mathcal {C}^{\sf op}$ let $\operatorname {Hom}_{\mathbb {k}}\left (X,G\right )(f)=\operatorname {Hom}_{\mathbb {k}}\left (X(f^{\sf op}),G\right ):\operatorname {Hom}_{\mathbb {k}}\left (X(p),G\right )\to \operatorname {Hom}_{\mathbb {k}}\left (X(q),G\right )$.

It is evident that $\operatorname {Hom}_{\mathbb {k}}\left (X,-\right )$ is a functor from the category ${_{\mathbb {k}}\mathsf {Mod}}$ to $\mathsf {Mod}_{\mathcal {C},\,A}$. This functor is left exact and preserves arbitrary products, so it has a left adjoint functor from $\mathsf {Mod}_{\mathcal {C},\,A}$ to ${_{\mathbb {k}}\mathsf {Mod}}$, which is denoted $-\otimes _{\mathcal {C},A} X$ and will play the role of the tensor product in our work. Then for each $Y\in \mathsf {Mod}_{\mathcal {C},\,A}$ there are natural isomorphisms

(1.6.1)\begin{align} \operatorname{Hom}_{\mathbb {k}}\left(Y\otimes_{\mathcal{C},A} X,G\right)\cong \operatorname{Hom}_{\mathcal{C}^{\sf op},A^{\sf op}}\left(Y,\operatorname{Hom}_{\mathbb {k}}\left(X,G\right)\right)\cong\operatorname{Hom}_{\mathcal{C},A}\left(X,\operatorname{Hom}_{\mathbb {k}}\left(Y,G\right)\right). \end{align}

Notation 1.7 Let $\mathsf {Prj}(\mathcal {C},\,A)$ (resp., $\mathsf {Inj}(\mathcal {C},\,A)$) be the subcategory of projective objects (resp., injective objects) in ${_{\mathcal {C},\,A}\mathsf {Mod}}$. In the case that $A=\mathbb {k}$, we write ${_{\mathcal {C}}\mathsf {Mod}}$ (resp., $\mathsf {Prj}(\mathcal {C})$, $\mathsf {Inj}(\mathcal {C})$, $\mathsf {Flat}(\mathcal {C})$ and $\mathsf {Cot}(\mathcal {C})$) instead of ${_{\mathcal {C},\,\mathbb {k}}\mathsf {Mod}}$ (resp., $\mathsf {Prj}(\mathcal {C},\,\mathbb {k})$, $\mathsf {Inj}(\mathcal {C},\,\mathbb {k})$, $\mathsf {Flat}(\mathcal {C},\,\mathbb {k})$ and $\mathsf {Cot}(\mathcal {C},\,\mathbb {k})$), and we write $-\otimes _{\mathcal {C}}-$ instead of $-\otimes _{\mathcal {C},\mathbb {k}}-$.

1.8 By [Reference Holm and Jørgensen14, Corollary 3.9], for every object $q$ in $\mathcal {C}$ there is an adjoint triple $(F_{q},\,E_{q},\,G_{q})$ as follows:

Proof. Statements (a) and (b) hold since the functors $F_q$ and $G_q$ are exact. For $X \in \mathsf {Flat}(\mathcal {C},\,A)$, one has $X \cong {\rm colim}_iP_i$ with $P_i \in \mathsf {Prj}(\mathcal {C},\,A)$, and deduce by (a) that

\[ X(q) = E_q(X) \cong E_q({\rm colim}_i P_i) \cong {\rm colim}_i E_q(P_i) = {\rm colim}_i P_i(q) \in\mathsf{Flat}(A), \]

so the statement (c) is true.

To verify the statement (d), we fix a flat left $A$-module $M$ and write $M = {\rm colim}_i T_i$ with each $T_i$ a projective left $A$-module. It follows that each $F_q(T_i)$ is projective and $F_q(M)\cong {\rm colim}_iF_q(T_i)$ is flat in ${_{\mathcal {C},\,A}\mathsf {Mod}}$. By lemma 1.9, one has

\[ \operatorname{Ext}_{A}^{1}\left(M,X(q)\right)\cong \operatorname{Ext}_{\mathcal{C},A}^{1}\left(F_q(M),X\right)=0. \]

Therefore, $X(q)$ is a cotorsion left $A$-module.

Proof. Take an arbitrary flat object $X$ in ${_{\mathcal {C},\,A}\mathsf {Mod}}$. Then by lemma 1.9

\[ \operatorname{Ext}_{\mathcal{C},A}^{1}\left(X,G_q(N)\right) \cong \operatorname{Ext}_{A}^{1}\left(X(q),N\right)=0, \]

as $X(q)$ is flat by proposition 1.10(c).

1.12 Let $(-)^{\natural }$ denote the forgetful functor from ${_{\mathcal {C},\,A}\mathsf {Mod}}$ to ${_{\mathcal {C}}\mathsf {Mod}}$. Then $(-)^{\natural }$ is an exact functor and there is an adjoint triple as follows (see [Reference Holm and Jørgensen14, cor. 3.5]):

Proof. In the following, we use the symbols ${\rm pd}_\mathcal {C}(-)$, $\mathrm {id}_\mathcal {C}(-)$ and ${\rm fd}_\mathcal {C}(-)$ to denote the projective, injective and flat dimension of objects in ${_{\mathcal {C}}\mathsf {Mod}}$, respectively.

(a). For each object $q$ in $\mathcal {C}$, one has $F_q(A)^\natural = \mathcal {C}(q,\,-) \otimes _\mathbb {k} A^\natural$. Since the functor

\[ \mathcal{C}(q,-) \otimes_\mathbb {k}-: {_{\mathbb {k}}\mathsf{Mod}} \to {_{\mathcal{C}}\mathsf{Mod}} \]

preserves projective objects and is exact, we deduce that

\[ {\rm pd}_\mathcal{C}(F_q(A)^\natural) \leqslant {\rm pd}_\mathbb {k} A^\natural = {\rm pd}_\mathbb {k} A. \]

Thus by [Reference Holm and Jørgensen14, prop. 3.12(a)], one has ${\rm pd}_\mathcal {C} X^\natural \leqslant {\rm pd}_\mathbb {k} A$.

(b). For each object $q$ in $\mathcal {C}$ and $I \in \mathsf {Inj}(A)$, one has $G_q(I)^\natural = \operatorname {Hom}_{\mathbb {k}}\left (\mathcal {C}(-,\,q),I^\natural \right )$. Since the functor $\operatorname {Hom}_{\mathbb {k}}\left (\mathcal {C}(-,\,q),-\right ): {_{\mathbb {k}}\mathsf {Mod}}\to {_{\mathcal {C}}\mathsf {Mod}}$ preserves injective objects and is exact, one has

\[ \mathrm{id}_\mathcal{C}(G_q(I)^\natural) \leqslant \mathrm{id}_\mathbb {k} I^\natural=\mathrm{id}_\mathbb {k} I \leqslant {\rm fd}_\mathbb {k} A, \]

where the last inequality holds by Avramov and Foxby [Reference Avramov and Foxby1, cor. 4.2(b)(I)]. It then follows from [Reference Holm and Jørgensen14, prop. 3.12(b)] that $\mathrm {id}_\mathcal {C} X^\natural \leqslant {\rm fd}_\mathbb {k} A$.

(c). Write $X = {\rm colim}_iP_i$ with each $P_i \in \mathsf {Prj}(\mathcal {C},\,A)$. Then $X^\natural = {\rm colim}_iP_i^\natural$. By statement (a), one has that ${\rm pd}_\mathcal {C}(P_i^\natural ) \leqslant {\rm pd}_\mathbb {k} A$ for each $i$, so ${\rm fd}_\mathcal {C}(X^\natural ) \leqslant {\rm pd}_\mathbb {k} A$.

Proof. By Holm and Jørgensen [Reference Holm and Jørgensen13, lem. 1.2], there is an exact sequence

\[ 0 \to \operatorname{Ext}_{\mathcal{C},A}^{1}\left(G \otimes_\mathbb {k} A,X\right) \to \operatorname{Ext}_{\mathcal{C}}^{1}\left(G,X^{\natural}\right) \to \operatorname{Hom}_{\mathcal{C},A}\left(L_1(- \otimes_\mathbb {k} A)(G),X\right). \]

Statement (a) holds after we show that $L_1(-\otimes _\mathbb {k} A)(G) = 0$. For this purpose, consider the exact sequence

\[ 0 \to K \to P \to G \to 0 \]

in ${_{\mathcal {C}}\mathsf {Mod}}$ with $P \in \mathsf {Prj}(\mathcal {C})$. Clearly, $L_1(- \otimes _\mathbb {k} A)(P) = 0$, so it is enough to show the exactness of the sequence

(1.14.1)\begin{equation} 0 \to K\otimes_\mathbb {k} A \to P\otimes_\mathbb {k} A \to G\otimes_\mathbb {k} A \to 0 \end{equation}

in ${_{\mathcal {C},\,A}\mathsf {Mod}}$.

Consider the exact sequence

\[ 0 \to G \to F_{{-}1} \to F_{{-}2} \to \cdots \]

with each $F_i \in \mathsf {Flat}(\mathcal {C})$. For each object $q$ in $\mathcal {C}$, the sequence

\[ 0 \to G(q) \to F_{{-}1}(q) \to F_{{-}2}(q) \to \cdots \]

is exact and each $F_i(q)$ is a flat $\mathbb {k}$-module by proposition 1.10(c). Since ${\rm fd}_\mathbb {k} A<\infty$ by the assumption, one gets that $\operatorname {Tor}_{1}^{\mathbb {k}}\left (G(q),A\right ) = 0$ by dimension shifting. Thus, the sequence

\[ 0 \to K(q) \otimes_\mathbb {k} A \to P(q) \otimes_\mathbb {k} A \to G(q) \otimes_\mathbb {k} A \to 0 \]

is exact, which guarantees the exactness of sequence (1.14.1).

Statement (b) can be proved dually using [Reference Holm and Jørgensen13, lem. 1.3] and proposition 1.10(b), observing that ${\rm pd}_\mathbb {k} A<\infty$, and $((-)^{\natural },\, \operatorname {Hom}_{\mathbb {k}}\left (A,-\right ))$ is an adjoint pair.

Proof. Take an object $F$ in $\mathsf {Flat}(\mathcal {C})$ and write $F = {\rm colim}_iP_i$ with each $P_i \in \mathsf {Prj}(\mathcal {C})$. Since the functor $- \otimes _{\mathbb {k}} A$ preserves projectives, each $P_i \otimes _{\mathbb {k}} A$ is contained in $\mathsf {Prj}(\mathcal {C},\,A)$. Thus, $F \otimes _{\mathbb {k}} A$ belongs to $\mathsf {Flat}(\mathcal {C},\,A)$, and by lemma 1.14(a) one deduces that

\[ \operatorname{Ext}_{\mathcal{C}}^{1}\left(F,X^{\natural}\right) \cong \operatorname{Ext}_{\mathcal{C},A}^{1}\left(F \otimes_\mathbb {k} A,X\right) = 0 \]

because $X \in \mathsf {Cot}(\mathcal {C},\,A)$. Consequently, $X^\natural$ is contained in $\mathsf {Cot}(\mathcal {C})$.

1.16 (Gorenstein homological objects)

Following [Reference Enochs and Jenda7], an object $X$ in ${_{\mathcal {C},\, A}\mathsf {Mod}}$ is called Gorenstein projective if there is an exact sequence

\[ \cdots \to P_{1}\to P_{0}\to P_{{-}1} \to \cdots \]

in ${_{\mathcal {C},\,A}\mathsf {Mod}}$ with each $P_{i}$ projective such that $X\cong {\operatorname {Coker}(P_{1}\to P_{0})}$ and the sequence remains exact after applying the functor $\operatorname {Hom}_{\mathcal {C},\,A}\left (-,P\right )$ for every projective object $P$ in ${_{\mathcal {C},\,A}\mathsf {Mod}}$.

Dually, an object $X$ in ${_{\mathcal {C},\, A}\mathsf {Mod}}$ is called Gorenstein injective if there is an exact sequence

\[ \cdots \to I_{1}\to I_{0}\to I_{{-}1}\to\cdots \]

in ${_{\mathcal {C},\,A}\mathsf {Mod}}$ with each $I_{i}$ injective such that $X\cong {\operatorname {Coker}(I_{1}\to I_{0})}$ and the sequence remains exact after applying the functor $\operatorname {Hom}_{\mathcal {C},\,A}\left (I,-\right )$ for every injective object $I$ in ${_{\mathcal {C},\,A}\mathsf {Mod}}$.

An object $X$ in ${_{\mathcal {C},\, A}\mathsf {Mod}}$ is called Gorenstein flat if there is an exact sequence

\[ \cdots \to F_{1}\to F_{0}\to F_{{-}1}\to\cdots \]

in ${_{\mathcal {C},\, A}\mathsf {Mod}}$ with each $F_{i}$ flat such that $X\cong {\operatorname {Coker}(F_{1}\to F_{0})}$ and the sequence remains exact after applying the functor $E\otimes _{\mathcal {C},A}-$ for every injective object $E$ in $\mathsf {Mod}_{\mathcal {C},\,A}$. Similarly, one can define projectively coresolved Gorenstein flat objects in ${_{\mathcal {C},\,A}\mathsf {Mod}}$ when flat objects in the above exact sequence are replaced by projective objects; see [Reference Šaroch and Št'ovíček22].

Let $\mathsf {GP}(\mathcal {C},\,A)$ (resp., $\mathsf {GI}(\mathcal {C},\,A)$, $\mathsf {GF}(\mathcal {C},\,A)$, $\mathsf {PGF}(\mathcal {C},\,A)$) be the subcategory of Gorenstein projective objects (resp., Gorenstein injective objects, Gorenstein flat objects, projectively coresolved Gorenstein flat objects) in ${_{\mathcal {C},\,A}\mathsf {Mod}}$. In the case that $A=\mathbb {k}$, we write $\mathsf {GP}(\mathcal {C})$ (resp., $\mathsf {GI}(\mathcal {C})$, $\mathsf {GF}(\mathcal {C})$, $\mathsf {PGF}(\mathcal {C})$) instead of $\mathsf {GP}(\mathcal {C},\,A)$ (resp., $\mathsf {GI}(\mathcal {C},\,A)$, $\mathsf {GF}(\mathcal {C},\,A)$, $\mathsf {PGF}(\mathcal {C},\,A)$).

Proof. It follows from lemma 1.14(a) that $\mathsf {E} = \{ G \otimes _\mathbb {k} A\ |\ G \in \mathsf {PGF}(\mathcal {C})\}^\perp$, so $(^{\perp }\mathsf {E},\, \mathsf {E})$ is a cotorsion pair in ${_{\mathcal {C},\,A}\mathsf {Mod}}$; see 1.1. It is clear that $(^{\perp }\mathsf {E},\, \mathsf {E})$ is hereditary as $\mathsf {E}$ is closed under cokernels of monomorphisms. By lemma 2.1(a), the cotorsion pair $(\mathsf {PGF}(\mathcal {C}),\, {\mathsf {PGF}(\mathcal {C})}^\perp )$ is generated by a set $\mathsf {S}$, that is, $\mathsf {S} \subseteq \mathsf {PGF}(\mathcal {C})$ and $\mathsf {S}^\perp = {\mathsf {PGF}(\mathcal {C})}^\perp$. One deduces that $\{G\otimes _\mathbb {k} A\ | G\in \mathsf {S}\}^\perp = \mathsf {E}$ again by proposition 1.14(a). Consequently, the cotorsion pair $(^{\perp }\mathsf {E},\, \mathsf {E})$ is generated by a set, so $(^{\perp }\mathsf {E},\, \mathsf {E})$ is complete; see e.g. Göbel and Trlifaj [Reference Göbel and Trlifaj11, thm. 6.11(b)]. Since the subcategory ${\mathsf {PGF}(\mathcal {C})}^\perp$ is thick by lemma 2.1(a), so is $\mathsf {E}$ clearly.

We prove that $\mathsf {E}$ contains the projective objects, which implies that $(^{\perp }\mathsf {E},\, \mathsf {E})$ is a projective cotorsion pair in ${_{\mathcal {C},\,A}\mathsf {Mod}}$ by [Reference Gillespie10, prop. 3.7]. For each $X \in \mathsf {Prj}(\mathcal {C},\,A)$, one has ${\rm pd}_\mathcal {C} X^{\natural }<\infty$ by proposition 1.13(a). It follows from lemma 2.1(a) that $\operatorname {Ext}_{\mathcal {C}}^{i}\left (G,P\right ) = 0$ for each $G \in \mathsf {PGF}(\mathcal {C})$, $P \in \mathsf {Prj}(\mathcal {C})$ and $i \geqslant 1$, so by dimension shifting one concludes that $X^{\natural }$ is in $\mathsf {PGF}(\mathcal {C})^\perp$. Thus, one has $X\in \mathsf {E}$, as desired.

Proof. By lemma 1.14(b), one has $\mathsf {T} = {^\perp \{\operatorname {Hom}_{\mathbb {k}}\left (A,G\right )\ | \ G \in \mathsf {GI}(\mathcal {C})\}}$, so $(\mathsf {T},\, \mathsf {T}^\perp )$ is a cotorsion pair in ${_{\mathcal {C},\,A}\mathsf {Mod}}$; it is hereditary as $\mathsf {T}$ is closed under kernels of epimorphisms. It is clear that the subcategory $\mathsf {T}$ is thick since ${^\perp \mathsf {GI}(\mathcal {C})}$ is thick by lemma 2.1(b). For each $X \in \mathsf {Inj}(\mathcal {C},\,A)$, one has $\mathrm {id}_\mathcal {C} X^{\natural }<\infty$ by proposition 1.13(b). It follows from lemma 2.1(b) that $\operatorname {Ext}_{\mathcal {C}}^{i}\left (I,G\right ) = 0$ for each $G \in \mathsf {GI}(\mathcal {C})$, $I \in \mathsf {Inj}(\mathcal {C})$ and $i \geqslant 1$, so by dimension shifting one gets that $X^{\natural }$ is in $^\perp \mathsf {GI}(\mathcal {C})$. Thus, one has $X \in \mathsf {T}$, and hence $\mathsf {T}$ contains all injective objects in ${_{\mathcal {C},\,A}\mathsf {Mod}}$.

Now we assume that the subcategory $^\perp \mathsf {GI}(\mathcal {C})$ is closed under pure quotients. Then $^\perp \mathsf {GI}(\mathcal {C})$ is also closed under pure submodules since $(^{\perp }\mathsf {GI}(\mathcal {C}),\,\mathsf {GI}(\mathcal {C}))$ is a hereditary cotorsion pair by lemma 2.1(b). The forgetful functor $(-)^{\natural }$ preserves colimits and pure exact sequence by [Reference Holm and Jørgensen14, lem. 5.7]. It follows that the subcategory $\mathsf {T}$ is closed under pure submodules and pure quotients, and furthermore is closed under filtered colimits since the subcategory $^\perp \mathsf {GI}(\mathcal {C})$ is closed under filtered colimits by lemma 2.1(b). Thus by [Reference Holm and Jørgensen14, thm. A.3], the cotorsion pair $(\mathsf {T},\, \mathsf {T}^\perp )$ is perfect. Moreover, by [Reference Gillespie10, prop. 3.6], $(\mathsf {T},\, \mathsf {T}^\perp )$ is an injective cotorsion pair.

Proof. By proposition 2.2, there is a set $\mathsf {S} \subseteq {^\perp \mathsf {E}}$ such that $\mathsf {S}^\perp = \mathsf {E}$. On the other hand, the cotorsion pair $(\mathsf {Flat}(\mathcal {C},\,A),\, \mathsf {Cot}(\mathcal {C},\,A))$ is generated by a set $\mathsf {R}$ (see 1.5), that is, $\mathsf {R} \subseteq \mathsf {Flat}(\mathcal {C},\,A)$ and $\mathsf {R}^\perp = \mathsf {Cot}(\mathcal {C},\,A)$. Thus, one has

\[ (\mathsf{R} \cup \mathsf{S})^\perp{=} \mathsf{R}^\perp{\cap} \mathsf{S}^\perp{=} \mathsf{Cot}(\mathcal{C},A) \cap \mathsf{E}, \]

which implies that $(^{\perp }(\mathsf {Cot}(\mathcal {C},\,A) \cap \mathsf {E}),\, \mathsf {Cot}(\mathcal {C},\,A) \cap \mathsf {E})$ is a complete cotorsion pair in ${_{\mathcal {C},\,A}\mathsf {Mod}}$ generated by a set; see [Reference Göbel and Trlifaj11, thm. 6.11(b)]. The subcategory $\mathsf {Cot}(\mathcal {C},\,A) \cap \mathsf {E}$ is closed under cokernels of monomorphisms as so are $\mathsf {Cot}(\mathcal {C},\,A)$ and $\mathsf {E}$ by proposition 2.2. Thus, the cotorsion pair

\[ (^{{\perp}}(\mathsf{Cot}(\mathcal{C},A) \cap \mathsf{E}), \mathsf{Cot}(\mathcal{C},A) \cap \mathsf{E}) \]

is hereditary.

We next prove the equality $^{\perp }(\mathsf {Cot}(\mathcal {C},\,A) \cap \mathsf {E}) \cap \mathsf {E} = \mathsf {Flat}(\mathcal {C},\,A)$. It is clear that $\mathsf {Flat}(\mathcal {C},\,A) = {^{\perp }\mathsf {Cot}(\mathcal {C},\,A)} \subseteq {^{\perp }(\mathsf {Cot}(\mathcal {C},\,A) \cap \mathsf {E})}$. On the other hand, if $X \in \mathsf {Flat}(\mathcal {C},\,A)$, then $X^{\natural }$ has finite flat dimension in ${_{\mathcal {C}}\mathsf {Mod}}$ by lemma 1.13(c), and so it follows from [Reference Šaroch and Št'ovíček22, thm. 4.11] that $X^{\natural }$ is in $\mathsf {PGF}(\mathcal {C})^\perp$. This implies that $X$ is in $\mathsf {E}$. Thus, one has

\[ \mathsf{Flat}(\mathcal{C},A) \subseteq{^{{\perp}} (\mathsf{Cot}(\mathcal{C},A) \cap \mathsf{E}) \cap \mathsf{E}}. \]

To prove the inclusion of the other direction, we take an arbitrary

\[ Y \in{^{{\perp}}(\mathsf{Cot}(\mathcal{C},A) \cap \mathsf{E}) \cap\mathsf{E}}. \]

Since $(\mathsf {Flat}(\mathcal {C},\,A),\, \mathsf {Cot}(\mathcal {C},\,A))$ is a complete cotorsion pair (see 1.5), there is an exact sequence

(2.4.1)\begin{align} 0 \to C \to F \to Y \to 0 \end{align}

with $C \in \mathsf {Cot}(\mathcal {C},\,A)$ and $F \in \mathsf {Flat}(\mathcal {C},\,A) \subseteq \mathsf {E}$ as proved before. Thus, $C$ is contained in $\mathsf {E}$ as $\mathsf {E}$ is thick by proposition 2.2, so $\operatorname {Ext}_{\mathcal {C},\,A}^{1}\left (Y,C\right )=0$. Consequently, sequence (2.4.1) splits, and so $Y$ belongs to $\mathsf {Flat}(\mathcal {C},\,A)$.

Proof. We mention that $(\mathsf {Flat}(\mathcal {C},\,A),\, \mathsf {Cot}(\mathcal {C},\,A))$ is a complete and hereditary cotorsion pair in ${_{\mathcal {C},\,A}\mathsf {Mod}}$; see 1.5. Thus, by Gillespie [Reference Gillespie9, thm. 1.1] and lemma 2.4, there is a hereditary abelian model structure

\[ (^{{\perp}}(\mathsf{Cot}(\mathcal{C},A)\cap\mathsf{E}), \mathsf{W}, \mathsf{Cot}(\mathcal{C},A)) \]

on ${_{\mathcal {C},\,A}\mathsf {Mod}}$, where the subcategory $\mathsf {W}$ can be described as follows:

\begin{align*} \mathsf{W}& =\{X\in{_{\mathcal{C},A}\mathsf{Mod}}\ |\ \exists\ s.e.s\ X\rightarrowtail U\twoheadrightarrow F\ with\ U\\ & \quad \in\mathsf{Cot}(\mathcal{C},A)\cap\mathsf{E}\ and\ F\in\mathsf{Flat}(\mathcal{C},A)\}. \end{align*}

We then prove that there is an equality $\mathsf {W}=\mathsf {E}$.

Take an object $X\in \mathsf {W}$. Then there is an exact sequence

\[ 0\to X\to U\to F\to 0 \]

in ${_{\mathcal {C},\,A}\mathsf {Mod}}$ with $U\in \mathsf {Cot}(\mathcal {C},\,A)\cap \mathsf {E}$ and $F\in \mathsf {Flat}(\mathcal {C},\,A)$. Thus, the sequence

\[ 0\to X^{\natural}\to U^{\natural}\to F^{\natural}\to 0 \]

is exact, where $U^{\natural }\in \mathsf {PGF}(\mathcal {C})^\perp$ and $F^{\natural }$ has finite flat dimension by proposition 1.13(c). So it is easy to check that $F^{\natural }$ is in $\mathsf {PGF}(\mathcal {C})^\perp$ as $\mathsf {Flat}(\mathcal {C})\subseteq \mathsf {PGF}(\mathcal {C})^\perp$ by lemma 2.1(c). The subcategory $\mathsf {PGF}(\mathcal {C})^\perp$ is thick by lemma 2.1(a), so $X^{\natural }$ is in $\mathsf {PGF}(\mathcal {C})^\perp$. This implies that $X$ is in $\mathsf {E}$. To prove the inclusion of the other direction, we take an object $Y\in \mathsf {E}$. By the completeness of the cotorsion pair $(\mathsf {Flat}(\mathcal {C},\,A),\, \mathsf {Cot}(\mathcal {C},\,A))$, there is an exact sequence

\[ 0\to Y \to U' \to F' \to 0 \]

in ${_{\mathcal {C},\,A}\mathsf {Mod}}$ with $U'\in \mathsf {Cot}(\mathcal {C},\,A)$ and $F'\in \mathsf {Flat}(\mathcal {C},\,A)$. Consider the exact sequence

\[ 0\to Y^{\natural} \to (U')^{\natural} \to (F')^{\natural} \to 0 \]

in ${_{\mathcal {C}}\mathsf {Mod}}$. The object $(F')^{\natural }$ has finite flat dimension by proposition 1.13(c), and so it is in $\mathsf {PGF}(\mathcal {C})^\perp$ as proved before. On the other hand, $Y^{\natural }$ is in $\mathsf {PGF}(\mathcal {C})^\perp$ as $Y\in \mathsf {E}$. Thus, by lemma 2.1(a), one has $(U')^{\natural }\in \mathsf {PGF}(\mathcal {C})^\perp$, and so $U'$ is in $\mathsf {E}$. This implies that $Y$ is in $\mathsf {W}$.

Now one gets a hereditary abelian model structure

\[ (^{{\perp}}(\mathsf{Cot}(\mathcal{C},A)\cap\mathsf{E}), \mathsf{E}, \mathsf{Cot}(\mathcal{C},A)) \]

on ${_{\mathcal {C},\,A}\mathsf {Mod}}$, as desired.

Setup 2.6 In the rest of this section we let $\mathcal {Q}$ be a small $\mathbb {k}$-linear category satisfying the following self-dual conditions specified in [Reference Holm and Jørgensen14, Setup 2.5]:

1. (1) Each $\mathbb {k}$-module $\mathcal {Q}(p,\,q)$ is finitely generated and projective.

2. (2) $\mathcal {Q}$ satisfies the local boundedness, that is for any object $q$ in $\mathcal {Q}$, there are only finitely many objects in $\mathcal {Q}$ mapping nontrivially into or out of $q$.

3. (3) There exists a Serre functor, that is a $\mathbb {k}$-linear auto-equivalence $\mathbb {S}: \mathcal {Q} \to \mathcal {Q}$ such that $\mathcal {Q}(p,\,q)\cong \operatorname {Hom}_{\mathbb {k}}\left (\mathcal {Q}(q,\,\mathbb {S}(p)),\mathbb {k}\right ).$

4. (4) For any object $q$ in $\mathcal {Q}$, the unit map $\mathbb {k}\to \mathcal {Q}(q,\,q)$ (via $x\mapsto x\cdot \mathrm {id}_q$) has a $\mathbb {k}$-module retraction; whence there is a $\mathbb {k}$-module decomposition $\mathcal {Q}(q,\,q)=\mathbb {k}\cdot \mathrm {id}_q\oplus \tau _q$.

Proof. The model structure in the statement can be obtained by theorem 2.5. As shown in remark 2.7, ${_{\mathcal {Q}}\mathsf {Mod}}$ is a locally Gorenstein category, so one has

\[ \mathsf{PGF}(\mathcal{Q})^\perp{=}\mathsf{GP}(\mathcal{Q})^\perp{=}\mathsf{Prj}(\mathcal{Q})_{<\infty}, \]

where $\mathsf {Prj}(\mathcal {Q})_{<\infty }$ is the subcategory of objects in ${_{\mathcal {Q}}\mathsf {Mod}}$ with finite projective dimension. Thus, one gets that

\[ \mathsf{E} = \{X \in {_{\mathcal{Q},A}\mathsf{Mod}}\ |\ X^{\natural}\ \text{has finite projective dimension in}\ {_{\mathcal{Q}}\mathsf{Mod}}\} \]

as desired.

3.1 (Frobenius functor)

Let $F: \sf C\to \sf D$ and $G: \sf D \to \sf C$ be two additive functors between abelian categories, and let $\alpha : \sf C \to \sf C$ and $\beta : \sf D \to \sf D$ be two auto-equivalences. Recall from [Reference Chen and Ren2] that $(F,\, G)$ is called a Frobenius pair (of type $(\alpha,\, \beta )$) between $\sf C$ and $\sf D$ if both $(F,\, G)$ and $(G,\, \beta F\alpha )$ are adjoint pairs, and the functor $F$ is said to be a Frobenius functor if it fits into a Frobenius pair $(F,\, G)$; in this case $G$ is also a Frobenius functor.

3.4 The product category $\prod _{q\in \mathcal {Q}}{_{A}\mathsf {Mod}}$ is a locally finitely presentable Grothendieck category admitting enough projectives and injectives. Let $(M_{q})_{q\in \mathcal {Q}}$ be an object in $\prod _{q\in \mathcal {Q}}{_{A}\mathsf {Mod}}$. Then it is clear that $(M_{q})_{q\in \mathcal {Q}}$ is projective (resp., injective, flat) if and only if each left $A$-module $M_{q}$ is projective (resp., injective, flat) for $q\in \mathcal {Q}$. Also, one can check that an object $(M_{q})_{q\in \mathcal {Q}}$ in $\prod _{q\in \mathcal {Q}}{_{A}\mathsf {Mod}}$ is Gorenstein projectiveFootnote ¹ (resp., Gorenstein injective, Gorenstein flat) if and only if each left $A$-module $M_{q}$ is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat) for $q\in \mathcal {Q}$.

Proof. For $X\in {_{\mathcal {Q},\,A}\mathsf {Mod}}$ and $(M_{q})_{q\in \mathcal {Q}}\in \prod _{q\in \mathcal {Q}}{_{A}\mathsf {Mod}}$ there are isomorphisms

\begin{align*} \operatorname{Hom}_{\mathcal{Q},A}\left(i^{*}(M_{q})_{q\in \mathcal{Q}},X\right)& \cong\prod_{q\in \mathcal{Q}}\operatorname{Hom}_{\mathcal{Q},A}\left(\mathcal{Q}(q,-)\otimes_{\mathbb {k}}M_{q},X\right)\\ & \cong\prod_{q\in \mathcal{Q}}\operatorname{Hom}_{A}\left(M_{q},X(q)\right)\\ & \cong\operatorname{Hom}_{\prod_{q\in\mathcal{Q}}{_{A}\mathsf{Mod}}}\left((M_{q})_{q\in\mathcal{Q}},(X(q))_{q\in\mathcal{Q}}\right)\\ & \cong\operatorname{Hom}_{\prod_{q\in\mathcal{Q}}{_{A}\mathsf{Mod}}}\left((M_{q})_{q\in\mathcal{Q}},i_{*}X\right), \end{align*}

where the second isomorphism holds by 1.8. On the other hand, one has

\begin{align*} \operatorname{Hom}_{\prod_{q\in\mathcal{Q}}{_{A}\mathsf{Mod}}}\left(i_{*}(X),(M_{q})_{q\in \mathcal{Q}}\right)& \cong\prod_{q\in\mathcal{Q}}\operatorname{Hom}_{A}\left(X(q),M_{q}\right)\\ & \cong\prod_{q\in\mathcal{Q}}\operatorname{Hom}_{\mathcal{Q},A}\left(X,\operatorname{Hom}_{\mathbb {k}}\left(\mathcal{Q}(-,q),M_{q}\right)\right)\\ & \cong\operatorname{Hom}_{\mathcal{Q},A}\left(X,\prod_{q\in\mathcal{Q}}\operatorname{Hom}_{\mathbb {k}}\left(\mathcal{Q}(-,q),M_{q}\right)\right)\\ & =\operatorname{Hom}_{\mathcal{Q},A}\left(X,i^{!}(M_{q})_{q\in\mathcal{Q}}\right), \end{align*}

where the second isomorphism holds again by 1.8. Thus, $(i^{*},\,i_{*},\,i^{!})$ is an adjoint triple.

We mention that the Serre functor $\mathbb {S}$ induces an equivalence

\[ \mathbb{S}_{*}:{_{\mathcal{Q},A}\mathsf{Mod}}\to {_{\mathcal{Q},A}\mathsf{Mod}} \]

with $\mathbb {S}_{*}(X)(q)=X(\mathbb {S}(q))$ and $\mathbb {S}_{*}^{-1}(X)(q)=X(\mathbb {S}^{-1}(q))$ for each $X\in {_{\mathcal {Q},\,A}\mathsf {Mod}}$ and $q\in \mathcal {Q}$. Thus, for each $r\in \mathcal {Q}$, one has

\begin{align*} i^{*}(M_{q})_{q\in\mathcal{Q}}(r)& =\coprod_{q\in\mathcal{Q}}\mathcal{Q}(q,r)\otimes_{\mathbb {k}}M_{q}\\ & \cong\coprod_{q\in\mathcal{Q}}\operatorname{Hom}_{\mathbb {k}}\left(\mathcal{Q}(r,\mathbb{S}(q)),\mathbb {k}\right)\otimes_{\mathbb {k}}M_{q}\\ & \cong\coprod_{q\in\mathcal{Q}}\operatorname{Hom}_{\mathbb {k}}\left(\mathcal{Q}(r,\mathbb{S}(q)),M_{q}\right)\\ & \cong\coprod_{q\in\mathcal{Q}}\operatorname{Hom}_{\mathbb {k}}\left(\mathcal{Q}(\mathbb{S}^{{-}1}(r),q),M_{q}\right)\\ & =\prod_{q\in\mathcal{Q}}\operatorname{Hom}_{\mathbb {k}}\left(\mathcal{Q}(\mathbb{S}^{{-}1}(r),q),M_{q}\right)\\ & =i^{!}(M_{q})_{q\in\mathcal{Q}}(\mathbb{S}^{{-}1}(r))\\ & =\mathbb{S}_{*}^{{-}1}i^{!}(M_{q})_{q\in\mathcal{Q}}(r), \end{align*}

where the first and third isomorphisms hold by the definition of Serre functors, the second isomorphism holds as $\mathcal {Q}(r,\,\mathbb {S}(q))$ is a finitely generated and projective $\mathbb {k}$-module, and the second equation follows from the fact that $\mathcal {Q}$ satisfies the local boundedness. Thus, one has $i^{*}\cong \mathbb {S}_{*}^{-1}\circ i^{!}$ by the naturality of the above isomorphisms, and so $i^{!}\cong \mathbb {S}_{*}\circ i^{*}.$ Since $(i^{*},\,i_{*})$ is an adjoint pair, it is easy to check that $(i^{!},\, i_{*}\circ \mathbb {S}_{*}^{-1})$ is an adjoint pair. This yields that $(i_{*},\, i^{!})$ is a Frobenius pair of type $(\mathbb {S}_{*}^{-1},\, \mathrm {id}_{\prod _{q\in \mathcal {Q}}{_{A}\mathsf {Mod}}})$.

Proof. It follows from lemma 3.5 that $i_{*}$ is a Frobenius functor, and it is faithful clearly. Thus, the desired conclusion holds by lemma 3.3.

Proof. We only prove the first isomorphism; the second one is proved similarly.

For each $p\in \mathcal {Q}$, one has

\begin{align*} G'_{q}(\operatorname{Hom}_{\mathbb {k}}\left(M,E\right))(p)& =\operatorname{Hom}_{\mathbb {k}}\left(\mathcal{Q}(q,p),\operatorname{Hom}_{\mathbb {k}}\left(M,E\right)\right)\\ & \cong\operatorname{Hom}_{\mathbb {k}}\left(\mathcal{Q}(q,p)\otimes_{\mathbb {k}}M,E\right)\\ & \cong\operatorname{Hom}_{\mathbb {k}}\left(\operatorname{Hom}_{\mathbb {k}}\left(\mathcal{Q}(p,\mathbb{S}(q)),\mathbb {k}\right)\otimes_{\mathbb {k}}M,E\right)\\ & \cong\operatorname{Hom}_{\mathbb {k}}\left(\operatorname{Hom}_{\mathbb {k}}\left(\mathcal{Q}(p,\mathbb{S}(q)),M\right),E\right)\\ & =\operatorname{Hom}_{\mathbb {k}}\left(G_{\mathbb{S}(q)}(M),E\right)(p), \end{align*}

where the second isomorphism holds as $\mathbb {S}$ is a Serre functor, and the third one holds since the $\mathbb {k}$-module $\mathcal {Q}(p,\,\mathbb {S}(q))$ is finitely generated and projective. It follows that $G'_{q}(\operatorname {Hom}_{\mathbb {k}}\left (M,E\right ))\cong \operatorname {Hom}_{\mathbb {k}}\left (G_{\mathbb {S}(q)}(M),E\right )$ by the naturality of the above isomorphisms.

Proof. For each $E\in {_{\mathbb {k}}\mathsf {Mod}}$, there are isomorphisms

\begin{align*} \operatorname{Hom}_{\mathbb {k}}\left(G'_{q}(N)\otimes_{\mathcal{Q},A}X,E\right)& \cong\operatorname{Hom}_{\mathcal{Q},A}\left(X,\operatorname{Hom}_{\mathbb {k}}\left(G'_{q}(N),E\right)\right)\\ & \cong\operatorname{Hom}_{\mathcal{Q},A}\left(X,G_{\mathbb{S}(q)}(\operatorname{Hom}_{\mathbb {k}}\left(N,E\right))\right)\\ & \cong\operatorname{Hom}_{A}\left(X(\mathbb{S}(q)),\operatorname{Hom}_{\mathbb {k}}\left(N,E\right)\right)\\ & \cong\operatorname{Hom}_{\mathbb {k}}\left(N\otimes_{A}X(\mathbb{S}(q)),E\right), \end{align*}

where the first isomorphism follows from (1.6.1), the second one holds by lemma 3.8, and the third one holds as $(E_{\mathbb {S}(q)},\,G_{\mathbb {S}(q)})$ is an adjoint pair; see 1.8. Thus, there is an isomorphism $G'_{q}(N)\otimes _{Q,A}X\cong N\otimes _{A}X(\mathbb {S}(q))$.

Proof. Fix an injective object $E$ in $\mathsf {Mod}_{\mathcal {Q},\,A}$. Let $I$ be a faithful injective right $A$-module. Then for all $q\in \mathcal {Q}$ the objects $G'_{q}(I)$ are injective and cogenerate $\mathsf {Mod}_{\mathcal {Q},\,A}$; see [Reference Holm and Jørgensen14, Proposition 3.12]. Thus, $E$ is a direct summand of an object of the form $\prod _{q\in \mathcal {Q}}\ G'_{q}(I).$ We mention that $\prod _{q\in \mathcal {Q}}\ G'_{q}(I) \cong \coprod _{q\in \mathcal {Q}} G'_{q}(I)$ by [Reference Holm and Jørgensen15, Proposition 3.7]. Since the functor $\operatorname {Tor}_{\mathcal {Q},\,A}^{i}\left (-,X\right )$ converts coproducts to coproducts, without loss of generality we may assume that $E$ has the form $G'_{q}(I)$. Let $\mathbf {P}$ be a projective resolution of $X$. Then for each $i\geq 1$ one has

\begin{align*} \operatorname{Tor}_{\mathcal{Q},A}^{i}\left(G'_{q}(I),X\right)& \cong \mathrm{H}_{i}(G'_{q}(I)\otimes_{\mathcal{Q},A}\mathbf{P})\\ & \cong \mathrm{H}_{i}(I\otimes_{A}\mathbf{P}(\mathbb{S}(q)))\\ & \cong\operatorname{Tor}_{A}^{i}\left(I,X(\mathbb{S}(q))\right)\\ & =0, \end{align*}

where the second isomorphism holds by lemma 3.9 and the equality holds as $X(\mathbb {S}(q))$ is Gorenstein flat. This completes the proof.

Proof. For the ‘only if’ part, let $X$ be a Gorenstein flat object in ${_{\mathcal {Q},\,A}\mathsf {Mod}}$. Then there is an exact sequence

\[ \mathbf{F}= \cdots\to F_{1}\to F_{0}\to F_{{-}1}\to F_{{-}2}\to\cdots \]

of flat objects in ${_{\mathcal {Q},\,A}\mathsf {Mod}}$ such that $X\cong \operatorname{Coker} (F_{1}\to F_{0})$ and the sequence remains exact after applying the functor $E\otimes _{\mathcal {Q},A}-$ for each injective object $E$ in $\mathsf {Mod}_{\mathcal {Q},\,A}$. Let $I$ be an injective right $A$-module. Then it follows from lemma 3.9 that there is an isomorphism $I\otimes _{A}\mathbf {F}(q)\cong G'_{\mathbb {S}^{-1}(q)}(I)\otimes _{\mathcal {Q},A}\mathbf {F}$ for each $q\in \mathcal {Q}$. We mention that $G'_{\mathbb {S}^{-1}(q)}(I)$ is an injective object in $\mathsf {Mod}_{\mathcal {Q},\,A}$. Then the $\mathbb {k}$-complex $G'_{\mathbb {S}^{-1}(q)}(I)\otimes _{\mathcal {Q},A}\mathbf {F}$ is exact, and so $I\otimes _{A}\mathbf {F}(q)$ is exact. On the other hand, $\mathbf {F}(q)$ is an exact sequence of flat left $A$-modules by proposition 1.10, which yields that $X(q)$ is Gorenstein flat for each $q\in \mathcal {Q}$.

For the ‘if’ part, let $X$ be an object in ${_{\mathcal {Q},\,A}\mathsf {Mod}}$ such that each left $A$-module $X(q)$ is Gorenstein flat for $q\in \mathcal {Q}$. Then $i_{*}(X)=(X(q))_{q\in \mathcal {Q}}$ is a Gorenstein flat object in the product category $\prod _{q\in \mathcal {Q}}{}_{A}\mspace {-1mu}\operatorname {Mod}$. Thus, there is an exact sequence

\[ 0\to i_{*}(X)\xrightarrow{f'}F\to Y\to 0 \]

in $\prod _{q\in \mathcal {Q}}{}_{A}\mspace {-1mu}\operatorname {Mod}$ with $F$ flat and $Y$ Gorenstein flat. Denote by $\eta$ and $\varepsilon$ the unit and counit of the adjoint pair $(i_{*},\,i^{!})$, respectively. Set $f=i^{!}(f')\circ \eta _{X}:X\to i^{!}(F)$. Since $i_{*}$ is faithful and $i^{!}$ is exact, it follows that $f$ is a monomorphism. Consequently, one gets an exact sequence

\[ 0\to X\xrightarrow{f}i^{!}(F)\to X'\to 0 \]

in ${}_{\mathcal {Q},\,A}\mspace {-1mu}\operatorname {Mod}$. Since $f'=\varepsilon _{F}\circ i_{*}(f)$, there is a pull-back diagram

which yields an exact sequence

\[ 0\to i_{*}i^{!}(F)\to i_{*}(X')\oplus F\to Y\to 0. \]

We mention that $i_{*}$ and $i^{!}$ preserve flats as $(i_{*},\, i^{!})$ is a Frobenius pair by lemma 3.5. Then it follows from [Reference Šaroch and Št'ovíček22, Corollary 4.12] that $i_{*}(X')$ is a Gorenstein flat object in the product category $\prod _{q\in \mathcal {Q}}{}_{A}\mspace {-1mu}\operatorname {Mod}$. Thus, for each injective object $E$ in $\mathsf {Mod}_{\mathcal {Q},\,A}$ one has $\operatorname {Tor}_{\mathcal {Q},\,A}^{i}\left (E,X'\right )=0$ for all $i\geq 1$ by lemma 3.10. Set $F_{0}=i^{!}(F)$ and repeat the above argument for $X'$. One obtains inductively an exact sequence

\[ 0\to X\to F_{0}\xrightarrow{d_{0}} F_{{-}1}\xrightarrow{d_{{-}1}} F_{{-}2}\to \cdots \]

in ${}_{\mathcal {Q},\,A}\mspace {-1mu}\operatorname {Mod}$ with each $F_{i}$ flat such that it remains exact after applying the functor $E\otimes _{\mathcal {Q},A}-$ for every injective object $E$ in $\mathsf {Mod}_{\mathcal {Q},\,A}$. Thus, it follows from lemma 3.10 that $X$ is an Gorenstein flat object in ${}_{\mathcal {Q},\,A}\mspace {-1mu}\operatorname {Mod}$.