1. Preliminaries

Resolution dimensions. Let ${\mathcal{B}} \subseteq {\mathcal{C}}$ and $C \in {\mathcal{C}}$ . The resolution dimension of C with respect to ${\mathcal{B}}$ (or the ${\mathcal{B}}$ -resolution dimension of C, for short), denoted ${{\text{resdim}}}_{{\mathcal{B}}}(C)$ , is the smallest integer $m \geq 0$ such that there exists an exact sequence

\begin{equation*}B_m \rightarrowtail B_{m-1} \to \cdots \to B_1 \to B_0 \twoheadrightarrow C,\end{equation*}

where $B_k \in {\mathcal{B}}$ for every integer $0 \leq k \leq m$ . If such m does not exist, we set ${{\text{resdim}}}_{{\mathcal{B}}}(C) \,:\!=\, \infty$ . Dually, we have the concept of coresolution dimension of C with respect to ${\mathcal{B}}$ , denoted by ${{\text{coresdim}}}_{{\mathcal{B}}}(C)$ . With respect to these two homological dimensions, we shall frequently consider the following subcategories of ${\mathcal{C}}$ :

\begin{align*}{\mathcal{B}}^\wedge_m & \,:\!=\, \{ C \in {\mathcal{C}} \text{ : } {{\text{resdim}}}_{{\mathcal{B}}}(C) \leq m \}, & & \text{and} & {\mathcal{B}}^\wedge & \,:\!=\, \bigcup_{m \geq 0} {\mathcal{B}}^\wedge_m, \\{\mathcal{B}}^\vee_m & \,:\!=\, \{ C \in {\mathcal{C}} \text{ : } {{\text{coresdim}}}_{{\mathcal{B}}}(C) \leq m \}, & & \text{and} & {\mathcal{B}}^\vee & \,:\!=\, \bigcup_{m \geq 0} {\mathcal{B}}^\vee_m.\end{align*}

Orthogonality with respect to extension bifunctors. In any abelian category ${\mathcal{C}}$ , we can define the extension bifunctors ${{\text{Ext}}}^i_{{\mathcal{C}}}(-,-) \colon {\mathcal{C}}^{\text{op}} \times {\mathcal{C}} \longrightarrow {\mathsf{Mod}}(\mathbb{Z})$ , with $i \geq 1$ , in the sense of Yoneda. We shall also identify ${{\text{Ext}}}^0_{{\mathcal{C}}}(-,-)$ with the hom bifunctor ${{\text{Hom}}}_{{\mathcal{C}}}(-,-)$ . The reader can check for instance [Reference Sieg43] for a detailed treatise on this matter.

Given ${\mathcal{A}}, {\mathcal{B}} \subseteq {\mathcal{C}}$ and $i \geq 0$ , the notation ${{\text{Ext}}}^i_{{\mathcal{C}}}({\mathcal{A}},{\mathcal{B}}) = 0$ will mean that ${{\text{Ext}}}^i_{{\mathcal{C}}}(A,B)$ vanishes for every $A \in {\mathcal{A}}$ and $B \in {\mathcal{B}}$ . Recall that the right i-th orthogonal complement of ${\mathcal{A}}$ is defined by ${\mathcal{A}}^{\perp_i} \,:\!=\, \{ N \in {\mathcal{C}} \mbox{ : } {{\text{Ext}}}^i_{{\mathcal{C}}}({\mathcal{A}},N) = 0 \}$ , and the total right orthogonal complement of ${\mathcal{A}}$ by ${\mathcal{A}}^\perp \,:\!=\, \bigcap_{i \geq 1} {\mathcal{A}}^{\perp_i}$ . Dually, we have the i-th and the total left orthogonal complements ${}^{\perp_i}{\mathcal{B}}$ and ${}^{\perp}{\mathcal{B}}$ of ${\mathcal{B}}$ , respectively.

Relative homological dimensions. Given ${\mathcal{X}} \subseteq {\mathcal{C}}$ and $M \in {\mathcal{C}}$ , the relative projective dimension of M with respect to ${\mathcal{X}}$ , denoted ${{\text{pd}}}_{{\mathcal{X}}}(M)$ , is the smallest integer $n \geq 0$ such that ${{\text{Ext}}}_{{\mathcal{C}}}^i(M,{\mathcal{X}}) = 0$ for every $i > n$ . If such n does not exist, we set ${{\text{pd}}}_{{\mathcal{X}}}(M) = \infty$ . Furthermore, the relative projective dimension of ${\mathcal{Y}} \subseteq {\mathcal{C}}$ with respect to ${\mathcal{X}}$ is defined as ${{\text{pd}}}_{{\mathcal{X}}}({\mathcal{Y}}) \,:\!=\, \mbox{sup}\{ {{\text{pd}}}_{{\mathcal{X}}}(Y) \text{ : } Y \in {\mathcal{Y}} \}$ . Dually, we denote by ${{\text{id}}}_{{\mathcal{X}}}(M)$ and ${{\text{id}}}_{{\mathcal{X}}}({\mathcal{Y}})$ the relative injective dimension of M and ${\mathcal{Y}}$ , respectively, with respect to ${\mathcal{X}}$ . It can be seen that ${{\text{pd}}}_{{\mathcal{X}}}({\mathcal{Y}}) = {{\text{id}}}_{{\mathcal{Y}}}({\mathcal{X}})$ . If ${\mathcal{X}} = {\mathcal{C}}$ , we just write ${{\text{pd}}}(M)$ , ${{\text{pd}}}({\mathcal{Y}})$ , ${{\text{id}}}(M)$ and ${{\text{id}}}({\mathcal{Y}})$ , for the (absolute) projective and injective dimensions.

Resolving subcategories. Let ${\mathcal{P}}$ and ${\mathcal{I}}$ denote the subcategories of projective and injective objects in ${\mathcal{C}}$ , respectively. It is said that a subcategory ${\mathcal{X}}$ is resolving if ${\mathcal{P}} \subseteq {\mathcal{X}}$ and if it is closed under extensions and under epi-kernels (that is, under taking kernels of epimorphisms between objects in ${\mathcal{X}}$ ). If the dual properties hold true, then ${\mathcal{X}}$ is said to be coresolving. A subcategory is left thick if it is closed under extensions, epi-kernels, and direct summands. Right thick subcategories are defined dually. Finally, a subcategory is thick if it is both left and right thick. For ${\mathcal{X}} \subseteq {\mathcal{C}}$ , we shall denote by ${\mathsf{Thick}}({\mathcal{X}})$ the smallest thick subcategory of ${\mathcal{C}}$ containing ${\mathcal{X}}$ .

Approximations. Let ${\mathcal{X}} \subseteq {\mathcal{C}}$ . An ${\mathcal{X}}$ -precover of $C \in {\mathcal{C}}$ is a morphism $f \colon X \to C$ with $X \in {\mathcal{X}}$ such that the induced homomorphism ${{\text{Hom}}}_{{\mathcal{C}}}(X',f) \colon {{\text{Hom}}}_{{\mathcal{C}}}(X',X) \to {{\text{Hom}}}_{{\mathcal{C}}}(X',C)$ is epic for every $X' \in {\mathcal{X}}$ . An ${\mathcal{X}}$ -precover $f \colon X \to C$ is special if it is epic and ${{\text{Ker}}}(f) \in {\mathcal{X}}^{\perp_1}$ . The dual concept is called (special) ${\mathcal{X}}$ -preenvelope.

Cotorsion pairs. Two subcategories ${\mathcal{X}}$ and ${\mathcal{Y}}$ of objects in ${\mathcal{C}}$ form a cotorsion pair $({\mathcal{X}},{\mathcal{Y}})$ if they are complete with respect to the orthogonality relation defined by the vanishing of the functor ${{\text{Ext}}}^1_{{\mathcal{C}}}(-,-)$ (see for instance [Reference Enochs and Jenda14, Reference Enochs and Jenda15, Reference Gillespie17, Reference Göbel and Trlifaj21]). For the purpose of this article, it comes handy to split this concept as follows.

Frobenius pairs. The concept of left and right Frobenius pairs was introduced in [Reference Becerril, Mendoza, Pérez and Santiago3, Def. 2.5] from the notion of (co)generators in Auslander–Buchweitz approximation theory. Given ${\mathcal{X}}, \omega \subseteq {\mathcal{C}}$ , recall that $\omega$ is said to be a relative cogenerator in ${\mathcal{X}}$ if $\omega \subseteq {\mathcal{X}}$ and if for every $X \in {\mathcal{X}}$ there exists a short exact sequence $X \rightarrowtail W \twoheadrightarrow X'$ where $W \in \omega$ and $X' \in {\mathcal{X}}$ .

We summarize in the following result the most important properties of Frobenius pairs that will be used in the sequel. The proofs of these properties can be found in [Reference Becerril, Mendoza, Pérez and Santiago3, Thms. 2.8, 2.11, 2.16, 3.6 3.7, Props. 2.7, 2.13 & 3.5].

Relative Gorenstein objects. Most of our examples in this article will be built from Gorenstein objects relative to certain pairs $({\mathcal{X}},{\mathcal{Y}})$ of subcategories of ${\mathcal{C}}$ (see Definition 1.7 below). Before specifying how these Gorenstein objects are defined, recall that a chain complex $X_\bullet = (X_m)_{m \in {\mathbb{Z}}} \in {\mathsf{Ch}}({\mathcal{C}})$ is said to be ${{\text{Hom}}}_{{\mathcal{C}}}(-,{\mathcal{Y}})$ -acyclic if the complex of abelian groups ${{\text{Hom}}}_{{\mathcal{C}}}(X_\bullet,Y) = ({{\text{Hom}}}_{{\mathcal{C}}}(X_m,Y))_{m \in {\mathbb{Z}}}$ is exact for every $Y \in {\mathcal{Y}}$ . ${{\text{Hom}}}_{{\mathcal{C}}}({\mathcal{Y}},-)$ -acyclic complexes are defined dually. The following concept is due to [Reference Becerril, Mendoza and Santiago4, Def. 3.2].

Following [Reference Becerril, Mendoza and Santiago4], let us denote by ${\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}$ and ${\mathcal{GI}}_{({\mathcal{X}},{\mathcal{Y}})}$ the subcategories of $({\mathcal{X}},{\mathcal{Y}})$ -Gorenstein projective and $({\mathcal{X}},{\mathcal{Y}})$ -Gorenstein injective objects of ${\mathcal{C}}$ , respectively. For example, ${\mathcal{GP}}_{({\mathcal{P}},{\mathcal{P}})}$ and ${\mathcal{GI}}_{({\mathcal{I}},{\mathcal{I}})}$ are precisely the subcategories of Gorenstein projective and Gorenstein injective objects of ${\mathcal{C}}$ , which we shall write as ${\mathcal{GP}}$ and ${\mathcal{GI}}$ , for simplicity. Moreover, Definition 1.6 also covers the following examples of relative Gorenstein projective and injective objects:

• Ding projective and Ding injective modules, in the sense of [Reference Gillespie19, Defs. 3.2 & 3.7], by setting $({\mathcal{X}},{\mathcal{Y}}) = ({\mathcal{P}}(R),{\mathcal{F}}(R))$ and $({\mathcal{X}},{\mathcal{Y}}) = (\mathrm{FP}\mbox{-}{\mathcal{I}}(R),{\mathcal{I}}(R))$ , respectively. Here, $\mathrm{FP}\mbox{-}{\mathcal{I}}(R)$ stands for the subcategory of FP-injective (or absolutely pure) R-modules.

• Gorenstein AC-projective and Gorenstein AC-injective modules, in the sense of [Reference Bravo, Gillespie and Hovey5, §5 & §8], by setting the pairs $({\mathcal{X}},{\mathcal{Y}}) = ({\mathcal{P}}(R),{\mathcal{L}}(R))$ and $({\mathcal{X}},{\mathcal{Y}}) = (\mathrm{FP}_\infty\mbox{-}{\mathcal{I}}(R),{\mathcal{I}}(R))$ , respectively. Here, ${\mathcal{L}}(R)$ and $\mathrm{FP}_\infty\mbox{-}{\mathcal{I}}(R)$ denote the subcategories of level and $\mathrm{FP}_\infty$ -injective (or absolutely clean) R-modules (see [Reference Bravo, Gillespie and Hovey5, Def. 2.6]). These subcategories of relative Gorenstein modules will be denoted by ${\mathcal{GP}}_{\mathrm{AC}}(R)$ and ${\mathcal{GI}}_{\mathrm{AC}}(R)$ , for simplicity.

• Gorenstein flat sheaves over a noetherian and semi-separated scheme X, by setting $({\mathcal{X}},{\mathcal{Y}}) = (\mathfrak{F}(X),\mathfrak{F}(X) \cap (\mathfrak{F}(X))^{\perp_1})$ . See Murfet and Salarian’s [Reference Murfet and Salarian35, Thm. 4.18]. The subcategory of Gorenstein flat sheaves over X will be denoted by $\mathfrak{GF}(X)$ . In particular, the latter holds in the affine case $X = \mathrm{Spec}(R)$ provided that R is a commutative noetherian ring.

Many useful properties of Gorenstein objects relative to $({\mathcal{X}},{\mathcal{Y}})$ are obtained in the case where $({\mathcal{X}},{\mathcal{Y}})$ is a GP-admissible or a GI-admissible pair [Reference Becerril, Mendoza and Santiago4, Defs. 3.1 & 3.6]. We recall this notion for further referring.

We summarize in the following result the most important properties of GP-admissible pairs and relative Gorenstein objects that will be used in the sequel. The proofs of these properties can be found in [Reference Becerril, Mendoza and Santiago4, Thms. 3.30, 3.32 & 3.34, Corolls. 3.15, 3.17, 3.25, 3.33 & 4.10].

2. Complete cut cotorsion pairs and cotorsion cuts

We present the following concept of cotorsion pairs relative to subcategories.

Now let us give some examples of complete cut cotorsion pairs which are not necessarily complete cotorsion pairs. Frobenius pairs and relative Gorenstein objects will be the main source to construct our first examples.

Proof. First, note by $\mathsf{(lFp1)}$ , $\mathsf{(lFp2)}$ and [Reference Becerril, Mendoza, Pérez and Santiago3, Thm. 2.11 & Prop. 2.13] that ${\mathcal{X}}$ , $\omega$ and $\omega^\wedge$ are closed under direct summands. Also, it is clear that the same holds for ${\mathcal{X}}^{\perp_1}$ . Moreover, ${\mathcal{X}}^\wedge$ is thick by [Reference Becerril, Mendoza, Pérez and Santiago3, Thm. 2.11].

1. (1) By the previous comments, we have that the pair $({\mathcal{X}},\omega^\wedge)$ and ${\mathcal{X}}^\wedge$ satisfy $\mathsf{(lccp1)}$ and $\mathsf{(rccp1)}$ . Moreover, from [Reference Becerril, Mendoza, Pérez and Santiago3, Thm. 2.8] we clearly obtain $\mathsf{(lccp3)}$ and $\mathsf{(rccp3)}$ . Finally, conditions $\mathsf{(lccp2)}$ and $\mathsf{(rccp2)}$ follow from [Reference Becerril, Mendoza, Pérez and Santiago3, Part 1. of Prop. 2.7], $\mathsf{(lccp1)}$ , $\mathsf{(rccp1)}$ , $\mathsf{(lccp3)}$ and $\mathsf{(rccp3)}$ . Hence, ${\mathcal{X}}^\wedge \in {\text{Cuts}}({\mathcal{X}},\omega^\wedge)$ . The assertion $\omega^\wedge \in {\text{Cuts}}({\mathcal{X}},\omega^\wedge)$ can be easily deduced from the previous.

2. (2) We already have conditions $\mathsf{(lccp1)}$ and $\mathsf{(rccp1)}$ for $(\omega,{\mathcal{X}}^{\perp_1})$ . On the one hand, for every $C \in \omega^\wedge$ it is clear the existence of a short exact sequence $K \rightarrowtail W \twoheadrightarrow C$ , where $W \in \omega$ and $K \in \omega^\wedge$ . Since $\omega^\wedge \subseteq {\mathcal{X}}^{\perp_1}$ by [Reference Becerril, Mendoza, Pérez and Santiago3, Part 1. of Prop. 2.7], we have that $K \in {\mathcal{X}}^{\perp_1}$ . Thus, $\mathsf{(lccp3)}$ follows. On the other hand, $\mathsf{(rccp3)}$ is immediate. Regarding $\mathsf{(lccp2)}$ , note that the containment $\omega \cap \omega^\wedge \subseteq {}^{\perp_1}({\mathcal{X}}^{\perp_1}) \cap \omega^\wedge$ follows by $\mathsf{(lFp3)}$ . The converse containment follows by $\mathsf{(lccp3)}$ and the fact that $\omega$ is closed under direct summands. Finally, condition $\mathsf{(rccp2)}$ follows by $\mathsf{(lFp3)}$ and $\mathsf{(rccp3)}$ .

3. (3) The implication $(\Leftarrow)$ follows from part 2. For the direct implication, suppose that ${\mathcal{X}}^\wedge \in {\text{lCuts}}(\omega,{\mathcal{X}}^{\perp_1})$ . Then, for every $C \in {\mathcal{X}}^\wedge$ there exists a short exact sequence $K \rightarrowtail W \twoheadrightarrow C$ with $W \in \omega$ and $K \in {\mathcal{X}}^{\perp_1}$ . By $\mathsf{(lFp1)}$ , $\mathsf{(lFp3)}$ and Proposition 1.5, we can note that $K \in {\mathcal{X}}^{\perp_1} \cap {\mathcal{X}}^\wedge = \omega^\wedge$ . It then follows that $C \in \omega^\wedge$ .

Recall from [Reference Becerril, Mendoza and Santiago4, Def. 3.3] that, for a pair $({\mathcal{X}},{\mathcal{Y}})$ of subcategories of ${\mathcal{C}}$ , the $({\mathcal{X}},{\mathcal{Y}})$ -Gorenstein projective dimension of an object $C \in {\mathcal{C}}$ , which we denote by ${{\text{Gpd}}}_{({\mathcal{X}},{\mathcal{Y}})}(C)$ , is defined as the ${\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}$ -resolution dimension of C. Note that setting $({\mathcal{X}},{\mathcal{Y}}) = ({\mathcal{P}}(R),{\mathcal{P}}(R))$ and $({\mathcal{X}},{\mathcal{Y}}) = ({\mathcal{P}}(R),{\mathcal{F}}(R))$ yields the Gorenstein projective and the Ding projective dimensions of an R-module C, which we denote by ${{\text{Gpd}}}(C)$ and ${{\text{Dpd}}}(C)$ for simplicity. The $({\mathcal{X}},{\mathcal{Y}})$ -Gorenstein injective, Gorenstein injective and Ding injective dimensions ${{\text{Gid}}}_{({\mathcal{X}},{\mathcal{Y}})}(C)$ , ${{\text{Gid}}}(C)$ and ${{\text{Did}}}(C)$ , are defined dually.

Concerning relative Gorenstein dimensions, we recall the following properties from [Reference Becerril, Mendoza and Santiago4, Coroll. 4.3].

We shall mention a couple of extra properties for the previous example after showing the following general result.

Proof. In what follows, some of the mentioned facts come from Proposition 1.5. In the first part, let us first assume condition (a). We need to verify that $\omega^\wedge = {\mathcal{X}}^{\perp_1}$ and that for every $C \in {\mathcal{C}}$ there exists a short exact sequence of the form $C \rightarrowtail H \twoheadrightarrow X$ with $H \in \omega^\wedge$ and $X \in {\mathcal{X}}$ . For the latter, let $C \in {\mathcal{C}}$ and consider a short exact sequence $K \rightarrowtail X \twoheadrightarrow C$ with $X \in {\mathcal{X}}$ and $K \in \omega^\wedge$ . On the other hand, since $\omega$ is a relative cogenerator in ${\mathcal{X}}$ , there is a short exact sequence $X \rightarrowtail W \twoheadrightarrow X'$ with $W \in \omega$ and $X' \in {\mathcal{X}}$ . Taking the push-out of $W \leftarrow X \to C$ yields the following solid diagram:

(i)

Note that $H \in \omega^\wedge$ . Then, the right-hand column in (i) is the desired short exact sequence for the right completeness of $({\mathcal{X}},\omega^\wedge)$ . In order to show $\omega^\wedge = {\mathcal{X}}^{\perp_1}$ , note that the containment $\omega^\wedge \subseteq {\mathcal{X}}^{\perp_1}$ is clear since ${{\text{id}}}_{{\mathcal{X}}}(\omega^\wedge) = {{\text{id}}}_{{\mathcal{X}}}(\omega) = 0$ . The converse containment follows from the right completeness and the fact that $\omega^\wedge$ is closed under direct summands. Hence, (a) $\Rightarrow$ (b) follows.

The implication (b) $\Rightarrow$ (c) follows by the left completeness of the cotorsion pair $({\mathcal{X}},\omega^\wedge)$ and the containment $\omega^\wedge \subseteq {\mathcal{X}}^\wedge$ . On the other hand, we know that $({\mathcal{X}},\omega^\wedge)$ is a ${\mathcal{X}}^\wedge$ -cotorsion pair (that is, a complete cotorsion pair in the exact subcategory ${\mathcal{X}}^\wedge$ ). Thus, the implication (c) $\Rightarrow$ (a) is clear.

Now for the assertion (2) ${\mathcal{X}}^\wedge_n \in {\text{Cuts}}({\mathcal{X}},{\mathcal{X}}^{\perp})$ , we already know that ${\mathcal{X}}$ is closed under direct summands. In order to show $\mathsf{(lccp2)}$ , note that the containment ${\mathcal{X}} \cap {\mathcal{X}}^\wedge_n \subseteq {}^{\perp_1}({\mathcal{X}}^\perp) \cap {\mathcal{X}}^\wedge_n$ is clear. For the converse, if we take $C \in {}^{\perp_1}({\mathcal{X}}^\perp) \cap {\mathcal{X}}^\wedge_n$ , then there exists a short exact sequence $K \rightarrowtail X \twoheadrightarrow C$ with $X \in {\mathcal{X}}$ and $K \in \omega^\wedge_{n-1}$ Footnote ¹. Note also that $K \in {\mathcal{X}}^\perp$ . Then, the previous sequence splits and so $C \in {\mathcal{X}}$ . On the other hand, for $\mathsf{(rccp2)}$ ${\mathcal{X}}^\perp \cap {\mathcal{X}}^\wedge_n = {\mathcal{X}}^{\perp_1} \cap {\mathcal{X}}^\wedge_n$ , the containment ( $\subseteq$ ) is clear. Now if $C \in {\mathcal{X}}^{\perp_1} \cap {\mathcal{X}}^\wedge_n$ we can find a short exact sequence $C \rightarrowtail H \twoheadrightarrow C'$ where $H \in \omega^\wedge_n$ and $C' \in {\mathcal{X}}$ , which is split and so C is a direct summand of H. This in turn implies that $C \in {\mathcal{X}}^\perp$ . The previous arguments also show $\mathsf{(lccp3)}$ and $\mathsf{(rccp3)}$ .

Proof. It follows after applying Proposition 2.8 to the pair $({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})},\omega)$ , which is left Frobenius by Proposition 1.9.

Remark 2.10. Although in all of our examples of GP-admissible pairs $({\mathcal{X}},{\mathcal{Y}})$ , the subcategory $\omega \,:\!=\, {\mathcal{X}} \cap {\mathcal{Y}}$ is closed under direct summands, another proof of Corollary 2.9 can be obtained without assuming this property. Indeed, keep in mind the properties from Propositions 1.9 and 2.5, and consider the pair $({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})},({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\perp)$ . The closure under direct summands of $({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\perp$ is clear, and the same property holds for ${\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}$ . Also, the following containments are clear:

\begin{align*}{\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})} \cap ({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\wedge_n & \subseteq {}^{\perp_1}(({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\perp) \cap ({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\wedge_n, \\({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\perp \cap ({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\wedge_n & \subseteq ({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^{\perp_1} \cap ({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\wedge_n.\end{align*}

On the other hand, for every $C \in {}^{\perp_1}(({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\perp) \cap ({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\wedge_n$ there exists a short exact sequence $K \rightarrowtail G \twoheadrightarrow C$ with $G \in {\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}$ and $K \in \omega^\wedge_{n-1}$ . Also, we can note that $\omega^\wedge_{n-1} \subseteq ({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^{\perp_1}$ . Then, $\mathsf{(lccp2)}$ and $\mathsf{(lccp3)}$ follow. The containment

\begin{equation*}({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\perp \cap ({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\wedge_n \supseteq ({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^{\perp_1} \cap ({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})})^\wedge_n\end{equation*}

follows as in the proof of part (2) of Proposition 2.8. Hence, $\mathsf{(rccp2)}$ follows, and $\mathsf{(rccp3)}$ is also a consequence from Propositions 1.9 and 2.5.

In [Reference Becerril, Mendoza, Pérez and Santiago3, Prop. 3.5], it is given an alternative description of relative cotorsion pairs. Following the spirit of this result, we present the following characterization for cut cotorsion pairs. Its proof is straightforward.

Getting new cotorsion cuts and pairs from old ones. In the following result, whose proof is straightforward and left to the reader, we assert that the class ${\mathrm{lCuts}}({\mathcal{A}},{\mathcal{B}})$ is closed under restrictions, arbitrary unions and intersections.

Maximal cotorsion cuts. From the union property, it is natural to think of the possibility of finding the largest cotorsion cut for a pair $({\mathcal{A}},{\mathcal{B}})$ . Indeed, assuming that ${\mathcal{A}}$ is closed under direct summands and that $0 \in {\mathcal{B}}$ (and so ${\text{lCuts}}({\mathcal{A}},{\mathcal{B}}) \neq \emptyset$ ), it is possible to define the largest cut for $({\mathcal{A}},{\mathcal{B}})$ as a certain union.

Definition 2.17. Let ${\mathcal{A}}, {\mathcal{B}} \subseteq {\mathcal{C}}$ with ${\mathcal{A}}$ closed under direct summands and $0 \in {\mathcal{B}}$ . The maximal left cotorsion cut of $\boldsymbol{({\mathcal{A}},{\mathcal{B}})}$ is the union

\begin{equation*}{\mathbb{S}}_l({\mathcal{A}},{\mathcal{B}}) \,:\!=\, \bigcup \{ {\mathcal{S}} \text{ : } {\mathcal{S}} \in {\mathrm{lCuts}}({\mathcal{A}},{\mathcal{B}}) \}.\end{equation*}

The maximal right cotorsion cut and the maximal cotorsion cut of $\boldsymbol{({\mathcal{A}},{\mathcal{B}})}$ are defined similarly, and will be denoted by ${\mathbb{S}}_r({\mathcal{A}},{\mathcal{B}})$ and ${\mathbb{S}}({\mathcal{A}},{\mathcal{B}})$ .

The subcategory ${\mathbb{S}}_l({\mathcal{A}},{\mathcal{B}})$ has the interesting property that, under some mild conditions, it can cover the subcategory of all objects satisfying $\mathsf{(lccp3)}$ . Let us denote this subcategory by ${\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ , that is, ${\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ is formed by all the objects $C \in {\mathcal{C}}$ for which there is an exact sequence $B \rightarrowtail A \twoheadrightarrow C$ with $A \in {\mathcal{A}}$ and $B \in {\mathcal{B}}$ . We can note that ${\mathbb{S}}_l({\mathcal{A}},{\mathcal{B}}) \subseteq {\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ , although the equality does not hold in general. Below we show that if ${\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ is a left cotorsion cut of $({\mathcal{A}},{\mathcal{B}})$ , then it has to be the maximal one.

Proof. Note first that ${\mathcal{A}} \subseteq {\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ since $0 \in {\mathcal{B}}$ . Then, ${\mathcal{A}} \cap {\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}}) = {\mathcal{A}}$ , and so the implication (c) $\Rightarrow$ (d) is clear. On the other hand, the implications (b) $\Rightarrow$ (c) and (d) $\Rightarrow$ (a) are trivial. Thus, we only focus on proving that (a) $\Rightarrow$ (b).

Suppose that ${{\text{Ext}}}^1_{{\mathcal{C}}}({\mathcal{A}},{\mathcal{B}}) = 0$ . Note that the containment ${\mathbb{S}}_l({\mathcal{A}},{\mathcal{B}}) \subseteq {\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ is clear. Now let $X \in {\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ . We prove that $({\mathcal{A}},{\mathcal{B}})$ is a complete left cotorsion pair cut along ${\mathcal{S}} \,:\!=\, {\mathbb{S}}_l({\mathcal{A}},{\mathcal{B}}) \cup \{ X \}$ . For this, we only need to prove $\mathsf{(lccp2)}$ . Consider the following two cases:

1. (1) $X \in {\mathcal{A}}$ : Since ${{\text{Ext}}}^1_{{\mathcal{C}}}({\mathcal{A}},{\mathcal{B}}) = 0$ we have that $X \in {}^{\perp_1}{\mathcal{B}}$ . Thus, ${\mathcal{A}} \cap \{ X \} = \{ X \} = {}^{\perp_1}{\mathcal{B}} \cap \{ X \}$ .

2. (2) $X \not\in {\mathcal{A}}$ : Since $X \in {\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ , there exists a non-split short exact sequence $B \rightarrowtail A \twoheadrightarrow X$ with $A \in {\mathcal{A}}$ and $B \in {\mathcal{B}}$ . It follows that $X \not\in {}^{\perp_1}{\mathcal{B}}$ , and so ${\mathcal{A}} \cap \{ X \} = \emptyset = {}^{\perp_1}{\mathcal{B}} \cap \{ X \}$ .

In both cases, we get the equality ${\mathcal{A}} \cap \{ X \} = {}^{\perp_1}{\mathcal{B}} \cap \{ X \}$ . Therefore, ${\mathbb{S}}_l({\mathcal{A}},{\mathcal{B}}) \cup \{ X \} \in {\text{lCuts}}({\mathcal{A}},{\mathcal{B}})$ , and so ${\mathbb{S}}_l({\mathcal{A}},{\mathcal{B}}) \cup \{ X \} = {\mathbb{S}}_l({\mathcal{A}},{\mathcal{B}})$ by maximality, proving that ${\mathbb{S}}_l({\mathcal{A}},{\mathcal{B}}) \supseteq {\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ .

A similar equivalence holds for the subcategory ${\mathcal{E}}_r({\mathcal{A}},{\mathcal{B}})$ of all objects satisfying $\mathsf{(rccp3)}$ , and for ${\mathcal{E}}({\mathcal{A}},{\mathcal{B}}) \,:\!=\, {\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}}) \cap {\mathcal{E}}_r({\mathcal{A}},{\mathcal{B}})$ .

Remark 2.19. As we mentioned earlier, ${\mathbb{S}}_l({\mathcal{A}},{\mathcal{B}}) \subseteq {\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ . In some cases this containment is strict and nontrivial, that is, we can find subcategories ${\mathcal{A}}$ and ${\mathcal{B}}$ such that $\{ 0 \} \subsetneq {\mathbb{S}}_l({\mathcal{A}},{\mathcal{B}}) \subsetneq {\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ . Indeed, let us consider ${\mathcal{A}} = {\mathcal{GP}}(R)$ and ${\mathcal{B}} = {\mathsf{Mod}}(R)$ . One can note that $({\mathcal{GP}}(R),{\mathsf{Mod}}(R))$ is a complete left cotorsion pair cut along ${\mathcal{P}}(R)^\wedge$ , and so ${\mathcal{P}}(R)^\wedge \subseteq {\mathbb{S}}_l({\mathcal{GP}}(R),{\mathsf{Mod}}(R))$ , that is, ${\mathbb{S}}_l({\mathcal{GP}}(R),{\mathsf{Mod}}(R)) \neq \{ 0 \}$ . On the other hand, by Theorem 2.18 we have that

\begin{equation*}{\mathbb{S}}_l({\mathcal{GP}}(R),{\mathsf{Mod}}(R)) = {\mathcal{E}}_l({\mathcal{GP}}(R),{\mathsf{Mod}}(R)) \Longleftrightarrow {{\mathrm{Ext}}}^1_R({\mathcal{GP}}(R),{\mathsf{Mod}}(R)) = 0,\end{equation*}

and there are rings over which the latter condition does not hold (see Example 2.11 (2)).

Compatibility between cotorsion cuts. So far the methods we have showed to obtain new cotorsion cuts are restricted to a fixed pair $({\mathcal{A}},{\mathcal{B}})$ of subcategories of ${\mathcal{C}}$ . In some cases, it is possible to get new pairs along new cuts. Specifically, we show in Proposition 2.21 below an extension of the union property of Proposition 2.14 (2), in the sense that it is possible to take the union of two different complete cut cotorsion pairs along the union of their cuts, provided that certain compatibility condition between the given pairs is satisfied.

The following result is straightforward and its proof is left to the reader.

More induced cotorsion cuts and examples. Previously we showed how to construct new cotorsion cuts from a given one, or from a family of cotorsion cuts (as in Propositions 2.14 and 2.21). In the last part of this section, we give some sufficient conditions on subcategories ${\mathcal{A}}, {\mathcal{B}}, {\mathcal{S}} \subseteq {\mathcal{C}}$ , without needing that ${\mathcal{S}} \in {\text{Cuts}}({\mathcal{A}},{\mathcal{B}})$ , which imply that $({\mathcal{A}},{\mathcal{B}})$ is a cotorsion pair cut along ${\mathcal{A}}^\wedge \cap {\mathcal{S}}$ . The subcategories ${\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ , ${\mathcal{E}}_r({\mathcal{A}},{\mathcal{B}})$ , and ${\mathcal{E}}({\mathcal{A}},{\mathcal{B}})$ will be useful to prove the following result.

Then, ${\mathcal{A}}^\wedge \cap {\mathcal{S}} \in {\text{Cuts}}({\mathcal{A}},{\mathcal{B}})$ .

Proof. First, we show that ${\mathcal{A}}^\wedge_n \subseteq {\mathcal{E}}_r({\mathcal{A}}, (\omega \cap {\mathcal{S}})^{\wedge})$ for every $n \geq 0$ (and so ${\mathcal{A}}^\wedge \subseteq {\mathcal{E}}_r({\mathcal{A}}, (\omega \cap {\mathcal{S}})^{\wedge})$ ) by using induction on n.

• Initial step: ${\mathcal{A}}^{\wedge}_0 \subseteq {\mathcal{E}}_r({\mathcal{A}}, (\omega \cap {\mathcal{S}})^{\wedge})$ follows by condition (3).

• Induction step: Suppose that $n \geq 1$ and that ${\mathcal{A}}^{\wedge}_{n-1} \subseteq {\mathcal{E}}_r({\mathcal{A}}, (\omega \cap {\mathcal{S}})^{\wedge})$ holds. Now consider a short exact sequence $L \rightarrowtail A \twoheadrightarrow M$ with $A \in {\mathcal{A}}$ and ${{\text{resdim}}}_{{\mathcal{A}}}(L) = n-1$ . By the induction hypothesis, there is a short exact sequence $L \rightarrowtail K \twoheadrightarrow A'$ with $A' \in {\mathcal{A}}$ and $K \in (\omega \cap {\mathcal{S}})^\wedge$ . Taking the pushout of $K \leftarrow L \to A$ yields the following solid diagram:

  (ii)

  Note that $E \in {\mathcal{A}}$ by condition (1), and so by condition (3) there exists a short exact sequence $E \rightarrowtail W \twoheadrightarrow A''$ with $W \in \omega \cap {\mathcal{S}}$ and $A'' \in {\mathcal{A}}$ . Now take the pushout of $W \leftarrow E \to M$ to obtain the following solid diagram:

  (iii)

  We have that $F \in (\omega \cap {\mathcal{S}})^\wedge$ , and so the right-hand column of (iii) implies that $M \in {\mathcal{E}}_r({\mathcal{A}},(\omega \cap {\mathcal{S}})^\wedge)$ .

The containment ${\mathcal{A}}^\wedge \subseteq {\mathcal{E}}_r({\mathcal{A}}, (\omega \cap {\mathcal{S}})^{\wedge})$ then holds true. Moreover, from the central row in (ii) and condition (4) we have that for every $M \in {\mathcal{A}}^\wedge$ with ${{\text{resdim}}}_{{\mathcal{A}}}(M) \geq 1$ there is a short exact sequence $B \rightarrowtail A \twoheadrightarrow M$ with $A \in {\mathcal{A}}$ and $B \in {\mathcal{B}}$ . Moreover, for the case $n = 0$ we can simply take $A = M$ and $B = 0$ in this sequence since $0 \in {\mathcal{B}}$ , as ${\mathcal{B}}$ is closed under direct summands. In other words, we also have that the containment ${\mathcal{A}}^\wedge \subseteq {\mathcal{E}}_l({\mathcal{A}},{\mathcal{B}})$ holds. Hence, from the latter along with ${\mathcal{A}}^\wedge \subseteq {\mathcal{E}}_r({\mathcal{A}}, (\omega \cap {\mathcal{S}})^{\wedge})$ and condition (4) we conclude that ${\mathcal{A}}^\wedge \subseteq {\mathcal{E}}({\mathcal{A}},{\mathcal{B}})$ . It is clear now that the pair $({\mathcal{A}},{\mathcal{B}})$ satisfies conditions $\mathsf{(lccp3)}$ and $\mathsf{(rccp3)}$ with respect to the subcategory ${\mathcal{A}}^\wedge \cap {\mathcal{S}}$ , and we also know from the hypotheses the validity of $\mathsf{(lccp1)}$ and $\mathsf{(rccp1)}$ . Finally, by condition (5) we have that ${\mathcal{A}} \cap ({\mathcal{A}}^\wedge \cap {\mathcal{S}}) \subseteq {}^{\perp_1}{\mathcal{B}} \cap ({\mathcal{A}}^\wedge \cap {\mathcal{S}})$ and that ${\mathcal{B}} \cap ({\mathcal{A}}^\wedge \cap {\mathcal{S}}) \subseteq {\mathcal{A}}^{\perp_1} \cap ({\mathcal{A}}^\wedge \cap {\mathcal{S}})$ , and the converse containments follow by $\mathsf{(lccp3)}$ and $\mathsf{(rccp3)}$ . Therefore, $({\mathcal{A}},{\mathcal{B}})$ is a complete cotorsion pair cut along ${\mathcal{A}}^\wedge \cap {\mathcal{S}}$ .

Let us apply the previous result to obtain another example of a complete cut cotorsion pair from Gorenstein objects relative to a GP-admissible pair.

Proposition 2.24. Let $({\mathcal{X}},{\mathcal{Y}})$ be a GP-admissible pair in ${\mathcal{C}}$ and $\omega \,:\!=\, {\mathcal{X}} \cap {\mathcal{Y}}$ , such that ${\mathcal{Y}}^\wedge$ , $\omega$ and ${\mathcal{X}} \cap {\mathcal{Y}}^\wedge$ are closed under direct summands. Then, $({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})},{\mathcal{Y}}^\wedge)$ is a complete cotorsion pair cut along ${\mathcal{X}}^\wedge$ . Moreover,

(iv) \begin{align}{\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})} \cap {\mathcal{Y}}^\wedge & = \omega = {\mathcal{X}} \cap {\mathcal{Y}}^\wedge = {\mathcal{Y}} \cap {\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}. \end{align}

Proof. We have by hypothesis that ${\mathcal{Y}}^\wedge$ is closed under direct summands. On the other hand, by [Reference Becerril, Mendoza and Santiago4, Coroll. 3.33] we also have that ${\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}$ is closed under extensions and direct summands. We then have that conditions (1) and (2) in Proposition 2.23 are valid. Now let us show that ${\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})} \cap {\mathcal{Y}}^\wedge \cap {\mathcal{X}}^\wedge$ is a relative cogenerator in ${\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}$ . By [Reference Becerril, Mendoza and Santiago4, Thm. 3.34 (c) & (d)] we know that the equality (iv) holds. Then, ${\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})} \cap {\mathcal{Y}}^\wedge \cap {\mathcal{X}}^\wedge = \omega \cap {\mathcal{X}}^\wedge = \omega$ , and so condition (3) in Proposition 2.23 follows as well by [Reference Becerril, Mendoza and Santiago4, Coroll. 3.25 (a)], while condition (4) is clear. Finally, the orthogonality relations in condition (5) of Proposition 2.23 are a consequence of [Reference Becerril, Mendoza and Santiago4, Coroll. 3.15].

Corollary 2.25. Let $({\mathcal{X}},{\mathcal{Y}})$ be a hereditary complete cotorsion pair in ${\mathcal{C}}$ and $\omega \,:\!=\, {\mathcal{X}} \cap {\mathcal{Y}}$ . Then, $({\mathcal{GP}}_{({\mathcal{X}},\omega)},\omega^\wedge)$ and $({\mathcal{GP}}_{({\mathcal{X}},\omega)},{\mathcal{Y}})$ are complete cotorsion pairs cut along ${\mathcal{X}}^\wedge$ . Moreover,

(v) \begin{align}{\mathcal{GP}}_{({\mathcal{X}},\omega)} \cap \omega^\wedge & = \omega = {\mathcal{X}} \cap \omega^\wedge. \end{align}

Proof. Recall from Example 1.8 (1) that $({\mathcal{X}},\omega)$ is a GP-admissible pair. Moreover, it is clear that ${\mathcal{X}}$ and $\omega$ are closed under direct summands, while the same holds for $\omega^\wedge$ by Proposition 1.5. Then, ${\mathcal{X}} \cap \omega^\wedge$ is also closed under direct summands, and Proposition 2.24 implies that $({\mathcal{GP}}_{({\mathcal{X}},\omega)},\omega^\wedge)$ is a complete cotorsion pair cut along ${\mathcal{X}}^\wedge$ and the equality (v).

For the second pair $({\mathcal{GP}}_{({\mathcal{X}},\omega)},{\mathcal{Y}})$ , the subcategories ${\mathcal{A}} \,:\!=\, {\mathcal{GP}}_{({\mathcal{X}},\omega)}$ , ${\mathcal{B}} \,:\!=\, {\mathcal{Y}}$ and ${\mathcal{S}} \,:\!=\, {\mathcal{X}}^\wedge$ satisfy the conditions in Proposition 2.23. Indeed, the first four conditions are clear. For (5), it suffices to note that ${\mathcal{Y}} \cap {\mathcal{X}}^\wedge = \omega^\wedge$ and ${\mathcal{GP}}_{({\mathcal{X}},\omega)} \cap {\mathcal{X}}^\wedge = {\mathcal{X}}$ , which follow from the assumptions and Proposition 1.5. Finally, for the equality (v) we have that

\begin{equation*}\qquad\qquad\ \ \ \ {\mathcal{GP}}_{({\mathcal{X}},\omega)} \cap \omega^\wedge = {\mathcal{GP}}_{({\mathcal{X}},\omega)} \cap \omega^\wedge \cap {\mathcal{X}}^\wedge = {\mathcal{X}} \cap \omega^\wedge = {\mathcal{X}} \cap {\mathcal{Y}} \cap {\mathcal{X}}^\wedge = \omega \cap {\mathcal{X}}^\wedge = \omega.\qquad\qquad\ \ \Box\end{equation*}

Intersections of the form $\omega \cap {\mathcal{S}}$ considered in Proposition 2.23 are not a mere technicality to obtain new cut cotorsion pairs. They are in fact an important component of the notions of cut Frobenius pairs and cut Auslander–Buchweitz contexts, which will be presented and studied in detail in the next section.

3. Cut Frobenius pairs and cut Auslander–Buchweitz contexts

As mentioned in the introduction, one of the main purposes of this article is to describe an interplay between cut Frobenius pairs, cut Auslander–Buchweitz contexts, and complete cut cotorsion pairs. This interplay will be the main topic of the next section. For the moment, we can note the following relation between certain Gorenstein complete cut cotorsion pairs and left Frobenius pairs.

Proof. The implications (a) $\Rightarrow$ (b) and (c) $\Rightarrow$ (b) are trivial, while (a) $\Rightarrow$ (c) follows by Proposition 1.5. Then, we only focus on proving (b) $\Rightarrow$ (a), for which one needs Proposition 1.9. So let us assume that the containment ${\mathcal{Y}} \subseteq {\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}$ is satisfied. We know that ${\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}$ is left thick, that is, condition $\mathsf{(lFp1)}$ holds. By hypothesis, we also have that $\omega$ is closed under direct summands. On the other hand, we have that $\omega = {\mathcal{Y}} \cap {\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})} = {\mathcal{Y}}$ , and then $\mathsf{(lFp2)}$ follows. Finally, the fact that ${\mathcal{Y}} = \omega$ is a ${\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}$ -injective relative cogenerator in ${\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}$ follows also by Proposition 1.9.

Some technical lemmas. Before giving the definition of cut Frobenius pairs, we need to prove some preliminary results. The idea is to find conditions under which $\omega^\wedge$ is closed under extensions, and for this the Induction Principle will be a frequently used argument.

Lemma 3.3. Let $\omega, {\mathcal{S}} \subseteq {\mathcal{C}}$ such that $\omega$ is closed under extensions and $\omega \cap {\mathcal{S}}$ is a relative generator in $\omega$ . Let $C \in {\mathcal{C}}$ for which there exists an exact sequence

(i) \begin{align}E_n & \overset{f_n}\rightarrowtail W_{n-1} \to \cdots \to W_1 \xrightarrow{f_1} W_0 \overset{f_0}\twoheadrightarrow C, \end{align}

for some $n \geq 1$ , with $E_{j+1} \,:\!=\, {{\mathrm{Ker}}}(f_j)$ and $W_j \in \omega$ for every $0 \leq j \leq n-1$ . Then, there exist short exact sequences

(ii) \begin{align}G_j & \rightarrowtail X_{j+1} \twoheadrightarrow E_{j+1}, & & \mathrm{and} & X_{j+1} & \rightarrowtail F_j \twoheadrightarrow X_j, \end{align}

where $X_0 \,:\!=\, C$ , $F_j \in \omega \cap {\mathcal{S}}$ and $G_j \in \omega$ for every $0 \leq j \leq n-1$ .

Proof. Let us prove this result by induction on j.

• Initial step: For the case $j = 0$ , since $\omega \cap {\mathcal{S}}$ is a relative generator in $\omega$ , there is a short exact sequence $G_0 \rightarrowtail F_0 \twoheadrightarrow W_0$ with $G_0 \in \omega$ and $F_0 \in \omega \cap {\mathcal{S}}$ . Taking the pullback of $E_1 \to W_0 \leftarrow F_0$ yields the result.

• Induction step: Now suppose that for $1 \leq j \leq n-2$ there are short exact sequences $G_j \rightarrowtail X_{j+1} \twoheadrightarrow E_{j+1}$ and $X_{j+1} \rightarrowtail F_j \twoheadrightarrow X_j$ , with $F_j \in \omega \cap {\mathcal{S}}$ and $G_j \in \omega$ . Consider also the $(j+1)$ -th splicer from the resolution (i), namely $E_{j+2} \rightarrowtail W_{j+1} \twoheadrightarrow E_{j+1}$ . Taking the pullback of $W_{j+1} \to E_{j+1} \leftarrow X_{j+1}$ yields the following solid diagram:

  

  Since $\omega$ is closed under extensions, $F^{\prime}_{j+1} \in \omega$ . By using again that $\omega \cap {\mathcal{S}}$ is a generator in $\omega$ , we have an exact sequence $G_{j+1} \rightarrowtail F_{j+1} \twoheadrightarrow F^{\prime}_{j+1}$ with $F_{j+1} \in \omega \cap {\mathcal{S}}$ and $G_{j+1} \in \omega$ . Taking the pullback of $E_{j+2} \to F^{\prime}_{j+1} \leftarrow F_{j+1}$ yields the result.

Proof. For ${{\text{resdim}}}_{\omega}(C) = 0$ , the result follows since $\omega$ is closed under extensions. So we may assume that ${{\text{resdim}}}_{\omega}(C) \geq 1$ . Then, there exists an exact sequence $W' \rightarrowtail W_0 \twoheadrightarrow C$ with $W_0 \in \omega$ and ${{\text{resdim}}}_{\omega}(W') = {{\text{resdim}}}_{\omega}(C) - 1$ . Taking the pullback of $B \to C \leftarrow W_0$ gives the desired inequality.

We are now ready to prove the main technical lemma of this section, where we give sufficient conditions so that the subcategory $\omega^\wedge$ is closed under extensions and direct summands. It is not enough to assume that $\omega$ is closed under extensions and direct summands. In addition, we need an auxiliary subcategory ${\mathcal{S}}$ so that $\omega \cap {\mathcal{S}}$ is a $\omega$ -projective relative generator in $\omega$ . Such ${\mathcal{S}}$ will play the role of a suitable cut to propose a relative version for the concept of left Frobenius pair in Definition 3.6 below.

Proof.

1. (1) First, we show that $\omega \cap {\mathcal{S}}$ is a relative generator in $\omega^\wedge$ . We use induction on $n \,:\!=\, {{\text{resdim}}}_{\omega}(C)$ for $C \in \omega^\wedge$ .

   • Initial step: This is clear.

   • Induction step: For the case $n \geq 1$ , we have an exact sequence

     \begin{equation*}W_n \rightarrowtail W_{n-1} \to \cdots \to W_1 \to W_0 \twoheadrightarrow C,\end{equation*}
     with $W_k \in \omega$ for every $0 \leq k \leq n$ . By Lemma 3.3 and its notation therein, we have that $X_n \in \omega$ since $G_{n-1}, E_n \,:\!=\, W_n \in \omega$ and $\omega$ is closed under extensions. Glueing together the splicer sequences (ii) in Lemma 3.3 gives rise to the exact sequence
     \begin{equation*}X_n \rightarrowtail F_{n-1} \to \cdots \to F_1 \to F_0 \twoheadrightarrow C,\end{equation*}
     with $F_k \in \omega \cap {\mathcal{S}}$ for every $0 \leq k \leq n-1$ . Thus, for the short exact sequence $X_1 \rightarrowtail F_0 \twoheadrightarrow C$ we have $F_0 \in \omega \cap {\mathcal{S}}$ and $X_1 \in \omega^\wedge$ , with ${{\text{resdim}}}_{\omega}(X_1) = n - 1$ ; that is, $\omega \cap {\mathcal{S}}$ is a relative generator in $\omega^\wedge$ .

     Now by [Reference Mendoza and Sáenz32, dual of Lem. 2.13 (a) Rmk. 1.2 (a)], we have that

     \begin{equation*}{{\text{pd}}}_{\omega^\wedge}(\omega \cap {\mathcal{S}}) = {{\text{id}}}_{\omega \cap {\mathcal{S}}}(\omega^\wedge) = {{\text{id}}}_{\omega \cap {\mathcal{S}}}(\omega) = {{\text{pd}}}_{\omega}(\omega \cap {\mathcal{S}}) = 0,\end{equation*}
     and so $\omega \cap {\mathcal{S}}$ is $\omega^\wedge$ -projective.

2. (2) Suppose we are given a short exact sequence $A \rightarrowtail B \twoheadrightarrow C$ with $A, C \in \omega^\wedge$ . Let us use induction on $n \,:\!=\, {{\text{resdim}}}_{\omega}(A)$ to show that $B \in \omega^\wedge$ .

   • Initial step: If ${{\text{resdim}}}_{\omega}(A) = 0$ , the result follows by Lemma 3.4.

   • Induction step: We may assume that ${{\text{resdim}}}_{\omega}(A) \geq 1$ . Suppose also that for any short exact sequence $A' \rightarrowtail B' \twoheadrightarrow C'$ with $C' \in \omega^\wedge$ and ${{\text{resdim}}}_{\omega}(A') \leq n-1$ , one has that $B' \in \omega^\wedge$ . By part (1) there is a short exact sequence $C' \rightarrowtail P \twoheadrightarrow C$ with $P \in \omega \cap {\mathcal{S}}$ and $C' \in \omega^\wedge$ . We take the pullback of $B \to C \leftarrow P$ to obtain the following solid diagram:

     

     Since ${{\text{pd}}}_{\omega^\wedge}(\omega \cap {\mathcal{S}}) = 0$ , the central row in this diagram splits, and then $E \simeq A \oplus P$ . Now by part (1) again, consider a short exact sequence $A' \rightarrowtail W_0 \twoheadrightarrow A$ with $W_0 \in \omega \cap {\mathcal{S}}$ and ${{\text{resdim}}}_{\omega}(A') = {{\text{resdim}}}_{\omega}(A) - 1$ . By taking the direct sum of this sequence and the identity complex on P, we get the short exact sequence $A' \rightarrowtail W_0 \oplus P \twoheadrightarrow A \oplus P$ , with $W_0 \oplus P \in \omega$ since $\omega$ is closed under extensions. Taking the pullback of $C' \to A \oplus P \leftarrow W_0 \oplus P$ and using the induction hypothesis gives the result.

3. (3) Given an object $C = C_1 \oplus C_2 \in \omega^\wedge$ with $n \,:\!=\, {{\text{resdim}}}_{\omega}(C)$ , the proof that $C_1, C_2 \in \omega^\wedge$ follows by induction on n and using an argument similar to the one appearing in [Reference Bravo, Gillespie and Pérez6, Proof of Prop. 5.3].

4. (4) The containment $\omega^\wedge \cap {\mathcal{S}} \supseteq (\omega \cap {\mathcal{S}})^\wedge$ is clear since ${\mathcal{S}}$ is closed under mono-cokernels. Now suppose we are given an object $S \in \omega^\wedge \cap {\mathcal{S}}$ , that is, an $S \in {\mathcal{S}}$ with a finite $\omega$ -resolution $W_n \rightarrowtail W_{n-1} \to \cdots \to W_0 \twoheadrightarrow S$ . If $n = 0$ , the result follows since $\omega$ is closed under isomorphisms. So we may assume that $n \geq 1$ . Since ${\mathcal{S}}$ is closed under epi-kernels, from Lemma 3.3 there are short exact sequences $X_{j+1} \rightarrowtail F_j \twoheadrightarrow X_j$ with $X_j \in {\mathcal{S}}$ and $F_j \in \omega \cap {\mathcal{S}}$ for all $0 \leq j \leq n-1$ . Moreover, $X_n \in \omega \cap {\mathcal{S}}$ since $G_{n-1}, E_n \,:\!=\, W_n \in \omega$ and $\omega$ is closed under extensions. Then, $X_n \rightarrowtail F_{n-1} \to \cdots \to F_0 \twoheadrightarrow S$ is a $(\omega \cap {\mathcal{S}})$ -resolution of S.

Cut Frobenius pairs. We are now ready to present the concept of Frobenius pairs cut along subcategories.

Of course any left Frobenius pair $({\mathcal{X}},\omega)$ , with $\omega$ closed under finite coproduts, is a left Frobenius pair cut along ${\mathcal{C}}$ , but not every relative left Frobenius pair is an absolute left Frobenius pair, as we show in Example 3.9 below.

Proposition 3.8. Let $({\mathcal{X}},{\mathcal{Y}})$ be a GP-admissible pair in ${\mathcal{C}}$ such that:

1. (1) ${\mathcal{X}}$ is closed under direct summands;

2. (2) ${\mathcal{Y}}$ is closed under extensions and direct summands; and

3. (3) ${\mathcal{X}} \cap {\mathcal{Y}}$ is a relative generator in ${\mathcal{Y}}$ .

Then, $({\mathcal{C}},{\mathcal{Y}}^\wedge)$ is a left Frobenius pair cut along ${\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}$ . Moreover, the following equality holds true:

(iii) \begin{align}{\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})} \cap {\mathcal{Y}}^\wedge & = {\mathcal{X}} \cap {\mathcal{Y}} = {\mathcal{X}} \cap {\mathcal{Y}}^\wedge. \end{align}

Proof. Let us verify each condition in Definition 3.6, although we need to do this in a specific order. Condition $\mathsf{(lcFp1)}$ is clearly satisfied. On the other hand, since ${\mathcal{Y}}$ is closed under extensions and direct summands (by (2)), and ${\mathcal{X}} \cap {\mathcal{Y}}$ is an ${\mathcal{Y}}$ -projective relative generator in ${\mathcal{Y}}$ (by (3) and $\mathsf{(GPa1)}$ in Definition 1.7), we have from Lemma 3.5 (2) and (3) that ${\mathcal{Y}}^\wedge$ is closed under extensions and direct summands, that is, condition $\mathsf{(lcFp4)}$ holds. Moreover, since also ${\mathcal{X}} \cap {\mathcal{Y}}$ and ${\mathcal{X}} \cap {\mathcal{Y}}^\wedge$ are closed under direct summands, we obtain the equality (iii) from Proposition 1.9. This, along with Lemma 3.5 (1), implies that ${\mathcal{Y}}^\wedge \cap {\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}$ is a ${\mathcal{Y}}^\wedge$ -projective relative generator in ${\mathcal{Y}}^\wedge$ , that is, we have condition $\mathsf{(lcFp3)}$ . Finally, condition $\mathsf{(lcFp2)}$ , that is, that $({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}, {\mathcal{Y}}^\wedge \cap {\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})}) = ({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})},{\mathcal{X}} \cap {\mathcal{Y}})$ is a left Frobenius pair in ${\mathcal{C}}$ , follows by Proposition 1.9.

Below we establish necessary and sufficient conditions under which the relative Frobenius pair from the previous example is left Frobenius in ${\mathsf{Mod}}(R)$ . This will be a consequence of the following general result.

Proof. The implication (a) $\Rightarrow$ (b) is straightforward. Now suppose that $\omega = {\mathcal{X}}^\perp \cap {\mathcal{X}}$ . This implies that $({\mathcal{X}},\omega)$ is a left Frobenius pair. By Proposition 1.5, we have that $\omega^\wedge = {\mathcal{X}}^\perp \cap {\mathcal{X}}$ , and so $\omega^\wedge$ is closed under direct summands. Finally, it is clear that $\omega^\wedge \subseteq {\mathcal{X}}$ is an ${\mathcal{X}}$ -injective relative cogenerator in ${\mathcal{X}}$ .

An immediate consequence of the previous result is the following.

From the previous two results, we can note that left Frobenius pairs $({\mathcal{X}},\omega)$ that induce a left Frobenius pair of the form $({\mathcal{X}},\omega^\wedge)$ are scarce. As a matter of fact, what we expect from a left Frobenius pair $({\mathcal{X}},\omega)$ is that $({\mathcal{X}},\omega^\wedge)$ is a complete cut cotorsion pair. This is precisely the case of Gorenstein left Frobenius pairs $({\mathcal{GP}}_{({\mathcal{X}},{\mathcal{Y}})},{\mathcal{Y}})$ in Proposition 3.1. We shall explore this in more detail in Section 4.

Let us now present one more example of cut Frobenius pairs in the context of quiver representations.

Example 3.12. Several facts in this example are extracted from [Reference Zhang and Xiong48, Ex. 5.3]. Let $\Lambda$ be the quotient path k-algebra given by the quiver

with relations $\alpha \beta = 0 = \beta \alpha$ . In the category ${\mathsf{mod}}(\Lambda)$ of finitely generated (left) $\Lambda$ -modules, the indecomposable projective $\Lambda$ -modules are:

\begin{align*}{\scriptsize \begin{array}{c} 1 \\ 2 \end{array}}, & & {\scriptsize \begin{array}{c} 2 \\ 1 \end{array}} & & \mathrm{and} & & {\scriptsize \begin{array}{l} 3 \\ 2 \\ 1. \end{array}}\end{align*}

On the other hand, the indecomposable injective $\Lambda$ -modules are:

\begin{align*}{\scriptsize \begin{array}{c} 3 \\ 2 \\ 1 \end{array}}, & & {\scriptsize \begin{array}{ccc} 1 & & 3 \\ & 2 \end{array}} & & \mathrm{and} & & {\scriptsize \begin{array}{c} 3. \end{array}}\end{align*}

It follows that the Auslander–Reiten quiver of ${\mathsf{mod}}(\Lambda)$ is given by:

where the two vertices 1 represent the same simple module.

Now let ${\mathcal{X}} \,:\!=\, {\mathrm{add}} ( \tiny{ \begin{array}{c} 1 \end{array}} \oplus \tiny{ \begin{array}{c} 2 \\ 1 \end{array}} \oplus \tiny{ \begin{array}{c} 2 \end{array} } \oplus \tiny{\begin{array}{c} 1 \\ 2 \end{array}} )$ be the subcategory of direct summands of finite coproducts of copies of the $\Lambda$ -module $\tiny{ \begin{array}{c} 1 \end{array} } \oplus \tiny{ \begin{array}{c} 2 \\ 1 \end{array}} \oplus \tiny{ \begin{array}{c} 2 \end{array} } \oplus \tiny{\begin{array}{c} 1 \\ 2 \end{array}}$ . Then, ${\mathcal{X}}$ is closed under extensions and a Frobenius subcategory of ${\mathsf{mod}}(\Lambda)$ . Moreover, the subcategory of the projective-injective objects in ${\mathcal{X}}$ is ${\mathcal{P}}({\mathcal{X}}) = {\mathrm{add}}(\tiny{\begin{array}{c} 1 \\ 2 \end{array}} \oplus \tiny{\begin{array}{c} 2 \\ 1 \end{array} })$ . Indeed, by using Auslander-Reiten theory, it can be shown that ${{\mathrm{Ext}}}^i_\Lambda({\mathcal{X}},{\mathcal{P}}({\mathcal{X}})) = 0$ for every $i \geq 1$ .

We assert that $({\mathsf{mod}}(\Lambda), {\mathcal{P}}({\mathcal{X}}))$ is a left Frobenius pair cut along ${\mathcal{X}}$ . Note first that ${\mathcal{X}}$ is not a resolving subcategory of ${\mathsf{mod}}(\Lambda)$ since it does not contain all indecomposable projective $\Lambda$ -modules. On the one hand, it is easy to see that conditions $\mathsf{(lcFp1)}$ , $\mathsf{(lcFp3)}$ and $\mathsf{(lcFp4)}$ in Definition 3.6 hold true as ${\mathcal{P}}({\mathcal{X}}) \subseteq {\mathcal{P}}(\Lambda)$ , and ${\mathcal{X}}$ is closed under extensions and direct summands. Since ${\mathcal{X}}$ is also a Frobenius subcategory of ${\mathsf{mod}}(\Lambda)$ , we only need to verify that ${\mathcal{X}}$ is closed under epi-kernels in order to conclude $\mathsf{(lcFp2)}$ . So suppose we are given an exact sequence $X \rightarrowtail Y \twoheadrightarrow Z$ in ${\mathsf{mod}}(\Lambda)$ with $Y, Z \in {\mathcal{X}}$ . Using that ${\mathcal{X}}$ has enough projective objects, taking the pullback of $Y \to Z \leftarrow P$ yields a solid diagram

with $P\in {\mathcal{P}}({\mathcal{X}})$ and $Z' \in {\mathcal{X}}$ . Note that $L \in {\mathcal{X}}$ . Moreover, since ${\mathcal{P}}({\mathcal{X}}) \subseteq {\mathcal{P}}(\Lambda)$ the central row in the previous diagram splits, and then $X \in {\mathcal{X}}$ .

Cut Auslander–Buchweitz contexts. In this part, we introduce the concept of cut Auslander–Buchweitz contexts and present some examples related to hereditary complete cotorsion pairs. Let us recall the following from [Reference Becerril, Mendoza, Pérez and Santiago3, Def. 5.1].

Let us now propose the following generalization of the previous definition.

In Example 3.16, we obtained the left weak AB-context $({\mathcal{GP}}_{({\mathcal{X}},\omega)}, {\mathcal{Y}})$ cut along ${\mathcal{X}}^\wedge$ , from a GP-admissible pair $({\mathcal{X}},{\mathcal{Y}})$ with ${\mathcal{X}}$ left thick and ${\mathcal{Y}}$ right thick. With a couple of extra assumptions on ${\mathcal{X}}$ and ${\mathcal{Y}}$ , we are able to characterize $({\mathcal{GP}}_{({\mathcal{X}},\omega)}, {\mathcal{Y}})$ as a left weak AB-context that is absolute (that is, cut along the whole category ${\mathcal{C}}$ ).

Proof. Let us show first that (a) $\Rightarrow$ (b). Since $({\mathcal{GP}}_{({\mathcal{X}},\omega)},{\mathcal{Y}})$ is a left weak AB-context in ${\mathcal{C}}$ , we have by [Reference Becerril, Mendoza, Pérez and Santiago3, Prop. 5.5] that ${{\text{id}}}_{{\mathcal{GP}}_{({\mathcal{X}},\omega)}}({\mathcal{Y}}) = 0$ . This fact, along with the assumption (a), implies that ${\mathcal{X}} \subseteq {\mathcal{GP}}_{({\mathcal{X}},\omega)} \subseteq {}^{\perp_1}{\mathcal{Y}} \subseteq {\mathcal{X}}$ , and thus ${\mathcal{X}} = {\mathcal{GP}}_{({\mathcal{X}},\omega)}$ , which is left thick since $({\mathcal{X}},\omega)$ is a GP-admissible pair (see Proposition 1.9). Then, ${\mathcal{Y}} \subseteq {\mathcal{GP}}^\wedge_{({\mathcal{X}},\omega)} = {\mathcal{X}}^\wedge$ by $\mathsf{(lABc3)}$ .

The implication (b) $\Rightarrow$ (c) is clear. Finally, let us show (c) $\Rightarrow$ (a). So suppose that $({\mathcal{X}},{\mathcal{Y}})$ is a left weak AB-context. Thus, we have that ${\mathcal{Y}}$ is right thick and that ${\mathcal{Y}} \subseteq {\mathcal{X}}^\wedge \subseteq {\mathcal{GP}}^\wedge_{({\mathcal{X}},\omega)}$ . It is only left to show that $({\mathcal{GP}}_{({\mathcal{X}},\omega)}, {\mathcal{GP}}_{({\mathcal{X}},\omega)} \cap {\mathcal{Y}})$ is a left Frobenius pair in ${\mathcal{C}}$ . We know that ${\mathcal{GP}}_{({\mathcal{X}},\omega)}$ is left thick by Proposition 1.9. For conditions $\mathsf{(lFp2)}$ and $\mathsf{(lFp3)}$ , it suffices to show that ${\mathcal{GP}}_{({\mathcal{X}},\omega)} \cap {\mathcal{Y}} = \omega$ . Indeed, using Propositions 1.5 and 1.9, the equality ${{\text{Ext}}}^1_{{\mathcal{C}}}({\mathcal{GP}}_{({\mathcal{X}},\omega)},\omega^\wedge) = 0$ and the fact that ${\mathcal{X}}$ is closed under direct summands, we can note that ${\mathcal{GP}}_{({\mathcal{X}},\omega)} \cap {\mathcal{X}}^\wedge = {\mathcal{X}}$ . This, along with the assumption ${\mathcal{Y}} \subseteq {\mathcal{X}}^\wedge$ , yields $\omega \subseteq {\mathcal{GP}}_{({\mathcal{X}},\omega)} \cap {\mathcal{Y}} \subseteq {\mathcal{X}} \cap {\mathcal{Y}} = \omega$ .