1. Background

Let $\textsf{A}$ be an additive category and $\textsf{B} \subset \textsf{A}$ a subcategory. For objects $a_1$ , $a_2$ of $\textsf{A}$ we will write $\textsf{A}(a_1,a_2) = {\text{Hom}}_\textsf{A}(a_1,a_2)$ . We define the left and right orthogonal categories of $\textsf{B}$ as follows:

\begin{align*}{}{}^\perp \textsf{B} \,:\!=\, \{a \in \textsf{A} \mid \textsf{A}(a,b) = 0 \text{ for all } b \in \textsf{B}\}\text{ and }\textsf{B}{}^\perp \,:\!=\, \{a \in \textsf{A} \mid \textsf{A}(b,a) = 0 \text{ for all } b \in \textsf{B}\}.\end{align*}

We will often use the shorthand $\textsf{A}(a,\textsf{B}) = 0$ to mean $\textsf{A}(a,b) = 0$ for all $b \in \textsf{B}$ ; similarly for the shorthand $\textsf{A}(\textsf{B},a) = 0$ .

Throughout this paper $\textsf{T}$ will be a triangulated category with shift functor $\Sigma \colon \textsf{T} \to \textsf{T}$ . For two subcategories $\textsf{A}$ , $\textsf{B}$ of $\textsf{T}$ , the full subcategory with objects $\{t\mid $ there exists a triangle $a \to t \to b \to \Sigma a$ with $a \in \textsf{A}$ and $b \in \textsf{B}\}$ will be denoted by $\textsf{A} * \textsf{B}$ . A full additive subcategory $\textsf{C}$ of $\textsf{T}$ is extension-closed if $\textsf{C} * \textsf{C} = \textsf{C}$ .

1·1. Approximations

Let $\textsf{A}$ be a subcategory of $\textsf{T}$ and let t be an object of $\textsf{T}$ . A morphism $f \colon t \to a$ with $a \in \textsf{A}$ is called:

1. (i) a left $\textsf{A}$ -approximation of t if $\textsf{T}(f,\textsf{A}) \colon \textsf{T}(a,\textsf{A}) \to \textsf{T}(t,\textsf{A})$ is surjective;

2. (ii) left minimal if any $g \colon a \to a$ such that $gf = f$ is an automorphism;

3. (iii) a minimal left $\textsf{A}$ -approximation of t if it is both left minimal and a left $\textsf{A}$ -approximation of t.

Left $\textsf{A}$ -approximations are sometimes called $\textsf{A}$ -pre-envelopes. If every object of $\textsf{T}$ admits a left $\textsf{A}$ -approximation then $\textsf{A}$ is said to be covariantly finite in $\textsf{T}$ . There is a dual notion of a (minimal) right $\textsf{A}$ -approximation (or an $\textsf{A}$ -precover); if every object of $\textsf{T}$ admits a right $\textsf{A}$ -approximation, then $\textsf{A}$ is said to be contravariantly finite in $\textsf{T}$ . The subcategory $\textsf{A}$ is said to be functorially finite if it is both covariantly finite and contravariantly finite.

Minimal approximations admit the following important property; see, for example, [ Reference Jørgensen18 ] for a triangulated version. We give the statement for left approximations; there is a dual statement for right approximations.

Lemma 1·1 (Wakamatsu lemma for triangulated categories). Let $\textsf{A}$ be an extension closed subcategory of $\textsf{T}$ and suppose $f \colon t \to a$ is a minimal left $\textsf{A}$ -approximation of t. Then in the triangle

\begin{align*}b \longrightarrow t \stackrel{f}{\longrightarrow} a \longrightarrow \Sigma b,\end{align*}

we have $b \in {}{}^\perp\textsf{A}$ .

1·4. The restricted Yoneda functor

Assume now that $\textsf{T}$ is essentially small, idempotent complete, Hom-finite, $\textbf{k}$ -linear and Krull-Schmidt, where $\textbf{k}$ is a commutative noetherian ring. In this situation, Hom-finiteness means that $\textsf{T}(a,b)$ is a finitely-generated $\textbf{k}$ -module for any $a,b\in \textsf{T}$ . In particular, the endomorphism ring of an object $\textsf{T}(s,s)$ is a noetherian ring. Suppose $\textsf{S}$ is a presilting subcategory of $\textsf{T}$ and let $\textsf{C} \,:\!=\, \textsf{S} * \Sigma \textsf{S}$ . We write $\textsf{Mod}\ \textsf{S}$ for the category of contravariant additive functors from $\textsf{S}$ to the category $\textsf{Mod}\ \textbf{k}$ and $\textsf{mod}\ \textsf{S}$ for the full subcategory of finitely presented functors; see [ Reference Auslander6 ]. Consider the restricted Yoneda functor

\begin{align*}F \colon \textsf{T} & \longrightarrow \textsf{Mod}\ \textsf{S} \\t & \longmapsto \textsf{T}(-,t)|_\textsf{S}.\end{align*}

By [ Reference Iyama and Yoshino17 , proposition 6·2], [ Reference Iyama16 , remark 3·1] the restricted Yoneda functor induces an equivalence of categories,

\begin{align*}F \colon (\textsf{S} * \Sigma \textsf{S})/\Sigma \textsf{S} \longrightarrow \textsf{mod}\, \textsf{S},\end{align*}

where $(\textsf{S} * \Sigma \textsf{S})/\Sigma \textsf{S}$ denotes the subfactor category in which morphisms factoring through an object of $\Sigma \textsf{S}$ are sent to zero. We will usually use this equivalence in case $\textsf{mod}\, \textsf{S}$ is an abelian category or, equivalently, when $\textsf{S}$ has weak kernels [ Reference Freyd12 ]. If $\textsf{S} = \textsf{add}(s)$ for some object $s\in \textsf{T}$ there is an equivalence $\textsf{mod}\, \textsf{S} \simeq \textsf{mod}\, E$ , where $E = \textsf{T}(s,s)$ ; see [ Reference Iyama16 , remark 4·1], and $\textsf{mod}\, \textsf{S}$ is automatically abelian.

1·5. Torsion pairs

A torsion pair in an abelian category $\textsf{H}$ consists of a pair of full subcategories $(\mathcal{T},\mathcal{F})$ of $\textsf{H}$ such that $\mathcal{T}{}^\perp = \mathcal{F}$ , ${}{}^\perp \mathcal{F} = \mathcal{T}$ , and for each object h of $\textsf{H}$ , there is a short exact sequence

(1·1) \begin{equation} 0 \longrightarrow t \longrightarrow h \longrightarrow f \longrightarrow 0\end{equation}

with $t \in \mathcal{T}$ and $f \in \mathcal{F}$ . The subcategory $\mathcal{T}$ is called the torsion class and the subcategory $\mathcal{F}$ is called the torsionfree class.

By virtue of the short exact sequence (1·1), it follows that $\mathcal{T}$ is contravariantly finite in $\textsf{H}$ and $\mathcal{F}$ is covariantly finite in $\textsf{H}$ . If, in addition, $\mathcal{T}$ is covariantly finite in $\textsf{H}$ , we say that $\mathcal{T}$ is a functorially finite torsion class. Note that, if $\textsf{H}\simeq\textsf{mod}\, A$ for an artin algebra A and $(\mathcal{T},\mathcal{F})$ is a torsion pair in $\textsf{H}$ , then $\mathcal{T}$ is covariantly finite in $\textsf{H}$ if and only if $\mathcal{F}$ is contravariantly finite in $\textsf{H}$ [ Reference Smalø31 , theorem]. Torsion pairs $(\mathcal{T},\mathcal{F})$ with functorially finite torsion class $\mathcal{T}$ and functorially finite torsionfree class $\mathcal{F}$ are called functorially finite; see e.g. [ Reference Adachi, Iyama and Reiten1 ].

If $\textsf{H}$ is noetherian, for example $\textsf{H} \simeq \textsf{mod}\, E$ for a noetherian ring E, then any subcategory closed under extensions and quotients is a torsion class of a torsion pair; see, e.g. [ Reference Assem, Simson and Skowroński5 , chapter VI] or [ Reference Liu and Stanley22 , proposition 3·5]. When $\textsf{H}$ is artinian, the dual statement holds for torsionfree classes.

2. HRS tilting of co-t-structures at complete cotorsion pairs

In this section $\textsf{T}$ will be an arbitrary triangulated category. The aim of this section is to prove Theorem A. We will need the following technical lemma.

Proof. We show the first equality holds; the second equality is analogous. The inclusion $\Sigma^{-1} \textsf{A} * \Sigma^{-1} \mathcal{X} \subseteq \textsf{A} * \Sigma^{-1} \mathcal{X}$ is immediate because $\Sigma^{-1} \textsf{A} \subseteq \textsf{A}$ . For the other inclusion, consider a decomposition of $t \in \textsf{A} * \Sigma^{-1} \mathcal{X}$ ,

\begin{align*}a \longrightarrow t \longrightarrow \Sigma^{-1} x \longrightarrow \Sigma a\end{align*}

with $a \in \textsf{A}$ and $x \in \mathcal{X}$ . Decompose a with respect to the co-t-structure $(\Sigma^{-1} \textsf{A},\Sigma^{-1} \textsf{B})$ to get a triangle $\Sigma^{-1} a^{\prime} \to a \to \Sigma^{-1} s \to a^{\prime}$ with $s \in \textsf{S} = \Sigma \textsf{A} \cap \textsf{B}$ . Since $(\mathcal{X},\mathcal{Y})$ is a cotorsion pair, by Remark 1·8, we have $\textsf{S} \subseteq \mathcal{X}$ . Applying the octahedral axiom to the two triangles gives

in which $x^{\prime} \in \mathcal{X}$ , giving a decomposition of $t \in \Sigma^{-1} \textsf{A} * \Sigma^{-1} \mathcal{X}$ . Hence .

Theorem 2·2. Suppose $\textsf{T}$ is a triangulated category, $(\textsf{A},\textsf{B})$ is a co-t-structure in $\textsf{T}$ , and $\textsf{C} = \Sigma^2 \textsf{A} \cap \textsf{B}$ is the extended coheart of $(\textsf{A},\textsf{B})$ . Then there is a bijection

\begin{align*}\{{co\mbox{-}t\mbox{-}structures }\ (\textsf{A}^{\prime},\textsf{B}^{\prime}) \ { with }\ \textsf{A} \subseteq \textsf{A}^{\prime} \subseteq \Sigma \textsf{A}\}& \stackrel{1-1}{\longleftrightarrow}\{{complete\ cotorsion\ pairs }\ (\mathcal{X},\mathcal{Y})\ { in }\ \textsf{C}\}. \\(\textsf{A}^{\prime},\textsf{B}^{\prime})& \shortmid\!\longrightarrow(\textsf{B}\cap \Sigma \textsf{A}^{\prime},\textsf{B}^{\prime}\cap \Sigma^2 \textsf{A}) \\(\textsf{add}(\Sigma^{-1} \textsf{A} * \Sigma^{-1} \mathcal{X}), \textsf{add}(\mathcal{Y} * \Sigma^2 \textsf{B}))& \longleftarrow\!\shortmid(\mathcal{X},\mathcal{Y}).\end{align*}

Proof. The proof of this theorem consists of three steps: first we construct the map

\begin{align*}\varphi \colon \{\text{co-t-structures } (\textsf{A}^{\prime},\textsf{B}^{\prime}) \text{ with } \textsf{A} \subseteq \textsf{A}^{\prime} \subseteq \Sigma \textsf{A}\}\to\{\text{complete cotorsion pairs } (\mathcal{X},\mathcal{Y}) \text{ in } \textsf{C}\};\end{align*}

then we construct the map $\psi$ in the opposite direction; then we prove that $\varphi\psi=\text{id}$ and $\psi\varphi=\text{id}$ .

Step 1: Let $(\textsf{A}^{\prime},\textsf{B}^{\prime})$ be a co-t-structure in $\textsf{T}$ such that $\textsf{A} \subseteq \textsf{A}^{\prime} \subseteq \Sigma \textsf{A}$ (equivalently $\Sigma \textsf{B} \subseteq \textsf{B}^{\prime} \subseteq \textsf{B}$ ) and consider the following subcategories of the extended coheart $\textsf{C}$ :

\begin{align*}\mathcal{X}\,:\!=\,\textsf{B}\cap \Sigma \textsf{A}^{\prime}\subseteq \textsf{C}\quad \text{and} \quad \mathcal{Y}\,:\!=\,\textsf{B}^{\prime}\cap \Sigma^2 \textsf{A} \subseteq \textsf{C}.\end{align*}

Note that $\mathcal{X}$ and $\mathcal{Y}$ are closed under summands, since so are $\textsf{A}, \textsf{A}^{\prime}, \textsf{B}$ and $\textsf{B}^{\prime}$ . We claim that $(\mathcal{X},\mathcal{Y})$ is a complete cotorsion pair in $\textsf{C}$ . Since $\mathcal{X} \subseteq \Sigma \textsf{A}^{\prime}$ and $\Sigma \mathcal{Y} \subseteq \Sigma \textsf{B}^{\prime}$ , we have $\mathbb{E}(\mathcal{X},\mathcal{Y})=\textsf{T}(\mathcal{X},\Sigma \mathcal{Y})=0$ and condition (i) of Definition 1·9 holds.

To find the $\mathbb{E}$ -triangle required for condition (ii), consider the following triangles for $c\in \textsf{C}$ given by taking approximation triangles coming from the co-t-structures $(\Sigma \textsf{A}, \Sigma \textsf{B})$ and $(\textsf{A}^{\prime},\textsf{B}^{\prime})$ , respectively,

(2·1) \begin{equation} b \longrightarrow \Sigma a\longrightarrow c \longrightarrow \Sigma b\quad \text{and} \quad a^{\prime}\longrightarrow \Sigma a \longrightarrow b^{\prime}\longrightarrow \Sigma a^{\prime},\end{equation}

where $a\in\textsf{A}$ , $b\in\textsf{B}$ , $a^{\prime}\in\textsf{A}^{\prime}$ and $b^{\prime}\in\textsf{B}^{\prime}$ . Applying the octahedral axiom, we get:

We first observe that $y \in \mathcal{Y}$ . In the triangle $b^{\prime}\to y \to \Sigma b\to \Sigma b^{\prime}$ the outer terms $b^{\prime}\in \textsf{B}^{\prime}$ and $\Sigma b\in \Sigma \textsf{B} \subseteq \textsf{B}^{\prime}$ , so $y\in \textsf{B}^{\prime}$ . In the triangle $c\to y \to \Sigma a^{\prime}\to \Sigma c$ the outer terms $c\in \Sigma^2 \textsf{A}$ and $\Sigma a^{\prime}\in \Sigma \textsf{A}^{\prime} \subseteq \Sigma^2 \textsf{A}$ , so $y\in \Sigma^2 \textsf{A}$ and thus $y\in \mathcal{Y}$ .

Since $\Sigma b\in \textsf{B}$ and $\Sigma c \in \textsf{B}$ , we get that $\Sigma^2 a \in \textsf{B}$ . In the triangle $b^{\prime}\to \Sigma a^{\prime} \to \Sigma^2 a\to \Sigma b^{\prime}$ the outer terms $b^{\prime}\in \textsf{B}^{\prime} \subseteq \textsf{B}$ and $\Sigma^2 a\in \textsf{B}$ , so $\Sigma a^{\prime}\in \Sigma \textsf{A}^{\prime} \cap \textsf{B} = \mathcal{X}$ . Thus, the triangle

\begin{align*}c\longrightarrow y \longrightarrow \Sigma a^{\prime} \longrightarrow \Sigma c\end{align*}

gives the $\mathbb{E}$ -triangle required for condition (ii) of Definition 1·9.

To find the $\mathbb{E}$ -triangle required for condition (iii), consider $c \in \textsf{C}$ as above and a triangle

\begin{align*}a^{\prime\prime}\longrightarrow b \longrightarrow b^{\prime\prime} \longrightarrow \Sigma a^{\prime\prime},\end{align*}

where b is the object in the triangle in (2·1), $a^{\prime\prime}\in\textsf{A}^{\prime}$ and $b^{\prime\prime}\in\textsf{B}^{\prime}$ . Observe that $\Sigma^{-1}c\in \Sigma \textsf{A}$ and $\Sigma a\in \Sigma \textsf{A}$ , so $b\in \Sigma \textsf{A} \subseteq \Sigma^2 \textsf{A}$ . Since $\Sigma a^{\prime\prime} \in \Sigma \textsf{A}^{\prime} \subseteq \Sigma^2 \textsf{A}$ , we get $b^{\prime\prime}\in \Sigma^2 \textsf{A}$ . Applying the octahedral axiom again, we get:

Clearly $x\in \mathcal{X}$ and $b^{\prime\prime}\in \mathcal{Y}$ , so the triangle

\begin{align*}b^{\prime\prime} \longrightarrow x \longrightarrow c \longrightarrow \Sigma b^{\prime\prime}\end{align*}

gives the $\mathbb{E}$ -triangle required for condition (iii) of Definition 1·9. Thus the assignment $\varphi \colon (\textsf{A}^{\prime},\textsf{B}^{\prime}) \mapsto (\mathcal{X},\mathcal{Y})$ defines a map from co-t-structures $(\textsf{A}^{\prime},\textsf{B}^{\prime})$ such that $\textsf{A} \subseteq \textsf{A}^{\prime} \subseteq \Sigma \textsf{A}$ to complete cotorsion pairs in $\textsf{C}$ .

Step 2: We now construct the map in the other direction. Let $(\textsf{A},\textsf{B})$ be a co-t-structure in $\textsf{T}$ and let $(\mathcal{X},\mathcal{Y})$ be a complete cotorsion pair in the extended coheart $\textsf{C}=\Sigma^2 \textsf{A} \cap \textsf{B}$ . Consider the following pair of subcategories of $\textsf{T}$ :

\begin{align*}(\textsf{A}^{\prime},\textsf{B}^{\prime})\,:\!=\,(\textsf{add}(\Sigma^{-1} \textsf{A} * \Sigma^{-1}\mathcal{X}), \textsf{add}(\mathcal{Y} * \Sigma^2 \textsf{B})).\end{align*}

Note that $\textsf{T}(\Sigma^{-1} \mathcal{X}, \Sigma^2 \textsf{B}) = 0$ since $\Sigma^{-1} \mathcal{X} \subseteq \Sigma \textsf{A} \subseteq \Sigma^2 \textsf{A}$ and $\textsf{T}(\Sigma^2 \textsf{A}, \Sigma^2 \textsf{B}) = 0$ because $(\textsf{A},\textsf{B})$ is a co-t-structure. Hence, $\textsf{T}(\textsf{A}^{\prime},\textsf{B}^{\prime}) = 0$ , verifying the orthogonality condition of the definition of a co-t-structure.

We claim that $\textsf{A} \subseteq \textsf{A}^{\prime}\subseteq \Sigma \textsf{A}$ , from which it follows that $\Sigma^{-1} \textsf{A}^{\prime} \subseteq \textsf{A}^{\prime}$ . By Lemma 2·1(i), we have $\Sigma^{-1} \textsf{A} * \Sigma^{-1} \mathcal{X} = \textsf{A} * \Sigma^{-1} \mathcal{X}$ , and therefore $\textsf{A} \subseteq \textsf{A} * \Sigma^{-1} \mathcal{X} \subseteq \textsf{A}^{\prime}$ . As $\mathcal{X} \subseteq \Sigma^2 \textsf{A}$ , again using Lemma 2·1(i), we have $\textsf{A}^{\prime} = \textsf{add}(\textsf{A} * \Sigma^{-1} \mathcal{X}) \subseteq \textsf{add}(\textsf{A} * \Sigma \textsf{A}) \subseteq \Sigma \textsf{A}$ since $\textsf{A} \subseteq \Sigma \textsf{A}$ and $\Sigma \textsf{A}$ is closed under extensions and summands. Similarly, using Lemma 2·1(ii), one can check that $\Sigma \textsf{B} \subseteq \textsf{B}^{\prime} \subseteq \textsf{B}$ , from which it follows that $\Sigma \textsf{B}^{\prime} \subseteq \textsf{B}^{\prime}$ .

It remains for us to construct the approximation triangle from the definition of the co-t-structure. Consider the following triangles for $t\in \textsf{T}$ :

\begin{align*}a_t \longrightarrow t \longrightarrow b_t \longrightarrow \Sigma a_t\quad \text{and} \quad \Sigma b \longrightarrow \Sigma^2 a\longrightarrow b_t \longrightarrow \Sigma^2 b,\end{align*}

where $a_t, a \in\textsf{A}$ and $b,b_t \in \textsf{B}$ . Since $\Sigma^2 a \in \Sigma^2 \textsf{A} \cap \textsf{B}=\textsf{C}$ , there is a triangle

\begin{align*}\Sigma^{-1}x \longrightarrow \Sigma^2 a \longrightarrow y \longrightarrow x\end{align*}

with $x \in \mathcal{X}$ and $y \in \mathcal{Y}$ coming from the $\mathbb{E}$ -triangle occurring in condition (ii) of the definition of complete cotorsion pair. Applying the octahedral axiom twice, we get:

where, in the left-hand diagram, we see $b^{\prime}\in \mathcal{Y} * \Sigma^{2} \textsf{B} \subseteq \textsf{B}^{\prime}$ , and in the right-hand diagram, we have $a^{\prime}\in \textsf{A} * \Sigma^{-1} \mathcal{X} \subseteq \textsf{A}^{\prime}$ . Thus the triangle

\begin{align*}a^{\prime} \longrightarrow t\longrightarrow b^{\prime} \longrightarrow \Sigma a^{\prime}\end{align*}

is an approximation triangle for t with respect to the co-t-structure $(\textsf{A}^{\prime},\textsf{B}^{\prime})$ . Thus the assignment $\psi \colon (\mathcal{X},\mathcal{Y}) \mapsto (\textsf{A}^{\prime}= \textsf{add}(\Sigma^{-1} \textsf{A} * \Sigma^{-1} \mathcal{X}),\textsf{B}^{\prime} = \textsf{add}(\mathcal{Y} * \Sigma^2 \textsf{B}))$ defines a map from complete cotorsion pairs in $\textsf{C}$ to co-t-structures $(\textsf{A}^{\prime},\textsf{B}^{\prime})$ such that $\textsf{A} \subseteq \textsf{A}^{\prime} \subseteq \Sigma \textsf{A}$ .

Step 3: We now show that the maps $\varphi$ and $\psi$ defined in Steps 1 and 2 are mutually inverse. Let $(\textsf{A},\textsf{B})$ be a co-t-structure in $\textsf{T}$ , let $(\mathcal{X},\mathcal{Y})$ be a complete cotorsion pair in the extended coheart $\textsf{C}=\Sigma^2 \textsf{A} \cap \textsf{B}$ and let $(\textsf{A}^{\prime},\textsf{B}^{\prime}) = \psi\big( (\mathcal{X},\mathcal{Y}) \big)$ be the co-t-structure constructed in Step 2. Let $(\mathcal{X}^{\prime},\mathcal{Y}^{\prime}) = \varphi \big( (\textsf{A}^{\prime},\textsf{B}^{\prime}) \big)$ be the complete cotorsion pair constructed from $(\textsf{A}^{\prime},\textsf{B}^{\prime})$ in Step 1. That is,

\begin{align*}\mathcal{X}^{\prime}\,:\!=\,\textsf{B}\cap \Sigma \textsf{A}^{\prime}\subseteq \textsf{C}\quad \text{and} \quad \mathcal{Y}^{\prime}\,:\!=\,\textsf{B}^{\prime}\cap \Sigma^2 \textsf{A} \subseteq \textsf{C}.\end{align*}

Since $\Sigma \mathcal{Y}^{\prime}\subseteq \textsf{add}(\Sigma \mathcal{Y} * \Sigma^3 \textsf{B}) $ and $\mathcal{X}\subseteq \Sigma^2 \textsf{A}$ , we get that $\textsf{T}(\mathcal{X},\Sigma\mathcal{Y}^{\prime})=0$ . For an object $y^{\prime}\in \mathcal{Y}^{\prime}$ , we can consider the triangle $y^{\prime}\to y \to x \to \Sigma y^{\prime}$ , where $x \in \mathcal{X}$ and $y \in \mathcal{Y}$ , coming from the complete cotorsion pair $(\mathcal{X},\mathcal{Y})$ . Since the map $x \to \Sigma y^{\prime}$ is zero, the triangle splits and y ^′ is a summand of y. Since $\mathcal{Y}$ is closed under summands, we get $\mathcal{Y}^{\prime}\subseteq \mathcal{Y}$ . Similarly, $\textsf{T}(\mathcal{X}^{\prime},\Sigma\mathcal{Y})=0$ , and so the splitting of the triangle $y\to y^{\prime} \to x^{\prime} \to \Sigma y$ , where $x^{\prime}\in \mathcal{X}^{\prime}$ and $y^{\prime} \in \mathcal{Y}^{\prime}$ , coming from the complete cotorsion pair $(\mathcal{X}^{\prime},\mathcal{Y}^{\prime})$ gives that $\mathcal{Y}\subseteq \mathcal{Y}^{\prime}$ . Since $\mathcal{X}=\textsf{C}\cap(^{\perp}\Sigma\mathcal{Y})=\textsf{C}\cap(^{\perp}\Sigma\mathcal{Y}^{\prime})=\mathcal{X}^{\prime}$ , the cotorsion pairs coincide and $\varphi\psi=\text{id}$ .

Let $(\textsf{A}^{\prime},\textsf{B}^{\prime})$ be a co-t-structure such that $\textsf{A}\subseteq \textsf{A}^{\prime}\subseteq \Sigma \textsf{A}$ (equivalently, $\Sigma \textsf{B}\subseteq \textsf{B}^{\prime}\subseteq \textsf{B}$ ). Consider the co-t-structure

\begin{align*}(\textsf{A}^{\prime\prime},\textsf{B}^{\prime\prime}) \,:\!=\, \psi \varphi \big( (\textsf{A}^{\prime}, \textsf{B}^{\prime}) \big) = (\textsf{add}(\Sigma^{-1} \textsf{A} * (\Sigma^{-1}\textsf{B} \cap \textsf{A}^{\prime} )), \textsf{add}((\textsf{B}^{\prime}\cap \Sigma^2 A) * \Sigma^2 \textsf{B})).\end{align*}

Clearly, $\textsf{A}^{\prime\prime}\subseteq \textsf{A}^{\prime}$ and $\textsf{B}^{\prime\prime}\subseteq \textsf{B}^{\prime}$ and, since both pairs of subcategories are co-t-structures, we get $(\textsf{A}^{\prime},\textsf{B}^{\prime})=(\textsf{A}^{\prime\prime},\textsf{B}^{\prime\prime})$ and $\psi\varphi=\text{id}$ . Thus we get the desired bijection.

Fig. 1. Schematic showing the construction of the intermediate co-t-structure $(\textsf{A}^{\prime},\textsf{B}^{\prime})$ in Theorem 2·2 from a complete cotorsion pair $(\mathcal{X},\mathcal{Y})$ in the extended coheart $\textsf{C}= \textsf{S}* \Sigma \textsf{S}$ of the co-t-structure $(\textsf{A},\textsf{B})$ .

The corollary below shows that Theorem 2·2 recovers the bijection between co-t-structures intermediate with respect to $(\textsf{A},\textsf{B})$ and silting subcategories $\textsf{S}^{\prime} \subseteq \textsf{S} * \Sigma \textsf{S}$ in [ Reference Iyama16 , theorem 2·3].

Proof. Let $\textsf{S}^{\prime} = \Sigma \textsf{A}^{\prime} \cap \textsf{B}^{\prime}$ be the coheart of the co-t-structure $(\textsf{A}^{\prime},\textsf{B}^{\prime})$ . By Theorem 2·2, $\mathcal{X} = \textsf{B} \cap \Sigma \textsf{A}^{\prime}$ and $\mathcal{Y} = \textsf{B}^{\prime} \cap \Sigma^2 \textsf{A}$ . Hence $\mathcal{W} = \mathcal{X} \cap \mathcal{Y} = \textsf{S}^{\prime} \cap \textsf{C}$ . But since $\textsf{S}^{\prime} = \Sigma \textsf{A}^{\prime} \cap \textsf{B}^{\prime} \subseteq \Sigma^2 \textsf{A} \cap \textsf{B} = \textsf{C}$ , we have $\mathcal{W} = \textsf{S}^{\prime}$ .

We finish this section with a straightforward example illustrating Theorem 2·2.

3. Cotorsion pairs versus torsion pairs

The aim of this section is to provide a direct proof of Theorem B. We restrict to the following setup so that we can apply the setup of Section 1·4.

Under the assumptions of Setup 3·1, if $\textsf{S} = \textsf{add}(s)$ , then $\textsf{mod}\, \textsf{S} \simeq \textsf{mod}\, E$ , where $E = \textsf{T}(s,s)$ is a noetherian ring, making $\textsf{mod}\, \textsf{S}$ abelian and noetherian.

Proposition 3·2. Suppose that the hypotheses of Setup 3·1 hold. Then the equivalence $F \colon \textsf{C}/\Sigma \textsf{S} \to \textsf{mod}\, \textsf{S}$ induces a well-defined map

\begin{align*}\Phi \colon \{ {cotorsion\ pairs\ in }\ \textsf{C}\} \longrightarrow & \{{torsion\ pairs\ in }\ \textsf{mod}\, \textsf{S}\}, \\(\mathcal{X},\mathcal{Y}) \longmapsto & (\mathcal{T} = F\mathcal{Y}, \mathcal{F} = \mathcal{T}{}^\perp)\end{align*}

which restricts to a well-defined map

Proof. Let $(\mathcal{X},\mathcal{Y})$ be a cotorsion pair in $\textsf{C}$ . We claim that the essential image $\mathcal{T} = F\mathcal{Y}$ is a torsion class in $\textsf{mod}\, \textsf{S}$ . Since $\textsf{mod}\, \textsf{S}$ is noetherian, by [ Reference Liu and Stanley22 , proposition 3·5] it is enough to show that $\mathcal{T}$ is closed under quotients and extensions.

We start by showing that $\mathcal{T} = F \mathcal{Y}$ is closed under quotients. Consider an exact sequence $t \stackrel{\varphi}{\longrightarrow} u \to 0$ in $\textsf{mod}\, \textsf{S}$ with $t \in \mathcal{T}$ . Lifting this to $\textsf{C}$ via F, there are objects $y\in \mathcal{Y}$ and $v\in \textsf{C}$ and a morphism $f \colon y \to v$ such that $Fy \simeq t$ , $Fv \simeq u$ and $Ff \simeq \varphi$ . Completing the morphism f to a distinguished triangle in $\textsf{T}$ gives

\begin{align*}c \longrightarrow y \stackrel{f}{\longrightarrow} v \stackrel{g}{\longrightarrow} \Sigma c.\end{align*}

Applying F to this triangle, we get the exact sequence

\begin{align*}\textsf{T}(-,y)|_\textsf{S} \stackrel{\textsf{T}(-,f)|_\textsf{S}}{\longrightarrow} \textsf{T}(-,v)|_\textsf{S} \stackrel{\textsf{T}(-,g)|_\textsf{S}}{\longrightarrow} \textsf{T}(-,\Sigma c)|_\textsf{S} \longrightarrow \textsf{T}(-,\Sigma y)|_\textsf{S}.\end{align*}

Since $\textsf{T}(-,f)|_\textsf{S} \simeq \varphi$ is an epimorphism, we have $\textsf{T}(-,g)|_\textsf{S} = 0$ . Moreover, $\Sigma y \in \Sigma \textsf{S} * \Sigma^2 \textsf{S}$ so that $\textsf{S}$ presilting implies that $\textsf{T}(-,\Sigma y)|_\textsf{S}=0$ . Hence, $\textsf{T}(-,\Sigma c)|_\textsf{S} = 0$ . In particular, it follows that $\Sigma c \in (\textsf{S} * \Sigma \textsf{S} * \Sigma^2 \textsf{S}) \cap \textsf{S}{}^\perp$ , in which case we get that $c \in \textsf{S} * \Sigma \textsf{S}$ . Now applying $\textsf{T}(\mathcal{X},{-})$ to the triangle above gives $\textsf{T}(\mathcal{X},\Sigma v) = 0$ , which means $v \in \mathcal{Y}$ because $(\mathcal{X},\mathcal{Y})$ is a cotorsion pair. Hence, $u \simeq Fv \in \mathcal{T}$ and $\mathcal{T}$ is closed under quotients.

Next we show that $\mathcal{T}$ is closed under extensions. Consider a short exact sequence

\begin{align*}0 \longrightarrow t^{\prime} \stackrel{\varphi}{\longrightarrow} t \longrightarrow t^{\prime\prime} \longrightarrow 0\end{align*}

in $\textsf{mod}\, \textsf{S}$ with $t^{\prime},t^{\prime\prime} \in \mathcal{T}$ . Lift the morphism $\varphi \colon t^{\prime} \to t$ to $\textsf{C}$ to obtain a morphism $f \colon y^{\prime} \to y$ such that $y^{\prime}\in\mathcal{Y}$ , $Ff \simeq \varphi$ , $Fy^{\prime} \simeq t^{\prime}$ and $Fy \simeq t$ . Extend f to a distinguished triangle to get

\begin{align*}y^{\prime} \stackrel{f}{\longrightarrow} y \longrightarrow c \longrightarrow \Sigma y^{\prime}.\end{align*}

Applying the restricted Yoneda functor to this triangle and noting that $\textsf{T}(-,\Sigma y^{\prime})|_\textsf{S} = 0$ gives a commutative diagram,

whence $Fc \simeq t^{\prime\prime}\simeq Fy^{\prime\prime}$ for some $y^{\prime\prime}\in \mathcal{Y}$ .

If $c \in \mathcal{Y} \subseteq \textsf{C}$ , then we are done. However, we do not know this to be the case. Reading off from the triangle above, $c \in \textsf{S} * \Sigma \textsf{S} * \Sigma^2 \textsf{S}$ , so we can consider a decomposition $m \to c \to n \to \Sigma m$ in which $m \in \textsf{S} * \Sigma \textsf{S}$ and $n \in \Sigma^2 \textsf{S}$ . Consider the octahedral diagram obtained from the two triangles below.

Applying the restricted Yoneda functor to a rotation of the lower horizontal triangle gives an isomorphism $Fe \stackrel{\sim}{\longrightarrow} Fy$ . Rotating the right-hand vertical triangle gives

\begin{align*}y^{\prime} \to e \to m \to \Sigma y^{\prime},\end{align*}

showing that $e \in \textsf{C}$ . Consider the octahedral diagram obtained using this triangle together with the triangle $e \to y \to n \to \Sigma e$ .

Note that from the previous diagram c ^′ is isomorphic to the cone of the map $y^{\prime}\xrightarrow{f} y$ , so $c^{\prime}\simeq c$ and $Fc^{\prime}\simeq Fc$ . Applying the restricted Yoneda functor to the two horizontal triangles gives,

In particular, there is an isomorphism $Fy^{\prime\prime} \stackrel{\sim}{\longrightarrow} Fm$ , so by the argument showing $\mathcal{T}$ is closed under quotients, we see that m lies in $\mathcal{Y}$ . Since $\mathcal{Y}$ is closed under extensions, it follows that $e \in \mathcal{Y}$ . Hence $Fe \simeq Fy \simeq t$ , showing that $t \in \mathcal{T}$ , as required.

Finally, we check that the map induced by the restricted Yoneda functor restricts as claimed. We need to show that $\mathcal{T} = F\mathcal{Y}$ is covariantly finite when $(\mathcal{X},\mathcal{Y})$ is a complete cotorsion pair. Let $m \in \textsf{mod}\, \textsf{S}$ . Suppose $c \in \textsf{C}$ is such that $Fc \simeq m$ . Without loss of generality we can assume $Fc = m$ . Consider a decomposition triangle of c with respect to the complete cotorsion pair $(\mathcal{X},\mathcal{Y})$ ,

\begin{align*}c \stackrel{f}{\longrightarrow} y \longrightarrow x \longrightarrow \Sigma c.\end{align*}

Note that $f \colon c \to y$ is a left $\mathcal{Y}$ -approximation of c in $\textsf{C}$ . We claim that $Ff \colon m \to Fy$ is a left $\mathcal{T}$ -approximation of m in $\textsf{mod}\, \textsf{S}$ . Suppose $\varphi \colon m \to t$ is a morphism in $\textsf{mod}\, \textsf{S}$ with $t \in \mathcal{T}$ . Then there exist $y^{\prime} \in \mathcal{Y}$ together with an isomorphism $e \colon Fy^{\prime} \stackrel{\sim}{\longrightarrow} t$ and $g \colon c \to y^{\prime}$ such that $ \varphi=eFg$ . Since f is a left $\mathcal{Y}$ -approximation of c in $\textsf{C}$ , there exists $h \colon y \to y^{\prime}$ such that $g = hf$ . Applying F to this composition gives $\varphi = eFg = eFh Ff$ , that is, Ff is a left $\mathcal{T}$ -approximation, as required.

The next lemma provides a useful criterion to detect when a cotorsion pair is complete.

Proof. If $(\mathcal{X},\mathcal{Y})$ is a complete cotorsion pair in $\textsf{C}$ then by definition such a triangle exists for $s \in \textsf{S}$ because such a triangle exists for each $c \in \textsf{C}$ .

Conversely, suppose $(\mathcal{X},\mathcal{Y})$ is a cotorsion pair in $\textsf{C}$ and that for each $s \in \textsf{S}$ , there is a triangle $s \to y \to x \to \Sigma s$ with $x \in \mathcal{X}$ and $y \in \mathcal{Y}$ . Let $c \in \textsf{C}$ and take a decomposition triangle, $s_2 \to s_1 \to c \to \Sigma s_2$ , and the triangle $s_2 \to y_2 \to x_2 \to \Sigma s_2$ given by the assumption. Applying the octahedral axiom to these triangles gives the following diagram.

We have $e \in \mathcal{X}$ since $\textsf{S} \subseteq \mathcal{X}$ , because $(\mathcal{X},\mathcal{Y})$ is a cotorsion pair. Thus, the triangle $y_2 \to e \to c \to \Sigma y_2$ provides the second triangle required for completeness in Definition 1·9.

To obtain the first triangle in Definition 1·9, we use the octahedral axiom again together with the triangle $s_1 \to y_1 \to x_1 \to \Sigma s_1$ given by the assumption:

Analogously, we have $d \in \mathcal{Y}$ because $\Sigma \textsf{S} \subseteq \mathcal{Y}$ , making $c \to d \to x_1 \to \Sigma c$ the required triangle.

We now define an inverse to the restricted map in Proposition 3·2. For a subcategory $\mathcal{T}$ of $\textsf{mod}\, \textsf{S}$ , denote by $F^{-1}(\mathcal{T})$ the full subcategory of $\textsf{C}$ whose objects are $\{c \in \textsf{C} \mid Fc \in \mathcal{T}\}$ .

Proposition 3·6. Let $\textsf{S}$ be a presilting subcategory of $\textsf{T}$ and $\textsf{C} = \textsf{S} * \Sigma \textsf{S}$ . There is a well-defined map

Proof. Let $(\mathcal{T},\mathcal{F})$ be a torsion pair in $\textsf{mod}\, \textsf{S}$ whose torsion class is functorially finite, $\mathcal{Y} = \{c \in \textsf{C} \mid Fc \in \mathcal{T}\}$ and $\mathcal{X} = {}{}^\perp (\Sigma \mathcal{Y})\cap \textsf{C}$ . The subcategories $\mathcal{Y}$ and $\mathcal{X}$ are closed under direct summands, since $\mathcal{T}$ is closed under direct summands and $\mathcal{X}$ is defined as an orthogonal subcategory. We set $\Theta\big( (\mathcal{T},\mathcal{F}) \big) = (\mathcal{X},\mathcal{Y})$ . We claim that $(\mathcal{X},\mathcal{Y})$ is a complete cotorsion pair. We start by showing that $\mathcal{Y}$ is closed under extensions (in $\textsf{C}$ ).

Let $y^{\prime} \stackrel{f}{\longrightarrow} y \longrightarrow y^{\prime\prime} \longrightarrow \Sigma y^{\prime}$ be an extension with $y^{\prime}, y^{\prime\prime} \in \mathcal{Y}$ . Applying F to this $\mathbb{E}$ -triangle in $\textsf{C}$ gives an exact sequence $Fy^{\prime} \stackrel{Ff}{\longrightarrow} Fy \longrightarrow Fy^{\prime\prime} \longrightarrow 0$ in $\textsf{mod}\, \textsf{S}$ since . We thus obtain a short exact sequence $0 \to \text{im}\ Ff \to Fy \to Fy^{\prime\prime} \to 0$ . Since $\mathcal{T}$ is a torsion class, it follows that $\text{im}\ Ff \in \mathcal{T}$ and $Fy \in \mathcal{T}$ . Hence, we conclude that $y \in \mathcal{Y}$ .

For $s \in \textsf{S}$ , we will construct a triangle $s \to y \to x \to \Sigma s$ in which $y \in \mathcal{Y}$ and $x \in \mathcal{X}$ , which will allow us to apply Lemma 3·4 to conclude that $(\mathcal{X},\mathcal{Y})$ is a complete cotorsion pair in $\textsf{C}$ . First, we need to check that Lemma 3·4 applies. Since $\mathcal{Y} \subseteq \textsf{C}$ , we have that $\textsf{T}(\textsf{S},\Sigma \mathcal{Y}) = 0$ so that $\textsf{S} \subseteq \mathcal{X}$ . Furthermore, if $s \in \textsf{S}$ then $F(\Sigma s) = 0 \in \mathcal{T}$ so that $\Sigma s \in \mathcal{Y}$ . By Remark 3·5(i), Lemma 3·4 applies.

Now let $s \in \textsf{S}$ . Consider a left $\mathcal{T}$ -approximation $\varphi \colon Fs \to t$ of Fs in $\textsf{mod}\, \textsf{S}$ . Replacing t by an isomorphic object we can assume without loss of generality that $t= Fy$ for some $y\in \mathcal{Y}$ . Let $f \colon s \to y$ be such that $Ff = \varphi$ . We claim that $f \colon s \to y$ is a left $\mathcal{Y}$ -approximation of s. Consider a morphism $g \colon s \to y^{\prime}$ with $y^{\prime} \in \mathcal{Y}$ . Applying F to f and g gives a diagram,

where the morphism $\psi \colon t \to Fy^{\prime}$ exists because $\varphi$ is a left $\mathcal{T}$ -approximation. Since the functor F is full, there exists $h \colon y \to y^{\prime}$ such that $Fh = \psi$ . We claim that $g = hf$ .

Applying F to $g - hf$ shows that $g - hf = 0$ in $\textsf{C}/\Sigma \textsf{S}$ . Hence, $g-hf$ factors through $\Sigma \textsf{S}$ .

Hence, $g - hf = 0$ in $\textsf{C}$ since $\textsf{S}$ is presilting. It follows that $f \colon s \to y$ is a left $\mathcal{Y}$ -approximation of $\textsf{S}$ . As $\textsf{T}$ is Krull–Schmidt, without loss of generality, we may assume that $f \colon s \to y$ is a minimal left $\mathcal{Y}$ -approximation and extend it to a distinguished triangle, $s \stackrel{f}{\longrightarrow} y \longrightarrow x \longrightarrow \Sigma s$ . By the Wakamatsu lemma for triangulated categories, Lemma 1·1, we see that $x \in {}{}^\perp (\Sigma \mathcal{Y})$ . Since y and $\Sigma s$ are in $\textsf{C}$ , we get that $x \in \textsf{C}$ , and so $x\in \mathcal{X}$ .

Theorem 3·7. Suppose the hypotheses of Setup 3·1 hold. Then, there is a bijection

Moreover, if in addition $\textsf{mod}\, \textsf{S}\simeq\textsf{mod}\, A$ for an artin algebra A, there is a bijection between complete cotorsion pairs in $\textsf{C}$ and functorially finite torsion pairs in $\textsf{mod}\, \textsf{S}$ .

Proof. We show that the maps $\Phi$ and $\Theta$ defined in Propositions 3·2 and 3·6 are mutually inverse. Let $(\mathcal{X},\mathcal{Y})$ be a complete cotorsion pair in $\textsf{C}$ . By Proposition 3·2 we have $\Phi \big( (\mathcal{X},\mathcal{Y}) \big) = (\mathcal{T},\mathcal{F})$ , where $\mathcal{T} = F\mathcal{Y}$ and $\mathcal{F} = \mathcal{T}{}^\perp$ in $\textsf{mod}\, \textsf{S}$ . Applying $\Theta$ to $(\mathcal{T},\mathcal{F})$ produces a complete cotorsion pair $(\mathcal{X}^{\prime},\mathcal{Y}^{\prime})$ in which $\mathcal{Y}^{\prime} = \{ c \in \textsf{C} \mid Fc \in \mathcal{T}\}$ . Clearly, $\mathcal{Y} \subseteq \mathcal{Y}^{\prime}$ . To see that $\mathcal{Y}^{\prime} \subseteq \mathcal{Y}$ take $y^{\prime} \in \mathcal{Y}^{\prime}$ and observe that there is an isomorphism $\varphi \colon Fy \to Fy^{\prime}$ in $\textsf{mod}\, \textsf{S}$ for some $y\in \mathcal{Y}$ by the definition of $\mathcal{T}$ . Now, applying the same argument used to show that $F\mathcal{Y}$ is closed under quotients in the proof of Proposition 3·2 shows that $\textsf{T}(\mathcal{X},\Sigma y^{\prime}) = 0$ , whence by completeness of the cotorsion pair $(\mathcal{X},\mathcal{Y})$ and the fact that any complete cotorsion pair is a cotorsion pair, we get $y^{\prime} \in \mathcal{Y}$ . As a cotorsionfree class determines a cotorsion pair uniquely, we get $(\mathcal{X}^{\prime},\mathcal{Y}^{\prime})=(\mathcal{X},\mathcal{Y})$ . The equality $\Phi \Theta = 1$ follows from $F(F^{-1}(\mathcal{T}))=\mathcal{T}$ . The last statement follows from [ Reference Smalø31 , theorem].