Proof. See [Reference Jasso and KvammeJK, Lemma 1.1].

Construction 2.9. The minimal d-cokernel of a morphism $f \colon X \rightarrow Y$ in a d-cluster tilting subcategory $\mathcal {M} \subseteq \mathcal {A}$ can be constructed as follows:

1. (1) Set $C_1=\operatorname {Coker} f$ and let $g_1 \colon C_1 \to M_1$ be the minimal left $\mathcal {M}$ -approximation of $C_1$ . Set ${f_1 \colon Y \to M_1}$ to be the composition $Y \rightarrow C_1 \xrightarrow {g_1} M_1$ .

2. (2) Repeat on $f_1 \colon Y \to M_1$ to construct $f_2 \colon M_1 \to M_2$ .

3. (3) Iterate the procedure, which must terminate and result in a d-cokernel by [Reference JassoJ2, Proposition 3.17].

Since each morphism $f_i$ is the composition of an epimorphism and a left minimal morphism, they are all left minimal, and hence, Lemma 2.8 shows that this is the minimal d-cokernel of f. Note that each $f_i$ can equivalently be described as the left minimal weak cokernel of the previous morphism.

Proof. The fact that a d-pushout diagram exists is precisely [Reference JassoJ2, Theorem 3.8]. It follows from [Reference JassoJ2, Proposition 4.8] that the bottom row is a d-extension with last term equal to Y. If this d-extension is minimal, then we are done. Otherwise, it has a minimal d-extension as a direct summand by Proposition 2.6. Composing the morphism in the statement with the projection onto this minimal d-extension gives another commutative diagram, which is a d-pushout by [Reference JassoJ2, Proposition 4.8]. The bottom row of this new diagram is a minimal d-extension as required.

Proof. By the definition of a d-torsion class, the morphism $U_M\to M$ is a right $\mathcal {U}$ -approximation. Since the morphism $tM\to M$ is a right $\mathcal {T}$ -approximation and $tM\in \mathcal {U}$ , it must also be a right $\mathcal {U}$ -approximation. Hence, the inclusions $U_M\to M$ and $tM\to M$ must factor through each other, which implies that $U_M\cong tM$ .

Proof. Consider the short exact sequence

$$\begin{align*}0 \rightarrow tM \xrightarrow{\iota} M \rightarrow f M \rightarrow 0 \end{align*}$$

associated to the torsion pair $(\mathcal {T},\mathcal {F})$ . The morphism $\iota $ is a right $\mathcal {T}$ -approximation of M, and $tM \in \mathcal {M}$ by assumption. As $X \in \mathcal {T}$ , there exists a morphism $\psi _X \colon X \to tM$ such that $\iota \psi _X = \phi _X$ . Notice that since $\phi _X$ is a left $\mathcal {M}$ -approximation of X, so is $\psi _X$ . Since $\phi _X$ is left minimal and $\iota $ is a monomorphism, this implies that M is isomorphic to $tM$ . Consequently, one obtains that M is contained in $\mathcal {T}$ , and hence also in $\mathcal {U}$ .

Proof. Let $\phi _X\colon X \rightarrow M$ be the minimal left $\mathcal {M}$ -approximation of X. By Lemma 3.3, we know that $M \in \mathcal {U}$ , and it follows that $\phi _X$ is also a left $\mathcal {U}$ -approximation. Since $\phi _X$ is left minimal, it is the minimal left $\mathcal {U}$ -approximation of X. Finally, since $\mathcal {M}$ is cogenerating, any left $\mathcal {M}$ -approximation is a monomorphism.

Proof. By Proposition 2.6, any equivalence class of d-extensions contains a unique (up to isomorphism) minimal d-extension, which is furthermore a direct summand of any other d-extension in the class. This immediately implies (1), since $\mathcal {U}$ is closed under direct summands. Part (2) is proved similarly, using that the minimal d-cokernel of a morphism f is a summand of any other d-cokernel of f.

Proof. The minimality of the d-extension (see Definition 2.5) gives that $e_i \in \operatorname {Rad}_{\mathcal {A}}(E_i, E_{i+1})$ for all $i=1, \dots , d-1$ . Moreover, the sequence

(2) $$ \begin{align} E_1 \xrightarrow{e_1} E_2 \xrightarrow{e_2} \cdots \xrightarrow{e_{d-1}} E_d \xrightarrow{g} Y \to 0 \end{align} $$

is a d-cokernel of f. If $g \in \operatorname {Rad}_{\mathcal {A}}(E_d,Y)$ , then this d-cokernel is minimal. Since $E_1 \in \mathcal {U}$ , the result then follows from $\mathcal {U}$ being closed under minimal d-quotients by Lemma 3.8.

Suppose hence that $g \notin \operatorname {Rad}_{\mathcal {A}}(E_d,Y)$ . Recall from Remark 2.7 that the sequence (2) is isomorphic to the direct sum of the minimal d-cokernel of f and shifted complexes of the form $N \xrightarrow {1_N} N$ . However, since $e_i \in \operatorname {Rad}_A(E_i,E_{i+1})$ for $i=1, \dots d-1$ , it follows that (2) is isomorphic to

$$ \begin{align*} E_1 \xrightarrow{e_1} E_2 \xrightarrow{e_2} \cdots \xrightarrow{e_{d-2}} E_{d-1} \xrightarrow{e_{d-1}'=\left(\begin{smallmatrix} h_1 \\ 0 \end{smallmatrix}\right)} E^{\prime}_d \oplus Y" \xrightarrow{\left(\begin{smallmatrix} g' & 0 \\ 0 & 1_{Y"} \end{smallmatrix}\right)} Y' \oplus Y" \to 0, \end{align*} $$

where $e_{d-1}'$ is $e_{d-1}$ composed with an isomorphism and

$$ \begin{align*} E_1 \xrightarrow{e_1} E_2 \xrightarrow{e_2} \cdots \xrightarrow{e_{d-2}} E_{d-1} \xrightarrow{h_1} E^{\prime}_d \xrightarrow{ g'} Y' \to 0 \end{align*} $$

is a minimal d-cokernel of f. In particular, the objects $E_2, \dots , E_{d-1},E_d'$ are in $\mathcal {U}$ as $E_1\in \mathcal {U}$ and $\mathcal {U}$ is closed under minimal d-quotients by Lemma 3.8. Since $Y'\oplus Y" = Y \in \mathcal {U}$ and $\mathcal {U}$ is closed under direct summands, we have that $Y"\in \mathcal {U}$ . Hence, $E_d=E_d' \oplus Y"\in \mathcal {U}$ , completing the proof.

Proof. As $\mathcal {U}$ is a d-torsion class in $\mathcal {M}$ , we have that $\mathcal {U} = \mathcal {T} \cap \mathcal {M}$ for some torsion class $\mathcal {T}$ in $\mathcal {A}$ as in Setup 3.1. Recall from Lemma 3.8 that it suffices to consider minimal d-extensions and minimal d-quotients.

We first show that $\mathcal {U}$ is closed under minimal d-quotients. Let $f \colon M \to U$ be a morphism in $\mathcal {M}$ with $U \in \mathcal {U}$ and consider its minimal d-cokernel

$$\begin{align*}M \xrightarrow{f} U \xrightarrow{f_1} V_1 \xrightarrow{f_2} \cdots \xrightarrow{f_{d-1}} V_{d-1}\xrightarrow{f_{d}} V_{d}\to 0. \end{align*}$$

By construction of the minimal d-cokernel (see Construction 2.9), we have that $V_i$ arises as the minimal left $\mathcal {M}$ -approximation of $\operatorname {Coker} f_{i-1}$ for all $i= 1,\ldots , d$ , where we set $f_0 = f$ . As $U \in \mathcal {U} = \mathcal {T} \cap \mathcal {M}$ and $\mathcal {T}$ is closed under quotients, Lemma 3.3 implies that $V_i \in \mathcal {U}$ for all $i= 1,\ldots , d$ . This shows that $\mathcal {U}$ is closed under minimal d-quotients.

We next prove that $\mathcal {U}$ is closed under minimal d-extensions. Consider a minimal d-exact sequence

$$\begin{align*}0 \to X \xrightarrow{e_0} E_1 \xrightarrow{e_1} \cdots \xrightarrow{e_{d-1}} E_d \xrightarrow{e_d} Y \to 0 \end{align*}$$

in $\mathcal {M}$ with X and Y in $\mathcal {U}$ . By Lemma 2.8, the morphism $e_i$ is left minimal for $i=0,\dots ,d-2$ . As $\mathcal {U}$ is a d-torsion class, we obtain the solid part of the diagram

where the bottom row is the d-exact sequence associated to $E_1$ by $\mathcal {U}$ . In particular, the object $tE_1 \in \mathcal {U}$ is the d-torsion subobject of $E_1$ with respect to $\mathcal {U}$ by Lemma 3.2, and the sequence

(3) $$ \begin{align}0 \rightarrow \operatorname{Hom}_{\mathcal{A}}(U,W_1) \rightarrow \cdots \rightarrow \operatorname{Hom}_{\mathcal{A}}(U,W_d) \rightarrow 0 \end{align} $$

is exact for every U in $\mathcal {U}$ . As $\iota $ is a right $\mathcal {T}$ -approximation and $X \in \mathcal {U} = \mathcal {T} \cap \mathcal {M}$ , there exists a morphism $g_0 \colon X \to tE_1$ making the left square commute. We can hence complete the diagram to a morphism g of d-exact sequences by using the factorisation property for weak cokernels; see Section 2.1.

Since (3) is exact and $Y\in \mathcal {U}$ , the morphism

$$\begin{align*}w_{d-1} \circ - \colon \operatorname{Hom}_{\mathcal{A}}(Y,W_{d-1}) \rightarrow \operatorname{Hom}_{\mathcal{A}}(Y,W_d)\end{align*}$$

is surjective. Hence, there exists a morphism $h_d\colon Y \to W_{d-1}$ with $w_{d-1} h_d=g_{d+1}$ . As the bottom row is d-exact, there is an exact sequence

$$\begin{align*}\operatorname{Hom}_{\mathcal{A}}(E_d,W_{d-2}) \rightarrow \operatorname{Hom}_{\mathcal{A}}(E_d, W_{d-1}) \rightarrow \operatorname{Hom}_{\mathcal{A}}(E_d,W_d). \end{align*}$$

Using the commutativity of the rightmost square, we get $w_{d-1}(g_d-h_de_d)=0$ , so $g_d-h_de_d$ is in the kernel of the second morphism. By exactness, there exists a morphism $h_{d-1}\colon E_d \to W_{d-2}$ such that $g_d-h_de_d = w_{d-2}h_{d-1}$ , or equivalently, $g_d= h_d e_d + w_{d-2}h_{d-1}$ .

We can repeat this process to obtain a homotopy of the map of complexes g. In particular, there are morphisms $h_0$ and $h_1$ such that $g_1 =1_{E_1} = \iota h_0 + h_1 e_1$ . This implies that $\iota h_0 e_0 = e_0$ , so $\iota h_0$ is an isomorphism by the left minimality of $e_0$ . The morphism $h_0$ is hence a split monomorphism, so $E_1$ is contained in $\mathcal {U}$ . By Lemma 3.10, this implies that $E_i \in \mathcal {U}$ for $i=2,\dots ,d$ , so the subcategory $\mathcal {U}$ is closed under minimal d-extensions.

Proof. Let $E_0$ denote the image of f, and let $\iota _0\colon E_0\to M_0$ be a minimal left $\mathcal {M}$ -approximation. The inclusion $E_0\to N$ lifts via $\iota _0$ to a morphism $g_0\colon M_0\to N$ . Let $E_1$ be the image of $g_0$ , and note that $E_0\subseteq E_1$ . Repeating this procedure, we get a subobject $E_i\subseteq N$ , a minimal left approximation $\iota _i\colon E_i\to M_i$ , and a lift $g_i\colon M_i\to N$ for each $i\geq 0$ . In particular, the $E_i$ ’s form an increasing sequence $E_0\subseteq E_1\subseteq E_2\subseteq \cdots \subseteq E_i\subseteq \cdots $ of subobjects of N.

Since $\mathcal {A}$ is of finite length, this sequence has to stabilise, say at $E_j$ . This implies that the image of $g_j\colon M_j\to N$ is $E_j$ . But then the inclusion $\iota _j\colon E_j\rightarrow M_j$ is a split monomorphism, and hence an isomorphism since it is also left minimal. This shows that $E_j\in \mathcal {M}$ . Now let $f_2\colon E_j\to N$ be the inclusion, and let $f_1\colon M\to E_j$ be the composite $M\to E_0\rightarrow E_j$ . Note that this is equal to the composite

By construction, the morphisms $M\to M_0$ and $M_i\to M_{i+1}$ for $i=0,\ldots , j-1$ are left minimal weak cokernels. Since $f_1$ is a composite of such morphisms and the isomorphism $M_j\cong E_j$ , this proves the claim.

Proof. Assume g is a weak cokernel of a morphism f. Then g is part of a d-cokernel of f, and if g is left minimal, then it is part of the minimal d-cokernel of f; see Construction 2.9. This proves the claim.

Proof. Consider the set of subobjects $U\subseteq M$ with $U\in \mathcal {U}$ . Note that this set is nonempty since $0\in \mathcal {U}$ .

We first prove that this set has a unique maximal element. Indeed, since $\mathcal {A}$ has finite length, we can choose $U\subseteq M$ with $U\in \mathcal {U}$ and where U is of maximal length with this property. Now let $V\subseteq M$ with $V\in \mathcal {U}$ , and consider the induced morphism U ⊕ V → M. By Proposition 3.13, there exists $W\subseteq M$ with $W\in \mathcal {M}$ such that $U\subseteq W\subseteq M$ and $V\subseteq W\subseteq M$ , and such that the induced morphism $U\oplus V\to W$ is a composite of left minimal weak cokernels. Since $\mathcal {U}$ is closed under d-quotients, it follows from Lemma 3.14 that $W\in \mathcal {U}$ . But since U is maximal with respect to the property that $U\in \mathcal {U}$ and $U\subseteq M$ , it follows that $U=W$ . This implies that V must be contained in U, and hence, U is the unique maximal subobject $U\subseteq M$ satisfying $U\in \mathcal {U}$ .

Now let $U'\to M$ be an arbitrary morphism with $U'\in \mathcal {U}$ . By Proposition 3.13, there exists an object $V'$ such that $U'\to M$ factors through $V'$ and where the morphism $U'\rightarrow V'$ is a composite of left minimal weak cokernels and the morphism $V'\rightarrow M$ is a monomorphism. It follows that $V'\in \mathcal {U}$ and $V'$ is a subobject of M.

By the maximality of U, we get that $V'\subseteq U\subseteq M$ , so the morphism $U'\to M$ also factors through U. This shows that the inclusion $U\subseteq M$ is a right $\mathcal {U}$ -approximation, which is minimal since it is a monomorphism. This proves the first claim.

Finally, the fact that $\mathcal {U}$ is contravariantly finite in $\mathcal {A}$ follows from the fact that $\mathcal {M}$ is contravariantly finite in $\mathcal {A}$ and $\mathcal {U}$ is contravariantly finite in $\mathcal {M}$ .

Proof. Consider an object $M \in \mathcal {M}$ . By the definition of a d-torsion class (see Definition 2.13), we need to show that there exists a d-exact sequence

$$\begin{align*}0 \rightarrow U_M \rightarrow M \rightarrow V_1 \rightarrow \cdots \rightarrow V_d \rightarrow 0 \end{align*}$$

where $U_M\in \mathcal {U}$ and the sequence $0 \rightarrow \operatorname {Hom}_{\mathcal {M}}(U,V_1) \rightarrow \cdots \rightarrow \operatorname {Hom}_{\mathcal {M}}(U,V_d) \rightarrow 0$ is exact for every U in $\mathcal {U}$ .

Since $\mathcal {U}$ is closed under d-quotients, Proposition 3.15 shows that we may take a minimal right $\mathcal {U}$ -approximation $f \colon U_M \to M$ which is a monomorphism. Taking the minimal d-cokernel of f gives a d-exact sequence

(4) $$ \begin{align} 0 \rightarrow U_M \xrightarrow f M \rightarrow V_1 \rightarrow \cdots \rightarrow V_d \rightarrow 0. \end{align} $$

Let $U\in \mathcal {U}$ . As f is a right $\mathcal {U}$ -approximation, we know that $\operatorname {Hom}_{\mathcal {M}}(U,f)$ is an epimorphism. Thus, it follows from d-exactness of the sequence (4) that

$$\begin{align*}0 \rightarrow \operatorname{Hom}_{\mathcal{M}}(U,V_1) \rightarrow \cdots \rightarrow \operatorname{Hom}_{\mathcal{M}}(U,V_{d-1}) \rightarrow \operatorname{Hom}_{\mathcal{M}}(U,V_d) \end{align*}$$

is exact. To finish our proof, we hence need to show that the rightmost morphism in this sequence is an epimorphism.

Consider $h_d\in \operatorname {Hom}_{\mathcal {M}}(U,V_d)$ and take a d-pullback of (4) along $h_d$ . This yields a commutative diagram

where the upper row is a d-exact sequence. By the dual of Lemma 2.10, this d-extension can be assumed to be minimal, and then closedness of $\mathcal {U}$ under d-extensions implies $W_i\in \mathcal {U}$ for all $i=0,\ldots ,d-1$ by Lemma 3.8. As f is a right $\mathcal {U}$ -approximation, the morphism $h_0$ factors through f, so $f_0$ is a split monomorphism. It follows, by [Reference JassoJ2, Proposition 2.6] and its dual, that $f_d$ is a split epimorphism, and hence, $h_d$ factors through $V_{d-1}$ . In particular, the morphism $\operatorname {Hom}_{\mathcal {M}}(U,V_{d-1}) \rightarrow \operatorname {Hom}_{\mathcal {M}}(U,V_d)$ is an epimorphism, as required.

Proof. The necessity follows from Proposition 3.11, while the sufficiency follows from Proposition 3.16.

Proof. By Theorem 3.17, the subcategory $\mathcal {U}$ is closed under d-extensions in $\mathcal {M}$ . The result hence follows by applying [Reference KlapprothKl, Corollary 4.15].

Proof. Assume that $\mathcal {U}$ is closed under d-quotients and under d-extensions with indecomposable first term. Let

(5) $$ \begin{align} 0\rightarrow X\rightarrow E_1 \rightarrow \cdots \rightarrow E_d \rightarrow Y \rightarrow 0 \end{align} $$

be a minimal d-extension with $X,Y\in \mathcal {U}$ . We want to show that $E_i \in \mathcal {U}$ for all $i=1,\dots ,d$ . By Lemma 3.10, it is sufficient to check that $E_1\in \mathcal {U}$ .

If X is indecomposable, we are done by assumption. Suppose, hence, that $X=X_1\oplus X_2$ , where $X_1$ is indecomposable and $X_2 \neq 0$ . We take a d-pushout of the d-extension along the projection $\pi \colon X\rightarrow X_1$ . This yields a commutative diagram

where the lower sequence is d-exact and can be assumed to be minimal by Lemma 2.10. As $X_1$ is indecomposable, we have $F_i\in \mathcal {U}$ for $1\leq i\leq d$ . By [Reference JassoJ2, Proposition 4.8 (ii)], we have a d-exact sequence

$$\begin{align*}0\to X\to X_1\oplus E_1\to F_1\oplus E_2\to \cdots \to F_{d-1}\oplus E_d\to F_d\to 0. \end{align*}$$

Since $X\cong X_1\oplus X_2$ , this complex can be written as the sum of the identity morphism

$$\begin{align*}0\to X_1\xrightarrow{1_{X_1}}X_1\to 0 \to \cdots \to 0 \to 0\to 0 \end{align*}$$

and a d-exact sequence

(6) $$ \begin{align} 0\to X_2\to E_1\to F_1\oplus E_2\to \cdots \to F_{d-1}\oplus E_d\to F_d\to 0. \end{align} $$

Next, we choose a decomposition $E_1\cong E_1"\oplus E_1'$ and $F_1\cong E_1^{\prime \prime }\oplus \tilde {F_1}$ such that the morphism $E_1\to F_1$ becomes

$$ \begin{align*} \begin{pmatrix} 1_{E_1"} & 0 \\ 0 & f \end{pmatrix}\colon E_1"\oplus E_1' \rightarrow E_1"\oplus \tilde{F_1} \end{align*} $$

with $f \in \operatorname {Rad}_{\mathcal {A}}(E_1', \tilde {F_1})$ . In particular, $E_1"$ is a summand of $F_1$ and is therefore contained in $\mathcal {U}$ . Hence, $E_1'\in \mathcal {U}$ if and only if $E_1\in \mathcal {U}$ . Furthermore, (6) can be written as a sum of the identity morphism

$$\begin{align*}0\to 0\to E_1"\xrightarrow{1_{E_1"}}E_1"\to 0 \to \cdots \to 0 \to 0\to 0 \end{align*}$$

and a d-exact sequence

$$ \begin{align*} 0\to X_2\to E^{\prime}_1\to \tilde{F_1}\oplus E_2\to \cdots \to F_{d-1}\oplus E_d\to F_d\to 0. \end{align*} $$

By minimality of (5), the morphism $E_1\to E_2$ is in the Jacobson radical, and hence, the induced morphism $E_1'\to E_2$ is also in the Jacobson radical. Combining this with the fact that $f\colon E_1'\to \tilde {F_1}$ is in the Jacobson radical, we get that $E_1'\to \tilde {F_1'}\oplus E_2$ is in the Jacobson radical. Hence, if we let

$$\begin{align*}0\to X_2\to E_1'\to F^{\prime}_1\to \cdots \to F^{\prime}_{d-1}\to F_d'\to 0 \end{align*}$$

be the minimal d-cokernel of $X_2\to E_1'$ , then this must also give a minimal d-extension. By Proposition 2.6, the term $F^{\prime }_d$ is a direct summand of $F_d$ and must therefore be in $\mathcal {U}$ . Obviously, the object $X_2$ has fewer indecomposable summands than X. If $X_2$ is indecomposable, we know that $E^{\prime }_1\in \mathcal {U}$ (which is equivalent to $E_1\in \mathcal {U}$ ), as the end terms $X_2$ and $F_d'$ of the above minimal d-extension are in $\mathcal {U}$ . If not, we repeat the argument to eventually show that $E_1\in \mathcal {U}$ .

Proof of Theorem 3.20.

Suppose that we have a minimal d-extension

$$ \begin{align*} 0 \to X \xrightarrow{e_0} E_1 \xrightarrow{e_1} E_2 \xrightarrow{e_2} \cdots \xrightarrow{e_{d-1}} E_d \xrightarrow{e_d} Y \to 0 \end{align*} $$

with $X, Y \in \mathcal {U}$ . By Lemma 3.21, we may assume that X is indecomposable. Let $Y= \bigoplus _{j=1}^t Y_j$ , where each $Y_j$ is indecomposable. For each inclusion $\iota _j \colon Y_j \to Y$ , consider a d-pullback diagram

(7)

By the dual of Lemma 2.10, the top d-extension can be chosen to be minimal. Consequently, each of the morphisms $f_{j,1}, \dots , f_{j,d-1}$ is in the Jacobson radical. Moreover, the middle objects $F_{j,1},\dots ,F_{j,d}$ are in $\mathcal {U}$ since $\mathcal {U}$ is closed under d-extensions between indecomposables.

Now look at the d-extension

$$ \begin{align*} 0 \to \bigoplus_{j=1}^t X \xrightarrow{f_0} \bigoplus_{j=1}^t F_{j,1} \xrightarrow{f_1} \cdots \xrightarrow{f_{d-1}} \bigoplus_{j=1}^t F_{j,d} \xrightarrow{f_d} \bigoplus_{j=1}^t Y_j \to 0 \end{align*} $$

given by the direct sum of all the upper d-extensions obtained as in (7). Consider the induced map

with $h_i = \left (\begin {smallmatrix} h_{1,i} & \cdots & h_{t,i} \end {smallmatrix}\right )$ for $i = 0,\dots ,d$ . This is a d-pushout diagram by [Reference JassoJ2, Proposition 4.8], and thus, the associated mapping cone

(8) $$ \begin{align} 0 \to \bigoplus_{j=1}^t X \xrightarrow{\left(\begin{smallmatrix} h_0 \\ -f_0 \end{smallmatrix}\right)} X \oplus \bigoplus_{j=1}^t F_{j,1} \xrightarrow{\left(\begin{smallmatrix} e_0 & h_1 \\ 0 & -f_1 \end{smallmatrix}\right)} E_1 \oplus \bigoplus_{j=1}^t F_{j,2} \rightarrow \cdots \rightarrow E_d \to 0 \end{align} $$

is a d-extension. Note that the term $X \oplus \bigoplus _{j=1}^t F_{j,1}$ lies in $\mathcal {U}$ . If (8) is given by the minimal d-cokernel of the first morphism, we are hence done by closure under minimal d-quotients. So suppose this d-cokernel is not minimal. By Remark 2.7, it is then isomorphic to the direct sum of the minimal d-cokernel and shifted complexes of the form $N \xrightarrow {1} N$ . In particular, if

$$ \begin{align*} X \oplus \bigoplus_{j=1}^t F_{j,1} \xrightarrow{\partial_1} M_1 \xrightarrow{\partial_2} M_2 \to \cdots \to M_d \to 0 \end{align*} $$

is the minimal d-cokernel of $\left (\begin {smallmatrix} h_0 \\ -f_0 \end {smallmatrix}\right )$ , then there is a commutative diagram

where the vertical maps are isomorphisms. Since $e_1 \in \operatorname {Rad}_{\mathcal {A}}(E_1, E_2)$ , the morphism

$$ \begin{align*} \phi \circ \begin{pmatrix} e_1 \\ 0 \end{pmatrix}\colon E_1 \rightarrow M_2 \oplus N_1 \oplus N_2 \end{align*} $$

also lies in the radical, and thus so does

$$ \begin{align*} \phi \circ \begin{pmatrix} e_1 \\ 0 \end{pmatrix} =\begin{pmatrix} \partial_2 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} a \\ c \end{pmatrix}= \begin{pmatrix} \partial_2\circ a \\ c \\ 0 \end{pmatrix}. \end{align*} $$

This shows that $c \in \operatorname {Rad}_{\mathcal {A}}(E_1, N_1 )$ . Now let $ \left (\begin {smallmatrix} \alpha & \beta \\ \gamma & \delta \end {smallmatrix}\right )$ denote the inverse of $\left (\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}\right )$ . We have $\alpha a + \beta c =1_{E_1}$ , or equivalently, $\alpha a = 1_{E_1} - \beta c$ . It follows from the definition of the radical that this is an isomorphism, as $c \in \operatorname {Rad}_{\mathcal {A}}(E_1, N_1 )$ . This implies that $E_1$ is a direct summand of $M_1$ . But $M_1 \in \mathcal {U}$ since $\mathcal {U}$ is closed under minimal d-quotients, and hence, $E_1$ also lies in $\mathcal {U}$ . It then follows from Lemma 3.10 that $E_2, \dots , E_d \in \mathcal {U} $ , so we can conclude that $\mathcal {U}$ is closed under d-extensions as required.

Proof. If $x_d=y_d$ , the only nonzero morphism (up to multiplication by a scalar) is the identity, which is a monomorphism. If $x_d < y_d$ , the result follows from Proposition 5.3 when looking at the extension between $M_x$ and $M_z$ , where $z=(x_1+1, \dots , x_d+1, y_d)$ .

Proof. Define $z=(z_0, \dots , z_d)$ such that

$$\begin{align*}z_j = \begin{cases} x_j & \text{if } j \neq i+1 \\ x_i & \text{if } j=i+1. \end{cases} \end{align*}$$

We then have $z \rightsquigarrow x$ , so there is a nonzero morphism $M_{z} \to M_x$ by Proposition 5.2. Because $x_i\leq z_{i+1}<x_{i+1}$ , the module $M_{y}$ is hence in $\operatorname {dq}(M_{x})$ by [Reference Herschend and JørgensenHJ, Lemma 3.8(2)], keeping in mind that this paper uses a different notation as we outlined in Remark 5.1.

Proof. Suppose $y \in \operatorname {os}_n^{d+1}$ satisfies $x \leq y$ and $x_d=y_d$ . Construct a sequence

$$\begin{align*}x=z^0, z^1, z^2, \dots, z^m=y\end{align*}$$

in $\operatorname {os}_n^{d+1}$ , where the element $z^{i+1}$ is constructed from $z^i$ as follows. If $z^i=y$ , then we are finished. Otherwise, there exists a maximal j such that $z_j^i < y_j$ , and we must have $j<d$ . Then $z_j^i+1 \leq y_j \leq y_{j+1}=z_{j+1}^i$ , and we define

$$\begin{align*}z^{i+1}_k = \begin{cases} z^i_k+1 & \text{if } k=j \\ z^i_k & \text{if } k \neq j. \end{cases} \end{align*}$$

Notice that $M_{z^{i+1}} \in \operatorname {dq}(M_{z^i})$ by Lemma 5.6 and that this process must terminate with $z^m=y$ . Thus, we get

$$\begin{align*}M_{y} \in \operatorname{dq}(M_{z^{m-1}}) \subseteq \dots \subseteq \operatorname{dq}(M_{x}), \end{align*}$$

as required.

Proof. Since each $C_i \in {\mathcal {M}}_n^d$ , we may assume that every $C_i$ is equal to a direct sum of modules of the form $M_z$ for $z\in \operatorname {os}_n^{d+1}$ . Choose $M_{z} \in \operatorname {add}(C_i)$ . Write the morphism $C_{i-1} \to C_i$ as

$$\begin{align*}\begin{pmatrix} f \\ g \end{pmatrix}\colon C_{i-1} \rightarrow M_{z} \oplus C_i'. \end{align*}$$

By the construction of minimal d-cokernels, this morphism factors through the cokernel K of the morphism $C_{i-2} \to C_{i-1}$ as indicated in the diagram

where $\left (\begin {smallmatrix} f' \\ g' \end {smallmatrix}\right )$ is a minimal left $\mathcal {M}^d_{n}$ -approximation of K.

There exists some $M_{w} \in \operatorname {add}(C_{i-1})$ with a nonzero morphism $M_{w} \to M_{z}$ . Indeed, if this is not the case, then $f=0$ . However, since $\pi $ is an epimorphism, this would imply $f'=0$ , contradicting $\left (\begin {smallmatrix} f' \\ g' \end {smallmatrix}\right )$ being left minimal. Consequently, we may write $C_{i-1} = D \oplus D'$ , where $D \neq 0$ and $D'$ is the largest summand of $C_{i-1}$ that maps to zero under f. This means that the morphism $\left (\begin {smallmatrix} f \\ g \end {smallmatrix}\right )$ may be written as

$$ \begin{align*} \begin{pmatrix} f_1 & 0 \\ g_1 & g_2 \end{pmatrix}\colon D \oplus D' \xrightarrow{} M_z \oplus C_i', \end{align*} $$

where $f_1$ is nonzero. For every $M_{v} \in \operatorname {add}(D)$ , there is a nonzero morphism to $M_z$ . Hence, Proposition 5.2 shows that $v \rightsquigarrow z$ . Since $D \neq 0$ , we may choose $M_y \in \operatorname {add}(D)$ with $y_d$ maximal (i.e. for all $M_{v} \in \operatorname {add}(D)$ , we have $v_d \leq y_d).$

With z and y fixed and knowing that $y \rightsquigarrow z$ , we may now consider $z'=(z_0,z_1, \dots , z_{d-1},y_d)$ . Notice that $z' \rightsquigarrow z$ . Proposition 5.2 and Lemma 5.5 thus yield that there exists a monomorphism $\iota \colon M_{z'} \to M_z$ . Moreover, by the maximality of $y_d$ , we have $v \rightsquigarrow z'$ for all $M_{v} \in \operatorname {add}(D)$ . Hence, by Proposition 5.2, there exists a morphism $h \colon D \rightarrow M_{z'}$ such that $f_1 = \iota \circ h$ . In particular, the solid part of the diagram

commutes. Since $\left (\begin {smallmatrix} \iota & 0 \\ 0 & 1 \end {smallmatrix}\right )$ is a monomorphism, this implies that the composition of the morphism $C_{i-2} \to C_{i-1}$ and $\left (\begin {smallmatrix} h & 0 \\ g_1 & g_2 \end {smallmatrix}\right )$ is zero. Thus, there is a morphism $t \colon K \to M_{z'} \oplus C_i'$ making the upper triangle in the diagram above commute.

Note that the lower triangle also commutes as $\pi $ is an epimorphism. Finally, since $\left (\begin {smallmatrix} f' \\ g' \end {smallmatrix}\right )$ is an $\mathcal {M}^d_{n}$ -approximation, there exists a morphism $s \colon M_z \oplus C_i' \to M_{z'} \oplus C_i'$ such that

$$ \begin{align*} t=s \circ \begin{pmatrix} f' \\ g' \end{pmatrix}. \end{align*} $$

It follows that

$$ \begin{align*} \begin{pmatrix} f' \\ g' \end{pmatrix} = \begin{pmatrix} \iota & 0 \\ 0 & 1 \end{pmatrix} \circ t = \begin{pmatrix} \iota & 0 \\ 0 & 1 \end{pmatrix} \circ s \circ \begin{pmatrix} f' \\ g' \end{pmatrix}, \end{align*} $$

and thus, $\left (\begin {smallmatrix} \iota & 0 \\ 0 & 1 \end {smallmatrix}\right ) \circ s$ is an isomorphism, since $\left (\begin {smallmatrix} f' \\ g' \end {smallmatrix}\right )$ is left minimal. Therefore, the monomorphism $\iota $ is also an epimorphism, so it must be an isomorphism, and we have $M_{z}\cong M_{z'}$ . This shows that $y_d=z_d$ , and hence, $M_y \in \operatorname {add}(C_{i-1})$ satisfies both $y \rightsquigarrow z$ (and thus, $y \leq z$ ) and $y_d=z_d$ .

We can now repeat the argument with $M_y$ and keep going until we get $M_{x} \in \operatorname {add}(C_0)$ with $x\leq y\leq z$ and $x_d=y_d=z_d$ .

Proof. By Corollary 5.7, it suffices to show that for any $M_y \in \operatorname {dq}(M_x)$ , we must have $x \leq y$ and $x_d=y_d$ . Let $i\geq 0$ be such that $M_y \in \operatorname {dq}(M_x)_i$ . If $i=0$ , we have $M_y \in \operatorname {dq}(M_x)_0 = \operatorname {add}(M_x)$ , so $M_y = M_x$ , and the statement holds. Assume hence, $i>0$ . By Lemma 5.8, there exists $M_z \in \operatorname {dq}(M_x)_{i-1}$ such that $z \leq y$ and $z_d=y_d$ .

Repeating this argument, we will eventually find some $M_w \in \operatorname {dq}(M_x)_0$ such that $w\leq y$ and $w_d=y_d$ . As $\operatorname {dq}(M_x)_0=\operatorname {add}(M_x)$ yields $M_w=M_x$ , the result follows.

Proof. We prove this by induction on the filtration of $\operatorname {dq}(\bigoplus _{x \in I} M_x)$ . If

$$\begin{align*}M_y \in \operatorname{dq}\,\left(\bigoplus_{x \in I} M_x\right)_{0}=\operatorname{add}\,\left(\bigoplus_{x \in I} M_x\right), \end{align*}$$

the statement clearly holds.

Now suppose $M_y \in \operatorname {dq}\left (\bigoplus _{x \in I} M_x\right )_i$ for some $i>0$ and that the result is known for all $M_z \in \operatorname {dq}\left (\bigoplus _{x \in I} M_x\right )_{i-1}$ . By construction, there must exist a minimal d-quotient

$$ \begin{align*} X \xrightarrow{f} Y \to C_1 \to \dots \to C_d \to 0 \end{align*} $$

in $\mathcal {M}^d_{n}$ of some morphism f such that $Y \in \operatorname {dq}\left (\bigoplus _{x \in I} M_x\right )_{i-1}$ and $M_y \in \operatorname {add}(C_j)$ for some $1\leq j \leq d$ . Hence, combining Lemma 5.8 with Corollary 5.7 shows that there exists

$$\begin{align*}M_z \in \operatorname{add}(Y) \subseteq \operatorname{dq}\,\left(\bigoplus_{x \in I} M_x\right)_{i-1} \end{align*}$$

such that $M_y \in \operatorname {dq}(M_z)$ . By the induction hypothesis, we have $M_z \in \operatorname {dq}(M_x)$ for some $x \in I$ , and thus, it follows that $M_y \in \operatorname {dq}(M_z) \subseteq \operatorname {dq}(M_x)$ , as required.

Proof. Since it is clear that $\operatorname {dq}(M_x) \subseteq \operatorname {dq}\left (\bigoplus _{x \in I} M_x\right )$ for each $x \in I$ , this is a direct consequence of Corollary 5.9 and Proposition 5.11.

Proof of Theorem 5.13.

By Corollary 5.12, condition (1) holds if and only if $\mathcal {U}_I$ is closed under d-quotients. It follows from Proposition 5.3 and Remark 5.4 that condition (2) is equivalent to $\mathcal {U}_I$ being closed under d-extensions by indecomposables. Moreover, if $\mathcal {U}_I$ is closed under d-quotients, it is closed under d-extensions by indecomposables if and only if it is closed under all d-extensions by Theorem 3.20. From Theorem 3.17, we know that $\mathcal {U}_I$ is a d-torsion class if and only if it is closed under d-extensions and d-quotients, which proves the claim.

Algorithm 5.15. Given a set of initial indecomposable modules in $\mathcal {M}^d_{n}$ , or equivalently, a subset of $\operatorname {os}^{d+1}_n$ , this algorithm computes the minimal d-torsion class containing those modules.

Input: Integers d ≥ 1, n ≥ 1 and a set X = {x ¹, …, x ^r }⊆ $\operatorname {os}^{d+1}_n$ .

1. (1) Let I = X.

2. (2) For each pair x, y ∈ I such that x ⇝ τ _d (y), add the (d + 1)-tuple (x ₀, …, x _d−1, y _d ) to I.

3. (3) For every x ∈ I, add the elements of dq(x) to I.

4. (4) If new elements were added to I in step (2) or (3), repeat from step (2). Otherwise, terminate the process.

Output: The set I.

Proof. We need to show that $\mathcal {U}_I$ is the minimal d-torsion class containing $M_{x^1}, \dots , M_{x^r}$ . Step (1) of the algorithm ensures that $M_{x^1}, \dots , M_{x^r}\in \mathcal {U}_I$ . By Proposition 5.11, step (3) implies that $\mathcal {U}_I$ is closed under d-quotients.

Consider two indecomposable modules $M_x, M_y \in \mathcal {U}_I$ with $\operatorname {Ext}^d_{A_n^d}(M_y,M_x) \neq 0$ . This means that the pair $x,y \in I$ satisfies $x\rightsquigarrow \tau _d(y)$ by Proposition 5.3. Moreover, if $M_z$ is a direct summand in one of the middle terms in the nontrivial d-extension from $M_x$ to $M_y$ described in Proposition 5.3, then either $z\in \operatorname {dq}(x)$ or $z\in \operatorname {dq}((x_{0}, \dots ,x_{d-1}, y_d))$ . Step (2) followed by step (3) thus ensures that $M_z \in \mathcal {U}_I$ , so $\mathcal {U}_I$ is closed under d-extensions with indecomposable end terms. By Theorem 3.20, this implies that $\mathcal {U}_I$ is closed under all d-extensions. We can hence conclude that $\mathcal {U}_I$ is a d-torsion class by Theorem 3.17.

We now know that $\mathcal {U}_I$ is a d-torsion class containing the modules $M_{x^1}, \dots , M_{x^r}$ . However, objects added to I in the algorithm correspond to either $M_{x^1}, \dots , M_{x^r}$ or to indecomposable direct summands obtained from minimal d-quotients or minimal d-extensions – see Remark 5.4 – and the result follows.

Algorithm 5.18. This algorithm computes all higher torsion classes associated to a higher Auslander algebra.

Input: Integers d ≥ 1, n ≥ 1.

1. (1) Let 𝔘 be the singleton set containing the empty set. Set l = 1.

2. (2) For all sets X consisting of l distinct (d + 1)-tuples in $\operatorname {os}^{d+1}_n$ , compute U(X) using Algorithm 5.15 and add it to 𝔘.

3. (3) If new elements were added to 𝔘 in step (2), increase l by one and repeat from step (2). Otherwise, terminate the process.

Output: The set 𝔘.

Proof. First, observe that the trivial d-torsion class $\{0\}=\mathcal {U}_{\emptyset }$ , and $\emptyset \in \mathfrak {U}$ . As ${\mathcal {M}}_n^d$ has finitely many indecomposable objects, any nontrivial d-torsion class $\mathcal {V}$ in ${\mathcal {M}}_n^d$ can be written as $\mathcal {V}=\mathcal {U}(M_{x^1}, \dots , M_{x^m})$ for some positive integer m.

Let r be the lowest positive integer for which any d-torsion class of the form $\mathcal {U}(M_{x^1}, \dots , M_{x^r})$ can also be written as $\mathcal {U}(M_{y^1}, \dots , M_{y^s})$ for some $s<r$ . By construction, the set $\mathfrak {U}$ indexes all d-torsion classes of the form $\mathcal {U}(M_{x^1}, \dots , M_{x^l})$ for $l<r$ .

Suppose that $\mathcal {V}$ is a nontrivial d-torsion class, and let $m\geq 0$ be minimal such that

$$\begin{align*}\mathcal{V}=\mathcal{U}(M_{x^1}, \dots, M_{x^m}). \end{align*}$$

We claim that $\mathcal {V}=\mathcal {U}_I$ for some $I\in \mathfrak {U}$ (i.e., that $m<r$ ). Indeed, if $m\geq r$ , consider the d-torsion class $\mathcal {V}'= \mathcal {U}(M_{x^1}, \dots , M_{x^{r}})$ . By the assumption on r, we can write $\mathcal {V}'$ as $\mathcal {U}(M_{z^1}, \dots , M_{z^s})$ for some $s< r$ . But then $\mathcal {V}$ is of the form $\mathcal {U}(M_{z^1}, \dots M_{z^s},M_{x^{r+1}},\dots M_{x^m})$ and can hence be generated by $m+s-r<m$ indecomposable modules. This contradicts the minimality of m, and the result follows.