Proof. Suppose for contradiction that \(K_{\smash{q}}(Y) \neq Y(q)\) holds for every \(q \in \operatorname{supp}\mspace{1mu}Y\). Since \(Y\) is non-zero, it has non-empty support, so we can choose some object \(q_1 \in \operatorname{supp}\mspace{1mu}Y\). By assumption we have \(K_{\smash{q_1}}(Y) \neq Y(q_1)\); this implies that there exists some \(g_1 \colon q_1 \to q_2\) in \(\mathfrak{r}\) with \(Y(g_1) \neq 0\); in particular, \(g_1 \neq 0\) and \(q_2 \in \operatorname{supp}\mspace{1mu}Y\). By assumption, \(K_{\smash{q_2}}(Y) \neq Y(q_2)\), so there exists a \(g_2 \colon q_2 \to q_3\) in \(\mathfrak{r}\) with \(Y(g_2) \neq 0\); in particular, \(g_2 \neq 0\) and \(q_3 \in \operatorname{supp}\mspace{1mu}Y\). Continuing in this manner, we construct an infinite sequence, \[\xymatrix@C=1.5pc{ q_1 \ar[r]^-{g_1} & q_2 \ar[r]^-{g_2} & q_3 \ar[r]^-{g_3} & \;\cdots },\] of non-zero morphisms in \(\mathfrak{r}\) where \(q_1,q_2,q_3,\ldots\) belong to \(\operatorname{supp}\mspace{1mu}Y\). As \(\operatorname{supp}\mspace{1mu}Y \subseteq \mathcal{L}\) holds, this contradicts condition (2) in Definition 4. Hence there exists some \(q \in \operatorname{supp}\mspace{1mu}Y\) with \(K_{\smash{q}}(Y)=Y(q)\). By [1] there is an adjunction \((S_{\mspace{-4mu}\smash{q}},K_{\smash{q}})\); write \(\varepsilon_q\) for the counit. The morphism \[\xymatrix{ S_{\mspace{-4mu}\smash{q}}(Y(q)) \,=\, S_{\mspace{-4mu}\smash{q}}K_{\smash{q}}(Y) \ar[r]^-{\varepsilon_q^Y} & Y }\] in \({}_{Q,A}\mspace{-1mu}\operatorname{Mod}\) is monic as \(\varepsilon_q^Y(q) \colon Y(q) \to Y(q)\) is the identity map and \(\varepsilon_q^Y(p) \colon 0 \to Y(p)\) is the zero map for \(p \neq q\). Hence \(S_{\mspace{-4mu}\smash{q}}(Y(q))\) is a subobject of \(Y\) in \({}_{Q,A}\mspace{-1mu}\operatorname{Mod}\). ◻

Proof. We start by constructing a filtration (?? ) that satisfies conditions (1), (2), and

Set \(X_0=0\). Let \(\beta\) be an ordinal and assume that \(X_\alpha\) has been constructed for every \(\alpha<\beta\). If \(\beta\) is a limit ordinal, define \(X_\beta\) as in (2). Now assume that \(\beta = \alpha+1\) is a successor ordinal. If \(X_\alpha = X\), then (1) holds with \(\lambda = \alpha\) and we are done; otherwise, \(Y_\alpha = X/X_\alpha\) is non-zero. As \(\operatorname{supp}\mspace{1mu}Y_\alpha \subseteq \operatorname{supp}\mspace{1mu}X \subseteq \mathcal{L}\) we can by Lemma 5 choose \(q_\alpha \in \operatorname{supp}\mspace{1mu}Y_\alpha\) such that \(S_{\mspace{-4mu}\smash{q_\alpha}}(Y(q_\alpha))\) is a subobject of the quotient \(Y_\alpha = X/X_\alpha\), and hence it has the form \(S_{\mspace{-4mu}\smash{q_\alpha}}(Y(q_\alpha)) = X_{\alpha+1}/X_\alpha\) for some \(X_\alpha \subseteq X_{\alpha+1} \subseteq X\). Thus, (3\('\)) holds. Note that since \(Y_\alpha(q_\alpha) \neq 0\), the inclusion \(X_\alpha \subset X_{\alpha+1}\) is strict, so eventually we get \(X_\lambda = X\) for some ordinal \(\lambda\), and hence (1) holds.

Below we prove that the following condition holds for every \(\alpha \le \lambda\).

Once this has been proved, the remaining property (3) follows from the already established property (3\('\)). Indeed, by (3\('\)) we have a \(q_\alpha \in \operatorname{supp}\mspace{1mu}X/X_\alpha\) and hence \(X_\alpha(q_\alpha) \neq X(q_\alpha)\); thus condition (\(*\)) implies that \(q_\alpha \notin \operatorname{supp}\mspace{1mu}X_\alpha\). This means that \(X_\alpha(q_\alpha)=0\), and consequently \[S_{\mspace{-4mu}\smash{q_\alpha}}(X(q_\alpha)/X_\alpha(q_\alpha)) \,=\, S_{\mspace{-4mu}\smash{q_\alpha}}(X(q_\alpha)) \;.\] Therefore, in view of (\(*\)), the property (3) follows from (3\('\)).

It remains to prove condition (\(*\)). We do this by transfinite induction on \(\alpha\). For \(\alpha=0\) there is nothing to prove as \(X_0=0\) and hence \(\operatorname{supp}\mspace{1mu}X_0 = \varnothing\). Now let \(\beta \le \lambda\) be an ordinal and assume that (\(*\)) holds for every \(\alpha<\beta\). First assume that \(\beta\) is a limit ordinal, in which case by (2). There is an equality, \[\label{eq:cup} \textstyle \operatorname{supp}\mspace{1mu}X_\beta \,=\, \bigcup_{\alpha<\beta} \operatorname{supp}\mspace{1mu}X_\alpha\;.\tag{1}\] Indeed, the inclusion “\(\subseteq\)” is trivial. On the other hand, if \(p \in \operatorname{supp}\mspace{1mu}X_{\alpha_0}\) for some \(\alpha_0<\beta\), then also \(p \in \operatorname{supp}\mspace{1mu}X_\alpha\) for every \(\alpha \ge \alpha_0\) since there is an inclusion \(X_{\alpha_0} \subseteq X_\alpha\). Hence, by the induction hypothesis one gets \(0 \neq X_\alpha(p) = X(p)\) for every \(\alpha_0 \le \alpha < \beta\), and consequently \[\textstyle X_\beta(p) \,=\, \varinjlim_{\,\alpha<\beta} X_\alpha(p) \,=\, \varinjlim_{\,\alpha_0 \le \alpha<\beta} X_\alpha(p) \,=\, X(p) \,\neq\, 0 \;.\] This simultaneously proves the other inclusion “\(\supseteq\)” in (1 ) and establishes condition (\(*\)) for the limit ordinal \(\beta\). Finally, consider the case where \(\beta = \alpha+1\) is a successor ordinal. By (3\('\)) there is a short exact sequence \[\label{eq:ses} 0 \longrightarrow X_\alpha \longrightarrow X_{\alpha+1} \longrightarrow S_{\mspace{-4mu}\smash{q_\alpha}}(Y_\alpha(q_\alpha)) \longrightarrow 0\tag{2}\] where \(Y_\alpha = X/X_\alpha\) and \(q_\alpha \in \operatorname{supp}\mspace{1mu}Y_\alpha\). This immediately yields: \[\operatorname{supp}\mspace{1mu}X_{\alpha+1} \,=\, \operatorname{supp}\mspace{1mu}X_\alpha \,\cup\, \operatorname{supp}\mspace{1mu}S_{\mspace{-4mu}\smash{q_\alpha}}(Y_\alpha(q_\alpha)) \,=\, \operatorname{supp}\mspace{1mu}X_\alpha \,\cup\, \{q_\alpha\}\;.\] For \(p \in \operatorname{supp}\mspace{1mu}X_\alpha\) the induction hypothesis yields \(X_\alpha(p) = X(p)\), and hence \(X_{\alpha+1}(p) = X(p)\) by the inclusions \(X_\alpha \subseteq X_{\alpha+1} \subseteq X\). As \(q_\alpha \in \operatorname{supp}\mspace{1mu}Y_\alpha\) we have \(X_\alpha(q_\alpha) \neq X(q_\alpha)\), so \(q_\alpha\) does not belong to \(\operatorname{supp}\mspace{1mu}X_\alpha\) as condition (\(*\)) holds for the ordinal \(\alpha\). This means that \(X_\alpha(q_\alpha)=0\) and hence \(Y_\alpha(q_\alpha) = X(q_\alpha)\). In view of this, the short exact sequence (2 ) now yields \[X_{\alpha+1}(q_\alpha) \,=\, S_{\mspace{-4mu}\smash{q}}(Y_\alpha(q_\alpha))(q_\alpha) \,=\, Y_\alpha(q_\alpha) \,=\, X(q_\alpha) \;,\] which establishes condition (\(*\)) for the successor ordinal \(\beta = \alpha+1\). ◻

Proof. If \(X\) belongs to \(\operatorname{Filt}\text{-}\mathcal{Z}\), then \(X\) has a filtration \[0 = X_0 \rightarrowtail X_1 \rightarrowtail X_2 \rightarrowtail \cdots \rightarrowtail X_\omega \rightarrowtail X_{\omega+1} \rightarrowtail \cdots \rightarrowtail X_\lambda = X\] satisfying the properties of Definition 7.

By transfinite induction we will prove that \(\operatorname{supp}\mspace{1mu}X_\alpha \subseteq \mathbb{L}\) for every \(\alpha \le \lambda\); note that with \(\alpha = \lambda\) we obtain \(\operatorname{supp}\mspace{1mu}X \subseteq \mathbb{L}\), as desired. For \(\alpha=0\) there is nothing to prove as \(X_0=0\) and hence \(\operatorname{supp}\mspace{1mu}X_0 = \varnothing\). Now let \(\beta \le \lambda\) be an ordinal and assume that \(\operatorname{supp}\mspace{1mu}X_\alpha \subseteq \mathbb{L}\) holds for every \(\alpha < \beta\). If \(\beta\) is a limit ordinal, then by (1), and hence \[\textstyle \operatorname{supp}\mspace{1mu}X_\beta \,=\, \bigcup_{\alpha<\beta} \operatorname{supp}\mspace{1mu}X_\alpha \,\subseteq\, \mathbb{L} \;,\] where the equality follows as in (1 ) and the inclusion holds by the induction hypothesis. If \(\beta = \alpha+1\) is a successor ordinal the exact sequence \(0 \to X_\alpha \to X_{\alpha+1} \to X_{\alpha+1}/X_\alpha \to 0\) yields \[\label{eq:3xsupp} \operatorname{supp}\mspace{1mu}X_{\alpha+1} \,=\, \operatorname{supp}\mspace{1mu}X_\alpha \;\cup\; \operatorname{supp}\mspace{1mu}(X_{\alpha+1}/X_\alpha) \;.\tag{3}\] By the induction hypothesis we have \(\operatorname{supp}\mspace{1mu}X_\alpha \subseteq \mathbb{L}\). By (2) one has \(X_{\alpha+1}/X_\alpha \in \mathcal{Z}\) and hence \(\operatorname{supp}\mspace{1mu}(X_{\alpha+1}/X_\alpha) \subseteq \mathbb{L}\) by assumption. Thus (3 ) implies that \(\operatorname{supp}\mspace{1mu}X_{\alpha+1} \subseteq \mathbb{L}\).

We proved that \(\operatorname{supp}\mspace{1mu}(\operatorname{Filt}\text{-}\mathcal{Z})\subset\operatorname{supp}\mspace{1mu}\mathcal{Z}\). The other inclusion \(\operatorname{supp}\mspace{1mu}\mathcal{Z}\subset\operatorname{supp}\mspace{1mu}(\operatorname{Filt}\text{-}\mathcal{Z})\) is straightforward. ◻

Proof. First assume that \(X\) belongs to \(\operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}}\); that is, \(X\) has a filtration \[\label{eq:filtration-1} 0 = X_0 \rightarrowtail X_1 \rightarrowtail X_2 \rightarrowtail \cdots \rightarrowtail X_\omega \rightarrowtail X_{\omega+1} \rightarrowtail \cdots \rightarrowtail X_\lambda = X\tag{4}\] satisfying the properties of Definition 7.

Since one has \(\operatorname{supp}\mspace{1mu}S_{\mspace{-4mu}\smash{q}}(A) = \{q\}\), every object in \(\mathcal{S}_{\mathcal{L}}\) has its support contained in \(\mathcal{L}\), so Lemma 8 implies that \(\operatorname{supp}\mspace{1mu}X \subseteq \mathcal{L}\).

Now fix \(q \in \mathcal{L}\). By (2) each quotient \(X_{\alpha+1}/X_\alpha\) has the form \(X_{\alpha+1}/X_\alpha = S_{\mspace{-4mu}\smash{q_\alpha}}(A)\) for some \(q_\alpha \in \mathcal{L}\) and hence \(X_{\alpha+1}(q)/X_\alpha(q) = S_{\mspace{-4mu}\smash{q_\alpha}}(A)(q) \in \{0,A\}\) by [1]. Thus, by evaluating the given filtration (4 ) at \(q\), we obtain a filtration of the \(A\)-module \(X(q)\) with quotients in \(\{0,A\}\); that is, \(X(q) \in \operatorname{Filt}\text{-}\{0,A\}\) in \({}_{A}\mspace{-1mu}\operatorname{Mod}\). It now easily follows that \(X(q)\) is free, cf. the remarks following Göbel and Trlifaj [19].

Conversely, assume \(\operatorname{supp}\mspace{1mu}X \subseteq \mathcal{L}\) and that the \(A\)-module \(X(q)\) is free for every \(q \in \mathcal{L}\) (equivalently, for every \(q \in Q\) since \(X(q)=0\) for \(q \notin \mathcal{L}\)). Consider the set \[\mathcal{X} \,=\, \{ S_{\mspace{-4mu}\smash{q}}(X(q)) \;| \;q \in \mathcal{L} \} \;.\] It follows from Proposition 6 that \(X \in \operatorname{Filt}\text{-}\mathcal{X}\), so to show that \(X\) is in \(\operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}}\), it suffices to prove the inclusion \[\label{eq:inclusion} \operatorname{Filt}\text{-}\mathcal{X} \,\subseteq\, \operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}} \;.\tag{5}\] Recall from [1] that \({}_{Q,A}\mspace{-1mu}\operatorname{Mod}\) is a Grothendieck category, so the theory developed in Št’ovı́ček [26] applies to this category. Since \(\mathcal{S}_{\mathcal{L}}\) is a set, \(\operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}}\) is by definition a deconstructible class, see [26]. Hence, to prove the inclusion (5 ), it suffices by [26] to argue that \(\mathcal{X} \subseteq \operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}}\) holds. By assumption, each module \(X(q)\) is free, so to finish the proof we must show that the object \(S_{\mspace{-4mu}\smash{q}}(A^{(I)})\) belongs to \(\operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}}\) for every \(q \in \mathcal{L}\) and every (index) set \(I\). However, this is trivial as \[S_{\mspace{-4mu}\smash{q}}(A^{(I)}) \,=\, S_{\mspace{-4mu}\smash{q}}(A)^{(I)} \,\in\, \operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}} \;,\] where “\(\in\)” follows as \(\operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}}\) is closed under coproducts, again by [26].

The fact that \(\operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}}\) is contained in \({}^{\perp_1}{}_{}\mathscr{E}\) follows from [3]. ◻

Proof. If \(X\) is a direct summand of an object \(F \in \operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}}\) then, in particular, there is an inclusion \(\operatorname{supp}\mspace{1mu}X \subseteq \operatorname{supp}\mspace{1mu}F\) and \(X(q)\) is a direct summand of \(F(q)\) for every \(q \in Q\). Hence the implication (i)\(\;\Rightarrow\;\)(ii) follows immediately from Proposition 9.

Conversely, assume that \(X\) satisfies condition (ii). As in the proof of Proposition 9 we have \(X \in \operatorname{Filt}\text{-}\mathcal{X}\) where \(\mathcal{X} = \{ S_{\mspace{-4mu}\smash{q}}(X(q)) \;| \;q \in \mathcal{L} \}\). As also noted in that proof, the class \(\operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}}\) is deconstructible. Hence Št’ovı́ček [26] shows that the class, say, \(\mathcal{F}\), of objects that satisfy condition (i) is deconstructible too. To see that \(X\) satisfies condition (i) it suffices to show the inclusion \(\operatorname{Filt}\text{-}\mathcal{X} \subseteq \mathcal{F}\), and to that end it is by [26] is enough to argue that \(\mathcal{X} \subseteq \mathcal{F}\). Thus, we must prove that every object of the form \(S_{\mspace{-4mu}\smash{q}}(X(q))\) where \(q \in \mathcal{L}\) satisfies condition (i). By assumption, the \(A\)-module \(X(q)\) is projective, and hence it is a direct summans of a free \(A\)-module \(A^{(I)}\) where \(I\) is some set. Thus, \(S_{\mspace{-4mu}\smash{q}}(X(q))\) is a direct summand of \(S_{\mspace{-4mu}\smash{q}}(A^{(I)})\), and the latter object belongs to \(\operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}}\), as already demonstrated in the proof of Proposition 9. Therefore, \(S_{\mspace{-4mu}\smash{q}}(X(q))\) satisfies condition (i), as desired.

The “in particular” statement follows immediately from the last assertion in Proposition 9 and the fact that class \({}^{\perp_1}{}_{}\mathscr{E}\) is closed under direct summands in \({}_{Q,A}\mspace{-1mu}\operatorname{Mod}\). ◻

Proof. Clearly, the class \({}^{\perp_1}{}_{}\mathscr{E}\) is closed under direct summands (= retracts) in \({}_{Q,A}\mspace{-1mu}\operatorname{Mod}\). Thus, to prove the assertion, it suffices by [28] to prove that \({}^{\perp_1}{}_{}\mathscr{E}\) is deconstructible. Recall from [1] that \(({}^{\perp_1}{}_{}\mathscr{E},{}_{}\mathscr{E})\) is a cotorsion pair generated by some set \(\mathcal{S}\); that is, \(\mathcal{S}^{\perp_1} = {}_{}\mathscr{E}\). Note that the Grothendieck category \({}_{Q,A}\mspace{-1mu}\operatorname{Mod}\) has a projective generator, e.g. \(G = \bigoplus_{q \in Q}F_{\mspace{-5mu}\smash{q}}(A)\) by [1]. As the set \(\mathcal{S} \cup \{G\}\) also generates the cotorsion pair \(({}^{\perp_1}{}_{}\mathscr{E},{}_{}\mathscr{E})\), it follows from [26] that \({}^{\perp_1}{}_{}\mathscr{E}\) is deconstructible. ◻

Proof. We know from Lemma 11 that the exact category \({}^{\perp_1}{}_{}\mathscr{E}\) is of Grothendieck type and hence also efficient. Since \({}^{\perp_1}{}_{}\mathscr{E}\) is Frobenius, it has enough projectives; in fact, the class of projectives in the exact category \({}^{\perp_1}{}_{}\mathscr{E}\) is precisely class \({}_{Q,A}\mspace{-1mu}\operatorname{Prj}\) of projectives in the Grothendieck abelian category \({}_{Q,A}\mspace{-1mu}\operatorname{Mod}\), see [1]. Hence Saorı́n and Št’ovı́ček [16] shows that the cotorsion pair \((\mathcal{X}_\mathcal{L},\mathcal{Y}_\mathcal{L})\) in \({}^{\perp_1}{}_{}\mathscr{E}\) generated by the set \(\mathcal{S}_{\mathcal{L}}\) is complete and that an object \(X \in {}^{\perp_1}{}_{}\mathscr{E}\) belongs to \(\mathcal{X}_\mathcal{L}\) if and only if it is a direct summand of an object \(C \in {}^{\perp_1}{}_{}\mathscr{E}\) appearing in a short exact sequence \[\label{eq:def95of95XL} 0 \to P \to C \to F \to 0\tag{6}\] with \(P \in {}_{Q,A}\mspace{-1mu}\operatorname{Prj}\) and \(F \in \operatorname{Filt}\text{-}\mathcal{S}_{\mathcal{L}}\). Notice however, that the short exact sequence ([eq:def_of_XL]) is a conflation in the Frobenius exact category \({}^{\perp_1}\mathscr{E}\), thus it splits. We conclude that \(X\) lies in \(\operatorname{Add}({}_{Q,A}\mspace{-1mu}\operatorname{Prj},\operatorname{Filt}\text{-}\mathcal{S}_\mathcal{L})\). The converse follows by Corollary 10.

To prove part (b), note that the first two equalities below hold by definition while the third one follows from [3]: \[\begin{align} \mathcal{Y}_\mathcal{L} &\,=\, (\mathcal{S}_\mathcal{L})^{\perp_1} \\ &\,=\, \{ Y \in {}^{\perp_1}{}_{}\mathscr{E} \;| \;\operatorname{Ext}_{Q,A}^{1}(S_{\mspace{-4mu}\smash{q}}(A),Y)=0 \text{ for every } q \in \mathcal{L} \} \\ &\,=\, \{ Y \in {}^{\perp_1}{}_{}\mathscr{E} \;| \;\mathbb{H}^{1}_{\textrm{\tiny[}q\textrm{\tiny]}}(Y)=0 \text{ for every } q \in \mathcal{L} \} \;. \qedhere \end{align}\] ◻

Proof. Since objects in \({}_{Q,A}\mspace{-1mu}\operatorname{Prj}\) are zero in \({}^{\perp_1}\underline{{}_{}\mathscr{E}}\), part (a) follows from Theorem 12, Proposition 9, and Corollary 10. Part (b) follows from (a). ◻

Proof. By Saorín and Št’ovı́ček theorem [16] one needs to prove that \((\mathcal{X}_\mathcal{L},\mathcal{Y}_\mathcal{L})\) is a complete and cohereditary cotorsion pair. The completeness of the cotorsion pair is proven in Theorem 12.

To prove that the aforementioned cotorsion pair is also cohereditary, it suffices to show that the \(\mathcal{X_{\mathcal{L}}}\) is closed under co-syzygies, that is, \(\Omega^{-1}\mathcal{X_{\mathcal{L}}}\subseteq\mathcal{X_{\mathcal{L}}}\). Consider an object \(X\) in \(\mathcal{X_{\mathcal{L}}}\), then by [3] there is a short exact sequence in \({}_{Q,A}\mspace{-1mu}\operatorname{Mod}\) (and in particular in \({}^{\perp_1}{}_{}\mathscr{E}\)) \[0\longrightarrow X\longrightarrow \mathbb{G}(X)\longrightarrow\mathbb{C}(X)\longrightarrow0,\] which is object-wise split, \(\mathbb{G}(X)=\myprod_{q\in\operatorname{supp}\mspace{1mu}X}G_q(X(q))\) and \(\mathbb{C}(X)\) is the cokernel of the morphism \(X\rightarrow \mathbb{G}(X)\), that is \(\mathbb{C}(X)=\Omega^{-1}(X)\). Moreover, by [3], one has that \[\operatorname{supp}\mspace{1mu}\mathbb{G}(X)\subseteq\bigcup_{q\in\operatorname{supp}\mspace{1mu}X}\mathsf{N}_{-}(q)\] and by Definition 4 since every \(q\in\operatorname{supp}\mspace{1mu}X\) is in \(\mathcal{L}\), it follows that \(\operatorname{supp}\mspace{1mu}\mathbb{G}(X)\subseteq\mathcal{L}\). Assume now that there is a \(p\in Q\) such that \(\mathbb{C}(X)(p)\neq0\). By the exactness of the split short exact sequence \[0\longrightarrow X(p)\longrightarrow \mathbb{G}(X)(p)\longrightarrow\mathbb{C}(X)(p)\longrightarrow0,\] in \({}_{A}\mspace{-1mu}\operatorname{Mod}\), it follows that \(\mathbb{G}(X)(p)\neq0\) and hence \(\operatorname{supp}\mspace{1mu}\mathbb{C}(X)\subseteq\operatorname{supp}\mspace{1mu}\mathbb{G}(X)\subseteq\mathcal{L}\). This proves that the cotorsion pair in question is cohereditary and by [16], the pair \((\Omega\,\underline{\mathcal{X}}_\mathcal{L},\underline{\mathcal{Y}}_\mathcal{L})\) forms a t-structure in the stable category \({}^{\perp_1}\underline{{}_{}\mathscr{E}}\). ◻

Proof. By the conditions imposed to \(\mathcal{L}\) in Definition 4, one has that \(\partial\mathcal{L}\) is a finite (since \(Q\) is locally bounded) acyclic (Definition 4(2)) quiver and thus its category of representations is equivalent to the module category of the finite-dimensional \(\Bbbk\)-algebra \(\Lambda:=\Bbbk(\partial\mathcal{L})/I\), where \(I\) is the admissible ideal of relations obtained by restricting the relations of \(Q\) to \(\partial\mathcal{L}\). For the admissibility of \(I\) it suffices to notice that since \(\partial\mathcal{L}\) is finite and acyclic there exists a maximal length of a path in \(\partial\mathcal{{L}}\). Consequently, denoting by \(J\) the arrow ideal of the path algebra \(\Bbbk(\partial\mathcal{L})\), we conclude that \(J\) is nilpotent for some integer \(m\ge2\). In particular, one has \(J^m\subseteq I\). On the other hand, \(I\) is obtained by restricting the relations of \(Q\) to the subquiver \(\partial\mathcal{L}\). Since the relations of \(Q\) lie in the square of the radical ideal every relation has length at least two, meaning \(I\subseteq J^2\).

Finally, since \(\partial\mathcal{L}\) is a finite acyclic quiver with \(\ell\) vertices for some positive integer \(\ell\), by [29] one has that the projective dimension of any \(\Lambda\)-module \(M\) (and thus also for the simple modules) is finite. More concretely, one has for every \(\Lambda\)-module \(M\), \(\operatorname{pd}_\Lambda(M)\le\ell-1\). It is now clear that \(\Lambda\) has finite global dimension. ◻

Proof. It is straightforward to check that the functor \(\vartheta\) is fully faithful and exact and thus it induces an injective map \(\operatorname{Ext}_{\partial\mathcal{L}}^{i}(M,N)\,\longrightarrow\,\operatorname{Ext}_{Q}^{i}(\vartheta(M),\vartheta(N))\) for any \(i>0\) and any \(M,N\in{}_{\partial\mathcal{L}}\mspace{-1mu}\operatorname{Mod}\). In order to show that this map is also surjective, consider the \(i\)-Yoneda extension \[\xi\,\colon=\quad 0\longrightarrow\vartheta(M)\longrightarrow E_1\longrightarrow\cdots\longrightarrow E_i\longrightarrow\vartheta(N)\longrightarrow 0.\]

It suffices to show that up to equivalence in \(\operatorname{Ext}_{Q}^{i}(\vartheta(M),\vartheta(N))\) (see for example [31]), one can choose for each \(j=1,\ldots,i\), objects \(E_j\) in \({}_{Q}\mspace{-1mu}\operatorname{Mod}\) that are only supported in \(\partial\mathcal{L}\). For this purpose, define \(\xi^*\) to be the exact sequence in \({}_{Q}\mspace{-1mu}\operatorname{Mod}\) obtain by \(\xi\) be setting \(E_j(q)=0\) for all \(q\in \mathcal{R}\) and \(\xi^{**}\) to be the exact sequence obtained by \(\xi^*\) by setting \(E_j(q)=0\) for all \(q\in \mathcal{L}\smallsetminus\partial\mathcal{L}\). One can show that there are maps \(\xi\longrightarrow\xi^*\longleftarrow\xi^{**}\) of exact sequences, implying that \(\xi\) and \(\xi^{**}\) represent the same element in \(\operatorname{Ext}_{Q}^{i}(\vartheta(M),\vartheta(N))\).

Moreover, every object in \({}_{\partial\mathcal{L}}\mspace{-1mu}\operatorname{Mod}\) bi-fibrant with respect to the projective (and injective) model of the \(Q\)-shaped derived category (see [3]). Therefore, an application of [3] together with the natural isomorphism obtained by the first part of the Lemma, gives the desired result. ◻

Proof. Let \(T\) be a tilting \(\Lambda\)-module. In Lemma 17, one has that there are no positive self extensions of \(\psi(T)\) in \(\mathbf{D}_{Q}(\Bbbk)\), since \(T\) is a tilting \(\Lambda\)-module. It remains to show that \(\psi(T)\) is a compact generator of \(\mathbf{D}_{Q}(\Bbbk)\). Notice that, since \(T\) considered as a functor \(\partial\mathcal{L}\longrightarrow {}_{\Bbbk}\mspace{-1mu}\operatorname{Mod}\) takes finitely many non-zero values in the category of finitely presented \(\Bbbk\)-vector spaces \({}_\Bbbk\operatorname{prj}\), its image \(\psi(T)\) is a compact (in fact perfect in the sense of [3]) object in \(\mathbf{D}_{Q}(\Bbbk)\).

For the generation, we prove that the smallest localising subcategory that contains the image of \(T\) is the whole \(\mathbf{D}_{Q}(\Bbbk)\), that is \(\operatorname{Loc}(\psi(T))\,=\,\mathbf{D}_{Q}(\Bbbk)\). For this purpose we show that the set \(\{S_q\in\mathbf{D}_{Q}(\Bbbk)\,|\,q\in Q\}\) is contained in \(\operatorname{Loc}(\psi(T))\), by considering the cases of stalks supported in \(\partial\mathcal{L}\), \(\mathcal{L}\) and \(\mathcal{R}\), respectively.

Boundary stalks: By the definition of tilting modules, there exists a finite co-resolution \[\label{eq:tilting32coresol} 0\longrightarrow\Lambda\longrightarrow T_0\longrightarrow T_1\longrightarrow\cdots\longrightarrow T_n\longrightarrow 0\tag{7}\] of the regular module \(\Lambda\), where \(T_i\) lie in the additive closure \(\operatorname{add}(T)\) of \(T\), that is they are direct summands of finite direct sums of \(T\). Applying the functor \(\psi\) to the co-resolution ([eq:tilting coresol]), one can realise the image of the regular module \(\psi(\Lambda)\) as a (co)cone of a map between direct summands of \(\psi(T)\), thus \(\psi(\Lambda)\) lies in \(\operatorname{Loc}(\psi(T))\). In particular the image of any projective \(\Lambda\)-module \(P\) lies in \(\operatorname{Loc}(\psi(T))\).

We claim that the image \(\psi(M)\) of any finitely presented \(\Lambda\)-module \(M\) in \({}_\Lambda\operatorname{mod}\), lies in \(\operatorname{Loc}(\psi(T))\). For this purpose, let \(M\) be a finitely presented \(\Lambda\)-module and let \[\label{eq:projresol} 0\longrightarrow P_m\longrightarrow P_{m-1}\longrightarrow\cdots\longrightarrow P_1\longrightarrow P_0\longrightarrow M\longrightarrow 0\,,\tag{8}\] be a projective resolution of \(M\). Recall that the global dimension of \(\Lambda\) is finite and thus the projective resolution (8 ) is bounded. Moreover, the resolution (8 ) of \(M\) gives rise to a family of short exact sequences \[0\longrightarrow\Omega^j(M)\longrightarrow P_{j-1}\longrightarrow\Omega^j(M)\longrightarrow0\,,\quad j=1,\ldots,m\,,\] where \(\Omega^j(M)\), \(j=1,\ldots,m\), denotes the \(j\)-th syzygy of \(M\). Notice that, \(\Omega^m(M)\) is a projective \(\Lambda\)-module and thus it lies \(\operatorname{Loc}(\psi(T))\) along with the rest of the projective \(\Lambda\)-modules that appear in the resolution (8 ). Inductively, one can show that each syzygy of \(M\) and hence also \(M\) itself lie in \(\operatorname{Loc}(\psi(T))\). In particular, the image of every simple \(\Lambda\)-module lies in \(\operatorname{Loc}(\psi(T))\), and thus (up to weak equivalence) stalk functors \(S_q\) with \(q\in\partial\mathcal{L}\) lie in the localising subcategory as well.

Stalks in \(\mathcal{L}\): Define a length function \(h_0\colon \mathcal{L}\to\mathbb{Z}_{\ge0}\) by \[h_0(q)=\operatorname{max}\{n\ge0\,|\, q=q_0\to q_1\to q_2\to\ldots\to q_n,\text{ with }q_n\in\partial\mathcal{L}\}\] where \(q=q_0\to q_1\to q_2\to\ldots\to q_n\) is a chain of non-zero pseudo-radical morphisms in \(\mathcal{L}\) and it is finite by 4(2).

Notice that, \(h_0\) takes values in the finite set \(\{0,1,\ldots,N-1\}\), where \(N\) denotes the nilpotency power of the pseudo-radical of \(Q\). Moreover, for every \(q\) in the boundary \(\partial\mathcal{L}\), the length \(h_0(q)=0\). We show that every stalk \(S_q\) for \(q\in\mathcal{L}\) such that \(h_0(q)\le N-1\) lies in \(\operatorname{Loc}(\psi(T))\) by induction to \(h_0(q)\). The base step \(h_0(q)=0\) is already shown above. Define for an integer \(0< n \le N-1\) the set \[\mathcal{L}_0^{<n}=\{q\in\mathcal{L}\,|\, h_0(q)<n\}\,,\] of all vertices of length at most \(n\), and assume that \(S_q\) lies in \(\operatorname{Loc}(\psi(T))\) for all \(q\in\mathcal{L}_0^{<n}\). Consider now all vertices \(q_i\in\mathcal{L}\), \(i\in I\) for a finite set \(I\) (by local boundedness), such that \(h_0(q_i)=n\). By [3], for the object \(S=\bigoplus_{i\in I}S_{q_i}\) in \({}_{Q}\mspace{-1mu}\operatorname{Mod}\) there is a short exact sequence \[\label{eq:f-ses} \mathfrak{f}\colon\quad0\longrightarrow \mathbb{K}(S)\longrightarrow\mathbb{F}(S)\longrightarrow S\longrightarrow0\,,\tag{9}\] where \(\mathbb{K}(S)=\bigoplus_{i\in I}\mathbb{K}(S_{q_i})\) and \(\mathbb{F}(S)=\bigoplus_{i\in I}\mathbb{F}(S_{q_i})\). By the dual argument of the proof Theorem 14, one has that \(\operatorname{supp}\mspace{1mu}\mathbb{K}(S)\) is contained in \(\bigcup_{i\in I}\mathsf{N}_{+}(q_i)\). In particular, since \(h_0(q_i)=n\), one has that \(\operatorname{supp}\mspace{1mu}\mathbb{K}(S)\) is in fact contained in \(\mathcal{L}_0^{<n}\), since \(\operatorname{supp}\mspace{1mu}\mathbb{K}(S)\subseteq\mathsf{N}_{+}(q)\setminus\bigcup_{i\in I}\{q_i\}\). This follows from the description of the functors \(S_{q_i}\) and \(\mathbb{F}(S_{q_i})\) in [1] and more concretely, the short exact sequence (9 ) for each summand is of the form \[0\to \mathfrak{r}(q_i,-)\to Q(q_i,-)\to Q(q_i,-)/\mathfrak{r}(q_i,-)\to0,\quad i\in I\] Notice however that since \(Q\) is acyclic by the Blanket Assumption in Section 3, \(\mathfrak{r}(q_i,q_i)=0\). Thus, \(\mathbb{K}(S_{q_i})(q_i)=0\) and \(\operatorname{supp}\mspace{1mu}\mathbb{K}(S_{q_i})\subseteq\mathsf{N}_{+}(q_i)\setminus\{q_i\}\) for every \(i\in I\). Denote for brevity by \(K_i\) the kernel \(\mathbb{K}(S_{q_i})\). Notice that the support of \(K_i\) is finite and \(K_i\) takes values in finitely presented \(\Bbbk\)-vector spaces. Thus it follows by Proposition 6 that there exists a finite filtration \[0 = K_i^0 \rightarrowtail K_i^1 \rightarrowtail K_i^2 \rightarrowtail \cdots \rightarrowtail K_i^r = K_i\] for some non-negative integer \(r\ge0\), such that for every \(j\le r\) there is a short exact sequence \[0\longrightarrow K_i^j\longrightarrow K_i^{j+1}\longrightarrow S_{q_j}\longrightarrow0\,\] for some object \(q_{j}\in\operatorname{supp}\mspace{1mu}K\setminus\operatorname{supp}\mspace{1mu}K_i^j.\) In particular, starting from the short exact sequence \[0\longrightarrow K_i^0\longrightarrow K_i^{1}\longrightarrow S_{q_0}\longrightarrow0\,\] and proceeding inductively, one can show that the images of each \(K_i^j\) lie in \(\operatorname{Loc}(\psi(T))\) and so does \(K_i\) since it is a finite extension of \(K_i^j\,\)’s, and thus also \(K=\bigoplus_{i\in I}K_i\) lies in \(\operatorname{Loc}(\psi(T))\). Applying \(\psi\) to the sequence \(\mathfrak{f}\) yields a triangle in \(\mathbf{D}_{Q}(\Bbbk)\), such that \(\mathbb{K}(S)\) and \(\mathbb{F}(S)\) (which is isomorphic to zero in the triangulated category) are in \(\operatorname{Loc}(\psi(T))\), and thus so is \(S\). Since \(\operatorname{Loc}(\psi(T))\) is closed under summands, this proves that every stalk \(S_q\) for \(q\in\mathcal{L}\) such that \(h_0(q)\le N-1\) lies in \(\operatorname{Loc}(\psi(T))\). We denote by \(\mathcal{L}_0^{\le N-1}\) the set of all such vertices.

Define now another length function \(h_1\colon \mathcal{L}\to\mathbb{Z}_{\ge0}\) by \[h_1(q)=\operatorname{max}\{n\ge0\,|\, q=q_0\to q_1\to q_2\to\ldots\to q_n,\text{ with }q_n\in\mathcal{L}_0^{\le N-1}\}\] where \(q=q_0\to q_1\to q_2\to\ldots\to q_n\) is a finite chain of non-zero pseudo-radical morphisms in \(\mathcal{L}\).

Notice that, yet again \(h_1\) takes values in the finite set \(\{0,1,\ldots,N-1\}\), and moreover, for every \(q\) in \(\mathcal{L}_0^{\le N-1}\), the length \(h_1(q)=0\). Applying the same arguments as above one can show that every stalk \(S_q\) for \(q\in\mathcal{L}\) such that \(h_1(q)\le N-1\) lies in \(\operatorname{Loc}(\psi(T))\). We denote by \(\mathcal{L}_1^{\le N-1}\) the set of all such vertices. Repeating this procedure, in each step we define a length function \(h_\kappa\colon\mathcal{L}\longrightarrow\mathbb{Z}_{\ge0}\) by \[h_\kappa(q)=\operatorname{max}\{n\ge0\,|\, q=q_0\to q_1\to q_2\to\ldots\to q_n,\text{ with }q_n\in\mathcal{L}_{\kappa-1}^{\le N-1}\}\] where \(q=q_0\to q_1\to q_2\to\ldots\to q_n\) is a finite chain of non-zero pseudo-radical morphisms in \(\mathcal{L}\) and \[\mathcal{L}_{\kappa-1}^{\le N-1}:=\{q\in\mathcal{L}\,|\, h_{\kappa-1}(q)=0\},\] proving that every stalk of \(\mathcal{L}_\kappa^{\le N-1}\) belongs to \(\operatorname{Loc}(\psi(T))\). Finally, notice that since \(Q\) and thus also \(\mathcal{L}\), are acyclic, locally bounded with a nilpotent pseudo-radical and \(\mathcal{L}\) is closed under predecessors (see Definition 4(1)), it is straightforward to check that every stalk in \(\mathcal{L}\) lies in \(\operatorname{Loc}(\psi(T))\).

Stalks in \(\mathcal{R}\): We already shown that every finitely presented \(\Lambda\)-module lies in \(\operatorname{Loc}(\psi(T))\) and thus so do the images of the indecomposable injective \(\Lambda\)-modules. Therefore, one can use the dual arguments by working with the injectives and the dual version of Proposition 6 (that is, every representation supported in \(\mathcal{R}\) is co-filtered by stalks in \(\mathcal{R}\)), to get the desired result. We point out that, \(\operatorname{Loc}(\psi(T))\) is not closed under limits however, as in the case of \(\mathcal{L}\), the cokernel in the dual of the short exact sequence (9 ) is going to admit only a finite filtration.

We conclude that \(\operatorname{Loc}(\psi(T))\) contains a set of compact generators, thus it equals \(\mathbf{D}_{Q}(\Bbbk)\), proving that \(\psi(T)\) generates, and thus it is a silting object. ◻

Proof. To show that the t-structure \((\check{T}^{\perp_{>0}},\check{T}^{\perp_{\le0}})\) coincides with the t-structure \(\Sigma\,\mathbf{t}=(\underline{\mathcal{X}}_\mathcal{L},\Sigma\underline{\mathcal{Y}_\mathcal{L}})\), we prove that the aisles coincide. In particular, notice that since \(\check{T}\) is a compact silting object, by [33] one has that \(\check{T}^{\perp_{>0}}=\operatorname{Susp}(\check{T})\), the smallest suspended subcategory of \(\mathbf{D}_{Q}(\Bbbk)\) closed under coproducts and direct summands that contains \(\check{T}\). It is worth to remark here that the silting objects considered in loc. cit. are large silting objects, however, by [9] every silting object in the sense of Aihara-Iyama (which are the silting objects that we consider here) can also be considered large silting objects. We thus show that \(\underline{\mathcal{X}}_\mathcal{L}=\operatorname{Susp}(\check{T})\).

One inclusion is clear, since \(T\) lies in \(\mathcal{X}_\mathcal{L}\) and \(\underline{\mathcal{X}}_\mathcal{L}\) is an aisle (thus closed under positive suspensions and coproducts), one has \(\operatorname{Susp}(\check{T})\subseteq\underline{\mathcal{X}}_\mathcal{L}\). For the converse inclusion, let \(X\) lie in \(\underline{\mathcal{X}}_\mathcal{L}\). By Proposition 6 \(X\) is filtered by stalk objects in \(\mathcal{L}\), thus it suffices to show that every stalk object in \(\mathcal{L}\) belongs in \(\operatorname{Susp}(\check{T})\), since the latter as an aisle is closed under homotopy colimts. However, this is already shown in the proof of Theorem 18. More concretely, every stalk object in the boundary \(\partial\mathcal{L}\) automatically lies in \(\operatorname{Susp}(\check{T})\) since they can be regarded as simple \(\Lambda\)-modules and \(T\) is a projective generator of \(\Lambda\). Then, assuming that every stalk in \(\mathcal{L}^{<n}_0\) lies in \(\operatorname{Susp}(\check{T})\) one can show that the kernel \(K\) from (9 ) is a finite extension of objects in \(\operatorname{Susp}(\check{T})\) and hence it belongs there as well. Consequently, following similar arguments as in the proof of Theorem 18, one has inductively that every stalk in \(\mathcal{L}\) is given by \(S_q\cong\Sigma K\) and hence it lies in \(\operatorname{Susp}(\check{T})\). We conclude that \(\underline{\mathcal{X}}_\mathcal{L}\subseteq\operatorname{Susp}(\check{T})\).

Finally, since \(T\) is a projective generator of \(\Lambda\), that is, via Morita’s Theorem [34] one computes that \[\mathscr{H}_{\check{T}}\,\cong\, \mathop{\mathrm{Mod}}\text{-}\operatorname{End}_{\mathbf{D}_{Q}(\Bbbk)}(\check{T})^{\text{op}}\,\cong\,{}_{\Lambda}\mspace{-1mu}\operatorname{Mod}.\] ◻

Proof. We begin by observing that in this case the bound quiver algebra \(\Bbbk Q^m/I\), where \(I\) is the ideal of relations described in Figure 2, is a representation finite self-injective artin algebra. Consider also the subcategory \({}_{Q^m}\mathop{\mathrm{mod}}:=\operatorname{Fun}(Q^m,{}_\Bbbk\mathop{\mathrm{mod}})\) of \({}_{Q^m}\mspace{-1mu}\operatorname{Mod}\) consisting of all the \(\Bbbk\)-linear functors from \(Q^m\) that take values in the finitely presented \(\Bbbk\)-modules. Notice that this subcategory is precisely the subcategory of finitely presented objects of \({}_{Q^m}\mspace{-1mu}\operatorname{Mod}\). Moreover, \({}_{Q^m}\mathop{\mathrm{mod}}\) is a Krull-Shcmidt category whose only non projective-injective indecomposable modules are the simples. Finally, the \(Q^m\)-shaped derived category in this case, is equivalent to the stable module category \({}_{Q^m}\mathop{\mathrm{\underline{Mod}}}\).

Assume now that \((\mathcal{C},\mathcal{D})\) is a cotorsion pair in \({}_{Q^m}\mathop{\mathrm{mod}}\) such that \(\mathcal{C}\) is closed under co-syzygies. Then \((\mathcal{C},\mathcal{D})\) is either the projective cotorsion pair \(({}_{Q^m}\operatorname{prj},{}_{Q^m}\mathop{\mathrm{mod}})\) or the injective cotorsion pair \(({}_{Q^m}\mathop{\mathrm{mod}}, {}_{Q^m}\operatorname{prj})\). This follows from the fact that if \(\mathcal{C}\) contains at least one simple representation, then it contains all of them since they all appear in the same totally acyclic projective resolution \[\begin{tikzcd} \cdots & {P_{m-1}} && {P_0} && {P_1} && {P_2} & \cdots \\ && {S_{m-1}} && {S_0} && {S_1} \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-4] \arrow[two heads, from=1-2, to=2-3] \arrow[from=1-4, to=1-6] \arrow[two heads, from=1-4, to=2-5] \arrow[from=1-6, to=1-8] \arrow[two heads, from=1-6, to=2-7] \arrow[from=1-8, to=1-9] \arrow[tail, from=2-3, to=1-4] \arrow[tail, from=2-5, to=1-6] \arrow[tail, from=2-7, to=1-8] \end{tikzcd}\]

Therefore, from [16] \({}_{Q^m}\mathop{\mathrm{\underline{mod}}}\) has no non-trivial t-structures. By a classical result in representation theory of artin algebras by Ringel and Tachikawa [36], since \({}_{Q^m}\mathop{\mathrm{mod}}\) is representation finite, every representation in \({}_{Q^m}\mspace{-1mu}\operatorname{Mod}\) can be written as a direct sum of finitely presented indecomposable \(Q^m\)-representations. Thus, every complete cotorsion pair in \({}_{Q^m}\mspace{-1mu}\operatorname{Mod}\) is of finite type and in particular, it restrict to a complete cotorsion pair in \({}_{Q^m}\mathop{\mathrm{mod}}\). It follows that the cotorsion pairs in \({}_{Q^m}\mspace{-1mu}\operatorname{Mod}\) are controlled by the cotorsion pairs in the category of finitely presented modules and since in \({}_{Q^m}\mathop{\mathrm{mod}}\) the only cotorsion pairs that lead to t-structures via Saorı́n-Št’ovı́ček’s correspondence Theorem 3 are the trivial ones, we conclude that \(\mathbf{D}_{Q^m}(\Bbbk)\) admits only stable t-structures, whose hearts are trivial and thus \(\mathbf{D}_{Q^m}(\Bbbk)\) does not have any silting objects. ◻