May 22, 2026
We classify connected finite acyclic graded quivers \(Q\) for which the graded path algebra \(kQ\), regarded as a formal dg algebra, is silting-discrete. We prove that \(kQ\) is silting-discrete if and only if it is derived-discrete, and that both conditions are equivalent to the underlying graph of \(Q\) being of type ADE, or of type \(\widetilde{A}\) with unequal clockwise and counter-clockwise total degrees. The key ingredient is an explicit construction of an infinite pre-simple-minded collection in \(\mathop{\mathrm{pvd}}kQ\) in the non-discrete case.
Gabriel’s theorem asserts that the path algebra \(kQ\) of a connected finite acyclic quiver \(Q\) over a field \(k\) admits only finitely many isomorphism classes of indecomposable modules if and only if the underlying graph of \(Q\) is a Dynkin diagram of type ADE. This celebrated result is one of the foundational theorems of representation theory: it shows that a finiteness condition on the module category is controlled entirely by the combinatorial shape of the quiver. Gabriel’s proof is constructive: when \(Q\) is not Dynkin, an infinite family of pairwise non-isomorphic indecomposable representations is exhibited explicitly.
Several weakenings of representation-finiteness have since been studied at the level of derived categories. Vossieck [1] introduced derived-discrete algebras: an algebra \(\Lambda\) is derived-discrete if for every cohomology dimension vector, \(\mathcal{D}^b(\relax\Lambda)\) contains only finitely many isomorphism classes of objects with that vector. He classified all such algebras over algebraically closed fields: a connected derived-discrete algebra is either piecewise hereditary of Dynkin type, or a gentle one-cycle algebra satisfying a certain combinatorial condition (the absence of a clock).
Keller-Vossieck [2] introduced silting objects as a generalization of tilting objects, and Aihara-Iyama [3] developed the mutation theory of silting objects. An algebra \(\Lambda\) is silting-discrete if for every \(l\ge 0\), the set of silting objects lying between \(\Lambda\) and \(\Sigma^l\Lambda\) in the silting partial order is finite. For an ordinary quiver \(Q\), \[\begin{align} Q\text{ is Dynkin }\Leftrightarrow kQ\text{ is derived-discrete }\Leftrightarrow kQ\text{ is silting-discrete.} \end{align}\]
In this paper, we study silting-discreteness for graded path algebras. A graded quiver \(Q\) is a quiver in which each arrow carries an integer degree, and the path algebra \(kQ\) then inherits a grading by total degree of paths; viewed with zero differential, \(kQ\) is a formal dg algebra. This class strictly contains ordinary path algebras. Our main result gives a complete classification.
Theorem 1. Let \(Q\) be a connected finite acyclic graded quiver. The following conditions are equivalent:
The underlying graph of \(Q\) is of type ADE, or of type \(\widetilde{A}\) with unequal clockwise and counter-clockwise total degrees;
\(\#\mathop{\mathrm{ind}}\left(\mathop{\mathrm{pvd}}^{[-n,0]}kQ\right)<\infty\) for every \(n\ge 0\);
\(kQ\) is silting-discrete.
The principal difficulty lies in proving \((3)\Rightarrow(1)\). By a field-extension argument (9), we may assume that \(k\) is uncountable. We prove the contrapositive: if condition (1) fails, we construct a pre-simple-minded collection \(\{L_\lambda\}_{\lambda\in k^*} \subseteq\mathop{\mathrm{pvd}}kQ\) indexed by \(k^*\), and apply a result of Hara-Wemyss [4], which bounds the cardinality of any pre-simple-minded collection in \(\mathop{\mathrm{pvd}}kQ\) by \(\#Q_0\) whenever \(kQ\) is silting-discrete. The existence of such a collection in the non-discrete case is established in 4, where three explicit families of graded quivers (21) are treated directly; successive applications of vertex deletion, vertex contraction, and sink/source mutation (2.3) reduce the general case to these three families.
Organization. 2 collects the prerequisites. 2.1 recalls silting theory and introduces pre-simple-minded collections. 2.2 reviews graded algebras and their relation to formal dg algebras. 2.3 introduces the three reduction operations on graded quivers and verifies that each preserves silting-discreteness. 3 proves 1, deferring the explicit construction of pre-simple-minded collections to 4.
Conventions and notation. We fix a field \(k\). All modules are right modules. All subcategories are full. For a graded quiver \(Q\), we write \(Q_0\) for the set of vertices and \(Q_1\) for the set of arrows, and use \(s(\alpha)\), \(t(\alpha)\), \(\deg(\alpha)\) for the source, target, and degree of an arrow \(\alpha\).
Acknowledgements. The author would like to express his sincere gratitude to his supervisor Akira Ishii for his continuous support and encouragement. The author also thanks Osamu Iyama for suggesting the problem of classifying silting-discrete graded path algebras.
Let \(\mathcal{T}\) be a Hom-finite Krull-Schmidt triangulated category with shift functor \(\Sigma\).
Definition 1 ([2], [3]). An object \(M\in\mathcal{T}\) is a silting object if
\(\mathop{\mathrm{Hom}}_\mathcal{T}(M,\Sigma^{>0}M)=0\), and
\(\mathop{\mathrm{thick}}M = \mathcal{T}\).
Two silting objects are equivalent if they have the same additive closure; we write \(\mathop{\mathrm{silt}}\mathcal{T}\) for the set of equivalence classes.
The set \(\mathop{\mathrm{silt}}\mathcal{T}\) carries a natural partial order: \(M\ge N\) if \(\mathop{\mathrm{Hom}}_\mathcal{T}(M,\Sigma^{>0}N)=0\) [3]. Silting mutation [3] replaces one indecomposable summand of a basic silting object via an approximation triangle to produce a new basic silting object; two basic silting objects are related by a mutation if and only if they are adjacent in the Hasse quiver of \((\mathop{\mathrm{silt}}\mathcal{T},\ge)\).
Definition 2. Let \(T_0\in\mathop{\mathrm{silt}}\mathcal{T}\). A silting object \(T\in\mathop{\mathrm{silt}}\mathcal{T}\) is reachable from \(T_0\) if there is a finite sequence of silting mutations connecting \(T_0\) to \(T\). We say \(\mathcal{T}\) is silting-discrete if for every \(l\ge 0\) the set \[\bigl\{T\in\mathop{\mathrm{silt}}\mathcal{T}\;\big|\; T_0 \ge T \ge \Sigma^l T_0\bigr\}\] is finite. Silting-discrete does not depend on the choice of \(T_0\).
Definition 3. Let \(A\) be a dg algebra.
\(\mathop{\mathrm{per}}(A):=\mathop{\mathrm{thick}}_{\mathcal{D}(A)}A_A\) is called the perfect derived category of \(A\).
\(\mathop{\mathrm{pvd}}(A):=\{X\in\mathcal{D}(A)\mid \sum_{i\in\mathbb{Z}}\dim H^i(X)<\infty\}\) is called the perfectly-valued derived category of \(A\).
For the definition of dg algebras and their derived categories, we refer the reader to [5]. For a subset \(I\subseteq\mathbb{Z}\), we put \[\begin{align} \mathop{\mathrm{pvd}}^I(A):=\{X\in\mathop{\mathrm{pvd}}(A)\mid H^j(X)=0\text{ if }j\notin I\}. \end{align}\]
In this paper, we consider finite-dimensional graded path algebras \(kQ\), and in this case \(\mathop{\mathrm{per}}kQ=\mathop{\mathrm{pvd}}kQ\). For a dg algebra \(A\), we say \(A\) is silting-discrete if \(\mathop{\mathrm{per}}A\) is silting-discrete.
We use pre-simple-minded collections to prove that certain graded path algebras are not silting-discrete.
Definition 4. A family \(\mathcal{L}=\{L_i\}_{i\in I}\subseteq\mathcal{T}\) is a pre-simple-minded collection if
\(\mathop{\mathrm{Hom}}_\mathcal{T}(\mathcal{L},\Sigma^{<0}\mathcal{L})=0\), and
\(\mathcal{L}\) is a semibrick: for every \(i,j\in I\), \[\mathop{\mathrm{Hom}}_\mathcal{T}(L_i,L_j)= \begin{cases} \text{division ring} & \text{if }i=j,\\ 0 & \text{if }i\ne j. \end{cases}\]
A pre-simple-minded collection \(\mathcal{L}\) is called simple-minded collection if \(\mathop{\mathrm{thick}}\mathcal{L}=\mathcal{T}\).
The relevance to silting-discreteness comes from the following result.
Lemma 5. Let \(A\) be a connective dg algebra with finite-dimensional cohomologies. If \(A\) is silting-discrete, then any pre-simple-minded collection can be completed to a simple-minded collection. In particular, every pre-simple-minded collection is finite.
Proof. This claim is proved in [4] for finite-dimensional algebras, and their proof carries over to the general case. ◻
Corollary 6. Assume \(k\) is uncountable. If \(\mathop{\mathrm{pvd}}A\) contains a pre-simple-minded collection \(\{L_\lambda\}_{\lambda\in k^*}\) indexed by \(k^*\), then \(A\) is not silting-discrete.
Definition 7. Let \(A=\bigoplus_{i\in\mathbb{Z}}A^i\) be a finite-dimensional graded algebra. We denote by \(\relax^\mathbb{Z}A\) the category of graded \(A\)-modules. For \(M,N\in\relax^\mathbb{Z}A\), we write \[\mathop{\mathrm{Hom}}_A^\mathbb{Z}(M,N) \;:=\; \mathop{\mathrm{Hom}}_{\relax^\mathbb{Z}A}(M,N) \;=\; \mathop{\mathrm{Hom}}_A(M,N)^0.\] We define \(M(i)\in\relax^\mathbb{Z}A\) by \(M(i)^j:=M^{i+j}\).
In the rest of this paper, we mainly consider negatively graded algebras. If \(A\) is negatively graded, then for every \(M\in\relax^\mathbb{Z}A\) the subspace \(M^{\le i}:=\bigoplus_{n\le i}M^n\) is a graded submodule, and we write \(M^{\ge i}\) for the quotient \(M/M^{\le i-1}\). For a subset \(I\subseteq\mathbb{Z}\) we put \[\begin{align} \relax^I A \;:=\; \{M\in\relax^\mathbb{Z}A\mid M^j=0\text{ for every }j\notin I\}. \end{align}\] For a graded quiver \(Q\), the path algebra \(kQ\) has a natural grading by total degree of paths, and every graded algebra may be regarded as a formal dg algebra (i.e.a dg algebra with zero differential). The following theorem of Kalck-Yang identifies the perfectly-valued derived category of \(kQ\), viewed as a formal dg algebra, with an orbit category of the bounded derived category of graded modules.
Theorem 8 ([6], Theorem 1.3). Let \(Q\) be an acyclic graded quiver. The functor taking total complexes induces an equivalence \[\begin{align} \mathop{\mathrm{\mathsf{Tot}}}\colon \mathcal{D}^b(\relax^\mathbb{Z}kQ)\,/\,\Sigma(-1) \;\overset{\sim}{\longrightarrow}\; \mathop{\mathrm{pvd}}kQ. \end{align}\] In particular, it induces a bijection \(\mathop{\mathrm{ind}}(\relax^{[-n,0]}kQ)\simeq\mathop{\mathrm{ind}}(\mathop{\mathrm{pvd}}^{[-n,0]}kQ)\).
Lemma 9. Let \(k\subseteq l\) be an extension of fields. The functor \((-)_l\colon\mathop{\mathrm{per}}kQ\to\mathop{\mathrm{per}}lQ\) induces a bijection between the isomorphism classes of reachable silting complexes. In particular, \(kQ\) is silting-discrete if and only if so is \(lQ\).
Proof. By [7], for every reachable basic silting object \(M=\bigoplus_{i=1}^n M_i\in\mathop{\mathrm{per}}kQ\) we have \(\mathop{\mathrm{top}}\mathop{\mathrm{End}}_{\mathop{\mathrm{per}}kQ}(M)\simeq k^n\), so \((M_i)_l\) is indecomposable. Since left approximations are preserved under \((-)_l\), the functor induces a bijection between the isomorphism classes of reachable silting complexes. The last claim then follows from [8] and [9]. ◻
Definition 10 (vertex contraction, vertex deletion). Let \(i\) be a vertex of \(Q\). We define graded quivers \(Q(i)\) and \(Q(\hat{i})\) as follows:
\(Q(i)_0=Q(\hat{i})_0=Q_0\setminus\{i\}\);
\(Q(\hat{i})_1 =\{\alpha\in Q_1\mid s(\alpha)\ne i\ne t(\alpha)\}\) with natural gradings;
\(Q(i)_1=Q(\hat{i})_1\sqcup \{\alpha\beta\mid s(\alpha)=i=t(\beta)\}\) with natural gradings.
Example 11. Consider a quiver \(Q=\)
Then we can compute that \(Q(2)=\)
and \(Q(\hat{2})=\)
Lemma 12. Let \(i\in Q_0\). If \(kQ\) is silting-discrete, then so are \(kQ(\hat{i})\) and \(kQ(i)\).
Proof. We have equivalences \(\mathop{\mathrm{per}}kQ(i)\simeq\mathop{\mathrm{thick}}(1-e_i)kQ\) and \(\mathop{\mathrm{per}}kQ(\hat{i})\simeq\mathop{\mathrm{per}}kQ/\mathop{\mathrm{thick}}e_ikQ\). The claim follows from [10]. ◻
Definition 13. A vertex \(i\in Q_0\) is a sink if
there are no arrows with source \(i\),
all arrows with target \(i\) carry the same degree.
Source vertices are defined analogously.
Remark 14. The notion of sink in 13 is more restrictive than the usual one. For example, consider a graded quiver \(Q=\)
The vertex \(2\) is a sink in the usual sense but not in the sense of 13, since not all arrows into \(2\) carry the same degree.
Let \(i\in Q_0\) be a sink vertex, and suppose every arrow \(\alpha\) with \(t(\alpha)=i\) has the same degree \(r\). Then there is a quasi-isomorphism \[\begin{align} k(\mu_i Q)\;\simeq\; \mathop{\mathrm{\mathbf{R}End}}_{kQ}\!\left( \bigoplus_{j\ne i}e_j kQ \;\oplus\; \Sigma^{-1}\mathop{\mathrm{cone}}\!\left[ \Sigma^{-r}\!\!\bigoplus_{t(\alpha)=i}\!\!e_{s(\alpha)}kQ \longrightarrow e_i kQ \right] \right), \end{align}\] where \(\mu_i Q\) is the graded quiver with \((\mu_i Q)_0=Q_0\) and \[\begin{align} (\mu_i Q)_1 = \{\alpha\mid t(\alpha)\ne i\} \sqcup \{\alpha^*\mid t(\alpha)=i\}, \end{align}\] and \(s(\alpha^*)=i\), \(t(\alpha^*)=s(\alpha)\), \(\deg(\alpha^*)=-r\). We call \(\mu_i Q\) the sink mutation of \(Q\) at \(i\); source mutations are defined dually.
Example 15. Consider a quiver \(Q=\)
Then the mutation of \(Q\) at the vertex \(2\) is \(\mu_2Q=\)
Lemma 16. Let \(i\in Q_0\) be a sink vertex. Then \(kQ\) is silting-discrete if and only if so is \(k(\mu_i Q)\).
Proof. The formal dg algebras \(kQ\) and \(k(\mu_i Q)\) are derived equivalent. ◻
Remark 17. It is clear that \(kQ\) is silting-discrete if and only if so is \(kQ^\text{op}\).
Definition 18. Let \(n\ge 0\). Define a quiver \(\widetilde{Q}^{[-n,0]}\) by
\(\widetilde{Q}^{[-n,0]}_0 =\{(i,l)\mid i\in Q_0,\;l\in[-n,0]\}\);
\(\widetilde{Q}^{[-n,0]}_1 =\bigl\{(i,l)\xrightarrow{\alpha_l}(j,l+\deg\alpha) \;\big|\; i\xrightarrow{\alpha}j,\; \{l,l+\deg\alpha\}\subseteq[-n,0]\bigr\}\).
Example 19. If \(Q=\)
, then \(\widetilde{Q}^{[-2,0]}=\)
. Since \(\widetilde{Q}^{[-2,0]}\) is not Dynkin, \(\#\mathop{\mathrm{ind}}\left(\relax k\widetilde{Q}^{[-2,0]}\right)=\infty\).
Remark 20. Assume that \(Q\) is non-positively graded. The category \(\relax^{[-n,0]}kQ\) has a progenerator \(\bigoplus_{i=0}^n kQ(i)^{\ge-n}\) whose endomorphism ring is \(k\widetilde{Q}^{[-n,0]}\), giving an equivalence \[\relax^{[-n,0]}kQ \;\simeq\; \relax k\widetilde{Q}^{[-n,0]}.\]
The proof of the following proposition is given in 4.
Proposition 21. Suppose \(Q\) has one of the following forms:
(\(n>0\));
(\(n,m>0\));
the \(n\)-Kronecker quiver (\(n\ge 3\)).
Then there is a pre-simple-minded collection \(\{L_\lambda\}_{\lambda\in k^*}\subseteq\mathop{\mathrm{pvd}}kQ\). In particular, if \(\#k\) is infinite, then \(kQ\) is not silting-discrete.
Theorem 22. Let \(Q\) be a connected acyclic graded quiver. The following conditions are equivalent:
The underlying graph of \(Q\) is of type ADE, or of type \(\widetilde{A}\) with unequal clockwise and counter-clockwise total degrees;
\(\#\mathop{\mathrm{ind}}\left(\mathop{\mathrm{pvd}}^{[-n,0]}kQ\right)<\infty\) for every \(n\in\mathbb{N}\);
\(kQ\) is silting-discrete.
Proof. Since \(Q\) is connected and acyclic, after a suitable degree shift we may assume that \(Q\) is non-positively graded and that the degree-\(0\) part \(Q^0\) is connected. Under this assumption, condition (1) is equivalent to
\((1)'\Rightarrow(2)\): The hypothesis implies that \(\widetilde{Q}^{[-n,0]}\) is a disjoint union of Dynkin quivers for every \(n\). By 20 and 8, \(\#\mathop{\mathrm{ind}}(\mathop{\mathrm{pvd}}^{[-n,0]}kQ)=\#\mathop{\mathrm{ind}}(\relax^{[-n,0]}kQ)=\#\mathop{\mathrm{ind}}(\relax k\widetilde{Q}^{[-n,0]})\), which is finite.
\((2)\Rightarrow(3)\): Immediate from the definitions.
\((3)\Rightarrow(1)'\): By 9 we may assume \(k=\overline{k}\). Since \(kQ^0\) is \(\tau\)-tilting finite, \(Q^0\) is of type ADE. We assume
and derive a contradiction with silting-discreteness.
Because \(Q^0\) is a connected tree, any two vertices of \(Q\) are connected by a unique undirected walk in \(Q^0\); write \(d_{Q^0}(i,j)\) for the length of this walk. Choose \(\alpha\in Q^{<0}\) such that
\(n:=d_{Q^0}(s(\alpha),t(\alpha))\) is minimal among all \(\alpha'\in Q^{<0}\);
subject to (i), \(N:=\#\{\beta\in Q^{<0}\mid s(\beta)=s(\alpha),\;t(\beta)=t(\alpha)\}\) is maximal.
Let \(W\subseteq Q^0\) denote the unique walk from \(s(\alpha)\) to \(t(\alpha)\). By 12 and 16, the three operations \[Q\mapsto Q(\hat{j}), \qquad Q\mapsto Q(j), \qquad Q\mapsto\mu_j Q\] each preserve silting-discreteness when applicable. We apply them successively to produce a quiver matching one of the forms (a), (b), (c) in 21, contradicting silting-discreteness.
Case \(N\ge 2\): Applying \(Q\mapsto Q(\hat{j})\) for every \(j\notin W\), we may assume \(Q^0\) is of type \(A_{n+1}\) with endpoints \(s(\alpha)\) and \(t(\alpha)\). If \(n=1\), then \(Q\) is the \((N+1)\)-Kronecker quiver, matching (c). Assume \(n>1\). We iteratively reduce \(n\) as follows. While \(n\ge 2\):
If some vertex \(i\) has both an incoming and an outgoing arrow, apply \(Q\mapsto Q(i)\); this removes \(i\) and decreases \(n\) by \(1\).
Otherwise every interior vertex is a sink or a source. Pick any such \(i\), apply \(Q\mapsto\mu_i Q\); the mutation flips the arrows at \(i\), so its neighbours now have both incoming and outgoing arrows. Return to the first step.
The procedure terminates at \(n=1\), giving the \((N+1)\)-Kronecker quiver.
Case \(N=1\): Since the underlying graph of \(Q\) is not of type \(\widetilde{A}\), there exists a vertex \(i_0\notin W\) adjacent to \(W\). Applying \(Q\mapsto Q(\hat{j})\) to every \(j\notin W\sqcup\{i_0\}\), we may assume \(Q_0=W\sqcup\{i_0\}\). The possible shapes of \(Q\) are exactly the following:
where solid edges are arrows of \(Q\), dashed edges represent (unoriented) paths in \(Q\), curved edges represent negative arrows, and curved double edges represent possibly multiple negative arrows. The same argument as in the case \(n=2\) reduces each of i)–v) to one of (a), (b), (c). ◻
We illustrate the reduction procedure with concrete examples.
Example 23. Consider a quiver \(Q=\)
\(Q\mapsto Q(\hat{3})\);
\(Q\mapsto Q(\hat{6})\);
\(Q\mapsto Q(2)\);
\(Q\mapsto Q(5)\);
The resulting quiver is \((c)\) in 21.
Example 24. Consider a quiver \(Q=\)
\(Q\mapsto Q(\hat{0})\);
\(Q\mapsto Q(1)\);
\(Q\mapsto Q(2)\);
\(Q\mapsto\mu_4Q\);
The resulting quiver is opposite to \((a)\) in 21.
Example 25. Consider a quiver \(Q=\)
\(Q\mapsto Q(2)\);
\(Q\mapsto \mu_4Q\);
\(Q\mapsto Q(3)\);
\(Q\mapsto Q(5)\);
The resulting quiver is \((b)\) in 21.
In this section, we prove 21 by constructing the required pre-simple-minded collections explicitly. Throughout this section, we freely identify \(\mathop{\mathrm{obj}}(\mathop{\mathrm{pvd}}^{[-n,0]}kQ)\), \(\mathop{\mathrm{obj}}(\relax^{[-n,0]}kQ)\), and \(\mathop{\mathrm{obj}}(\relax k\widetilde{Q}^{[-n,0]})\) via the equivalences of 8 and 20.
Lemma 26. Let \(\mathcal{L}=\{L_i\}_{i\in I}\subseteq\relax kQ\). The following are equivalent:
\(\mathcal{L}\) is a pre-simple-minded collection in \(\mathop{\mathrm{pvd}}kQ\);
\(\mathcal{L}\) is a semibrick in \(\relax kQ\), and \(\mathop{\mathrm{Hom}}_{kQ}(L,L)^{<0}=0=\mathop{\mathrm{Ext}}^1_{kQ}(L,L)^{<0}\).
Proof. By 8 there are isomorphisms \[\mathop{\mathrm{Hom}}_{\mathop{\mathrm{pvd}}kQ}(L,\Sigma^i L) \;\simeq\; \bigoplus_{s+t=i}\mathop{\mathrm{Ext}}^s_{\relax kQ}(L,L(t)) \;\simeq\; \mathop{\mathrm{Hom}}_{kQ}(L,L)^i \oplus \mathop{\mathrm{Ext}}^1_{kQ}(L,L)^{i-1},\] where the second isomorphism uses the fact that \(kQ\) is hereditary. Hence
\(\mathop{\mathrm{Hom}}_{\mathop{\mathrm{pvd}}kQ}(L_i,L_j)=\delta_{ij}k\) iff \(\mathop{\mathrm{Hom}}^\mathbb{Z}_{kQ}(L_i,L_j)=\delta_{ij}k\) and \(\mathop{\mathrm{Ext}}^1_{kQ}(L_i,L_j)^{-1}=0\);
\(\mathop{\mathrm{Hom}}_{\mathop{\mathrm{pvd}}kQ}(L_i,\Sigma^{<0}L_j)=0\) iff \(\mathop{\mathrm{Hom}}_{kQ}(L_i,L_j)^{<0}=0=\mathop{\mathrm{Ext}}^1_{kQ}(L_i,L_j)^{<-1}\).
◻
Lemma 27. Let \(L_i,L_j\in\relax kQ\) and let \(0\to P_1\to P_0\to L_i\to 0\) be a projective resolution of \(L_i\). If \(\mathop{\mathrm{Hom}}_{kQ}(P_0,L_j)^{<0}\simeq \mathop{\mathrm{Hom}}_{kQ}(P_1,L_j)^{<0}\) as graded vector spaces, then \(\mathop{\mathrm{Hom}}_{kQ}(L_i,L_j)^{<0}=0\) implies \(\mathop{\mathrm{Ext}}^1_{kQ}(L_i,L_j)^{<0}=0\).
We prove 21(a). It suffices to treat the case \(n=1\).
Proposition 28. Let \(Q=\begin{tikzcd} 1\ar[r,yshift=0.7ex]\ar[r,yshift=-0.7ex,"-1"'] & 2\ar[r] & 3 \end{tikzcd}\), and set \(L_\lambda=\)
\(\in\relax^{[-3,0]}kQ^\text{op}\). Then \(\{L_\lambda\}_{\lambda\in k^*}\subseteq\mathop{\mathrm{pvd}}kQ^\text{op}\) is a pre-simple-minded collection.
Proof. One checks directly that \(\mathop{\mathrm{Hom}}^\mathbb{Z}_{kQ^\text{op}}(L_\lambda,L_\mu)=\delta_{\lambda\mu}k\). We show \(\mathop{\mathrm{Hom}}_{kQ^\text{op}}(L_\lambda,L_\lambda)^{<0}=0\). Let \(f\in\mathop{\mathrm{Hom}}^\mathbb{Z}_{kQ^\text{op}}(L_\lambda,L_\lambda(-1))\). We have \(f_{(2,-3)}=f_{(1,-2)}=f_{(3,-2)}=f_{(3,0)}=0\). Thus \(f_{(1,-1)}=f_{(2,-1)}\) has the form \(\begin{pmatrix}a&b\\0&0\end{pmatrix}\) and \(f_{(1,0)}=f_{(2,0)}\) has the form \(\begin{pmatrix}0\\c\end{pmatrix}\). From \(\begin{pmatrix}1&-1\end{pmatrix}\begin{pmatrix}a&b\\0&0\end{pmatrix} =f_{(3,-1)}\begin{pmatrix}1&0\end{pmatrix}\) we get \(b=0\) and \(f_{(3,-1)}=a\). From \(\begin{pmatrix}a&0\\0&0\end{pmatrix}\begin{pmatrix}1\\\lambda\end{pmatrix} =\begin{pmatrix}0\\c\end{pmatrix}\) we get \(a=0=c\). Hence \(f=0\). One similarly checks \(\mathop{\mathrm{Hom}}_{kQ^\text{op}}(L_\lambda,L_\lambda)_{<-1}=0\).
We have the projective resolution \(0\to P^\lambda_1\to P^\lambda_0\to L_\lambda\to 0\) with
\(P^\lambda_0=P_1\oplus P_1(1)^2\oplus P_1(2)^2\),
\(P^\lambda_1=P_2(1)\oplus P_2(2)^2\oplus P_2(3) \oplus P_3\oplus P_3(1)\oplus P_3(2)\oplus P_3(3)\).
A direct computation gives \[\mathop{\mathrm{Hom}}_{kQ^\text{op}}(P^\lambda_0,L_\mu)^{<0} \;=\; k(1)^6\oplus k(2)^2 \;=\; \mathop{\mathrm{Hom}}_{kQ^\text{op}}(P^\lambda_1,L_\mu)^{<0}.\] By 27, \(\mathop{\mathrm{Ext}}^1_{kQ^\text{op}}(L_\lambda,L_\mu)^{<0}=0\), and by 26 the collection is pre-simple-minded. ◻
We prove 21(b). It suffices to treat \(\gcd(n,m)=1\).
Proposition 29. Let \(Q=\begin{tikzcd} 1\ar[r,yshift=0.7ex]\ar[r,yshift=-0.7ex,"-m"'] & 2 & 3\ar[l,yshift=0.7ex]\ar[l,yshift=-0.7ex,"-n"] \end{tikzcd}\) with \(\gcd(m,n)=1\) and \(m\le n\). Set
\(L_\lambda=\)
\(\in\relax^{[-m-n+1,0]}kQ\).
Then \(\{L_\lambda\}_{\lambda\in k^*}\subseteq\mathop{\mathrm{pvd}}kQ^\text{op}\) is a pre-simple-minded collection.
Proof. One checks directly that \(\mathop{\mathrm{Hom}}^\mathbb{Z}_{kQ^\text{op}}(L_\lambda,L_\mu)=\delta_{\lambda\mu}k\). We show \(\mathop{\mathrm{Hom}}_{kQ^\text{op}}(L_\lambda,L_\lambda)^{<0}=0\). Let \(f\in\mathop{\mathrm{Hom}}^\mathbb{Z}_{kQ^\text{op}}(L_\lambda,L_\lambda(-h))\) for \(h>0\). One verifies:
\(f_{(3,-l)}=0\;\Rightarrow\; f_{(2,-l)}=0\) for \(0\le l<m\);
\(f_{(2,-l)}=0\;\Rightarrow\; f_{(1,-l)}=0\) for \(0\le l<m+n\);
\(f_{(1,-l)}=0\;\Rightarrow\; f_{(2,-l-m)}=0\) for \(0\le l<n\);
\(f_{(2,-l)}=0\;\Rightarrow\; f_{(3,-l+n)}=0\) for \(n\le l<m+n\).
Since \(f_{(3,-m+1)}=0\), these implications force \(f=0\). The projective resolution \(0\to P^\lambda_1\to P^\lambda_0\to L_\lambda\to 0\) has
\(P^\lambda_0=\bigoplus_{l=0}^{m+n-1}P_1(l)\oplus\bigoplus_{l=0}^{m-1}P_3(l)\),
\(P^\lambda_1=\bigoplus_{l=0}^{m+n-1}P_2(l)\oplus\bigoplus_{l=n}^{m+n-1}P_2(l)\).
A direct computation gives \[\begin{align} \mathop{\mathrm{Hom}}_{kQ^\text{op}}(P^\lambda_0,L_\mu)^{<0} =\bigoplus_{l=1}^{m+n-1}k(l)^{m+n-l}\oplus\bigoplus_{l=1}^{m-1}k(l)^{m-l} =\mathop{\mathrm{Hom}}_{kQ^\text{op}}(P^\lambda_1,L_\mu)^{<0}. \end{align}\] [lem:trivial,lem:i.e.] then give the result. ◻
Definition 30. For a sequence of integers \(a_0,a_1,\ldots,a_k\), we write \(\mathcal{K}_{a_0,a_1,\ldots,a_k}\) for the \((k+1)\)-Kronecker algebra whose arrows carry degrees \(a_0,a_1,\ldots,a_k\).
We prove 21(c) first for 3-arrow Kronecker quivers.
Proposition 31. Let \(0<m\le n\) with \(\gcd(m,n)=1\). Set
\(L_\lambda=\)
\(\in\relax^{[-m-n+1,0]}k\mathcal{K}_{0,-m,-n}\).
Then \(\{L_\lambda\}_{\lambda\in k^*}\subseteq\mathop{\mathrm{pvd}}k\mathcal{K}_{0,-m,-n}^\text{op}\) is a pre-simple-minded collection.
Proof. One checks directly that \(\mathop{\mathrm{Hom}}^\mathbb{Z}_{k\mathcal{K}_{0,-m,-n}^\text{op}}(L_\lambda,L_\mu)=\delta_{\lambda\mu}k\). We show \(\mathop{\mathrm{Hom}}_{k\mathcal{K}_{0,-m,-n}^\text{op}}(L_\lambda,L_\lambda)^{<0}=0\). Let \(f\in\mathop{\mathrm{Hom}}^\mathbb{Z}_{k\mathcal{K}_{0,-m,-n}^\text{op}}(L_\lambda,L_\lambda(-h))\) for \(h>0\). One verifies:
\(f_{(2,-l)}=0\;\Rightarrow\; f_{(1,-l+m)}=0\) for \(m\le m+n\);
\(f_{(1,-l)}=0\;\Rightarrow\; f_{(2,-l)}=0\) for \(0\le l<n\);
\(f_{(1,-l)}=0\;\Rightarrow\; f_{(2,-l-n)}=0\) for \(0\le l<m\).
Since \(f_{(2,-m-n+1)}=0\), we get \(f=0\).
We have a projective resolution \[0\;\to\;\bigoplus_{l=m}^{2n-1}P_2(l) \;\to\;\bigoplus_{l=0}^{n-1}P_1(l) \;\to\; L_\lambda\;\to\; 0.\] A direct computation gives \[\mathop{\mathrm{Hom}}_{k\mathcal{K}_{0,-m,-n}^\text{op}}\!\left(\bigoplus_{l=0}^{n-1}P_1(l),L_\mu\right)^{<0} =\bigoplus_{l=1}^{n-1}k(l)^{m-l} =\mathop{\mathrm{Hom}}_{k\mathcal{K}_{0,-m,-n}^\text{op}}\!\left(\bigoplus_{l=m}^{2n-1}P_2(l),L_\mu\right)^{<0}.\] [lem:trivial,lem:i.e.] give the result. ◻
Lemma 32. Let \(k\ge 2\) and \(0<a_1\le a_2\le\cdots\le a_k\). If \(a_1+a_2\ge a_3\), then there is a pre-simple-minded collection \(\{L_\lambda\}_{\lambda\in k^*}\subseteq\mathop{\mathrm{pvd}}k\mathcal{K}_{0,-a_1,-a_2,\ldots,-a_k}^\text{op}\).
Proof. Let \(L_\lambda\in\relax^{(-a_1-a_2,0]}k\mathcal{K}_{0,-a_1,-a_2}^\text{op} \subseteq\relax^{(-a_1-a_2,0]}k\mathcal{K}_{0,-a_1,-a_2,\ldots,-a_k}^\text{op}\) be the module from 31. By assumption, the projective resolution of \(L_\lambda\) takes the form \[0\;\to\;\bigoplus_{l=a_1}^{2a_2-1}P_2(l)\oplus P \;\to\;\bigoplus_{l=0}^{a_2-1}P_1(l) \;\to\; L_\lambda\;\to\; 0,\] where \(P\in\mathop{\mathrm{add}}P_2(\ge a_1+a_2)\). Since \[\mathop{\mathrm{Hom}}_{k\mathcal{K}_{0,-a_1,\ldots,-a_k}^\text{op}}(P(\ge a_1+a_2),L_\mu)^{<0} =\mathop{\mathrm{Hom}}_{kQ^\text{op}}(P_2,L_\mu)_{\le -a_1-a_2}=0,\] 27 26 give the result. ◻
Proposition 33. Let \(k\ge 2\) and \(0<a_1\le a_2\le\cdots\le a_k\). Then there is a pre-simple-minded collection \(\{L_\lambda\}_{\lambda\in k^*}\subseteq\mathop{\mathrm{pvd}}k\mathcal{K}_{0,-a_1,-a_2,\ldots,-a_k}^\text{op}\).
Proof. Case \(a_{k-2}-a_{k-3}\ge a_k-a_{k-1}\): By [5], there is an equivalence \[(-)^!\;:=\;\mathop{\mathrm{\mathbf{R}Hom}}(-,S_1\oplus S_2)\colon \mathop{\mathrm{pvd}}k\mathcal{K}_{0,-a_1,\ldots,-a_k}^\text{op} \;\simeq\; (\mathop{\mathrm{pvd}}k\mathcal{K}_{1,a_1+1,\ldots,a_k+1}^\text{op})^\text{op}\] sending \(S_i\mapsto P_i\). Since \((a_k-a_{k-1})+(a_k-a_{k-2})\le a_k-a_{k-3}\), 32 produces a pre-simple-minded collection \(\{L^!_\lambda\}_{\lambda\in k^*}\subseteq\mathop{\mathrm{pvd}}k\mathcal{K}_{1,a_1+1,\ldots,a_k+1}^\text{op}\) with \(L^!_\lambda\in\mathop{\mathrm{add}}\Sigma^{(-a_k+a_{k-2},0]}P_1\ast \mathop{\mathrm{add}}\Sigma^{(-a_k-a_{k-1}+a_{k-2},0]}P_2\). Applying \((-)^!\) yields a pre-simple-minded collection \(\{L_\lambda\}_{\lambda\in k^*}\subseteq\mathop{\mathrm{pvd}}k\mathcal{K}_{0,-a_1,\ldots,-a_k}^\text{op}\) with \(L_\lambda\in\mathop{\mathrm{add}}\Sigma^{[0,a_k+a_{k-1}-a_{k-2})}S_2\ast \mathop{\mathrm{add}}\Sigma^{[0,a_k-a_{k-2})}S_1\).
Case \(a_{k-2}-a_{k-3}<a_k-a_{k-1}\): Let \(l\ge 2\) be the largest index with \(a_{l-2}-a_{l-3}\ge a_l-a_{l-1}\) (setting \(a_{-1}=-\infty\)). Since \(a_{l+1}\ge a_l+a_{l-1}-a_{l-2}\), the argument above gives a pre-simple-minded collection \(\{L_\lambda\}_{\lambda\in k^*}\subseteq\mathop{\mathrm{pvd}}k\mathcal{K}_{0,-a_1,\ldots,-a_k}^\text{op}\) with \(L_\lambda\in\mathop{\mathrm{add}}\Sigma^{[0,a_l+a_{l-1}-a_{l-2}-1]}S_2\ast \mathop{\mathrm{add}}\Sigma^{[0,a_l-a_{l-2}-1]}S_1\). ◻
R. Fushimi, Department of mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
E-mail address: fushimi.riku.h9@s.mail.nagoya-u.ac.jp