WHEN IS THE SILTING-DISCRETENESS INHERITED?

Abstract

We explore when the silting-discreteness is inherited. As a result, one obtains that taking idempotent truncations and homological epimorphisms of algebras transmit the silting-discreteness. We also study classification of silting-discrete simply-connected tensor algebras and silting-indiscrete self-injective Nakayama algebras. This paper contains two appendices; one states that every derived-discrete algebra is silting-discrete, and the other is about triangulated categories whose silting objects are tilting.

1. Introduction

Silting-discreteness is a finiteness condition on a triangulated category, namely, a silting-discrete triangulated category admits only finitely many silting objects in any interval of silting objects [5]; actually, the set of silting objects has a poset structure [9]. In the case, we can fully grasp the whole picture of silting objects, and such a triangulated category has so nice structure. For example, a silting-discrete triangulated category satisfies a Bongartz-type condition; that is, any presilting object is partial silting [10]. For an ST-pair $(\mathcal {C}, \mathcal {D})$ , $\mathcal {C}$ is silting-discrete if and only if $\mathcal {D}$ is t-discrete if and only if every bounded t-structure in $\mathcal {D}$ is algebraic; moreover, in the case, the stability space of $\mathcal {D}$ is contractible [4] (see also [47]).

In general, it is very hard to check if a given triangulated category is silting-discrete. Thus, an extremely powerful tool was introduced [1], [5], [10], and it is applied to the perfect derived category of a finite-dimensional algebra over an algebraically closed field; we know several algebras with silting-discrete perfect derived categories, say silting-discrete algebras [1], [3], [5], [8]–[10], [16], [17], [21], [27]. Typical examples of silting-discrete algebras are local algebras, piecewise hereditary algebras of Dynkin type, and representation-finite symmetric algebras. It was also shown by [56] that for an ST-triple $(\mathcal {C}, \mathcal {D}, T)$ if $\mathcal {C}$ is T-discrete, then it is silting-discrete, which implies that any derived-discrete algebra is silting-discrete; we will also observe the result in Appendix A. We obtain from [4] that if $(\mathcal {C}, \mathcal {D})$ is a $(d+1)$ -Calabi–Yau pair for $d=1,2$ , then $\mathcal {C}$ is silting-discrete if and only if the endomorphism algebra of a silting object is $\tau $ -tilting finite, which leads to the fact that for the derived preprojective algebra $\Gamma :=\Gamma _{d+1}(Q)$ of a finite quiver Q with $H^0(\Gamma )$ finite-dimensional (note that it is a nonpositive dg algebra), the perfect derived category of $\Gamma $ is silting-discrete if and only if Q is Dynkin.

The aim of this paper is to investigate when the silting-discreteness is inherited. For example, derived equivalences certainly transmit the silting-discreteness. Tensoring a local algebra also conveys the property [7]. One of the main theorems of this paper (Theorem 2.10) states that a full triangulated subcategory of a silting-discrete triangulated category is also silting-discrete. Moreover, we show that a homological epimorphism of (nonpositive) dg algebras induces the silting-discreteness (Theorem 2.15). These lead us to the following results.

Theorem 1 (Corollaries 2.11 and 2.22).

Let $\Lambda $ be a finite-dimensional algebra over an algebraically closed field. Assume that $\Lambda $ is silting-discrete. Then we have the following.

1. (1) For an idempotent e of $\Lambda $ , the truncation $e\Lambda e$ is silting-discrete.

2. (2) If e is a stratifying idempotent of $\Lambda $ , then $\Lambda /\Lambda e\Lambda $ is silting-discrete.

We remark that quotients of algebras always inherit the $\tau $ -tilting finiteness [26], but not necessarily the silting-discreteness (§2.5).

The Bongartz-type condition is also one of the most important subjects in silting theory. We naturally ask whether or not the Bongartz-type condition always holds. ¹ For example, silting-discrete triangulated categories satisfy the Bongartz-type condition and every 2-term presilting object can be completed to a silting object [5], [10]. Note that the tilting version (i.e., is a pretilting object partial tilting?) often fails [40], [48] (see Example B.8). Theorem 1 and its proof provide a new Bongartz-type lemma under a certain condition (see Corollary 2.17 for a general setting, which recovers the 2-term version).

Theorem 2 (Corollary 2.18(2)).

Let e be an idempotent of $\Lambda $ . Then a presilting object of the perfect derived category of $\Lambda $ generating the thick closure of $e\Lambda $ is partial silting.

As a corollary of Theorem 1, we classify silting-discrete simply connected tensor algebras; note that the tensor algebra of three nonlocal algebras is never silting-discrete [7]. Also, we obtain a surprising result, which says that a representation-finite self-injective algebra is not necessarily silting-discrete.

Theorem 3 (Proposition 3.1, Theorems 3.4 and 3.6).

1. (1) A simply connected algebra is silting-discrete if and only if it is piecewise hereditary of Dynkin type.

2. (2) Let A and B be finite-dimensional nonlocal simply connected algebras over an algebraically closed field K. Then the following are equivalent:

   1. (i) $A\otimes _KB$ is silting-discrete;

   2. (ii) it is piecewise hereditary of type $D_4, E_6$ , or $E_8$ ;

   3. (iii) it is derived equivalent to a commutative ladder of degree $\leq 4$ .

3. (3) The self-injective Nakayama algebra with n simple modules of length r is not silting-discrete if (i) $r=3, 4$ and $n\geq 11$ ; (ii) $r=5,6$ and $n\geq r+8$ ; or (iii) $r\geq 7$ and $n\geq 2r+1$ .

2. Main results

Throughout this paper, let $\mathcal {T}$ be a triangulated category which is Krull–Schmidt, K-linear for an algebraically closed field K, and Hom-finite. For an object T of $\mathcal {T}$ , its thick closure is denoted by $\operatorname {\mathsf {thick}} T$ ; that is, it is the smallest thick subcategory of $\mathcal {T}$ containing T.

Silting objects play a central role in this paper.

Definition 2.1. Let T be an object of $\mathcal {T}$ .

1. (1) We say that T is presilting (pretilting) if it satisfies $\operatorname {Hom}_{\mathcal {T}}(T, T[i])=0$ for any $i>0\ (i\neq 0)$ .

2. (2) A presilting (pretilting) object T is said to be silting (tilting) provided $\mathcal {T}=\operatorname {\mathsf {thick}}\ T$ .

3. (3) A partial silting (partial tilting) object is defined to be a direct summand of some silting (tilting) object.

Notation 2.2. We use the following notations.

• • Denote by $\operatorname {\mathsf {silt}}\mathcal {T}$ the set of isomorphism classes of basic silting objects of $\mathcal {T}$ .

• • We always suppose that $\operatorname {\mathsf {silt}}\mathcal {T}$ is nonempty.

• • For a silting object A of $\mathcal {T}$ and $d>0$ , $\mathsf {d}_{A}\operatorname{-silt}\mathcal {T}:=\{T\in \operatorname {\mathsf {silt}}\mathcal {T}\ |\ A\geq T\geq A[d-1] \}$ .

• • Let S be a presilting object of $\mathcal {T}$ . The subset $\operatorname {\mathsf {silt}}_S\mathcal {T}$ of $\operatorname {\mathsf {silt}}\mathcal {T}$ consists of all silting objects having S as a direct summand.

As terminology, for presilting objects T and U of $\mathcal {T}$ , we use $T\geq U$ if $\operatorname {Hom}_{\mathcal {T}}(T, U[i])=0$ for every $i>0$ . This actually gives a partial order on $\operatorname {\mathsf {silt}}\mathcal {T}$ [9, Th. 2.11].

We define the silting-discreteness of $\mathcal {T}$ , which is the main theme of this paper.

Definition 2.3.

1. (1) We say that $\mathcal {T}$ is silting-discrete if it has a silting object A such that for any $d>0$ , the set $\mathsf{d}_{A}\operatorname{-silt}\mathcal {T}$ is finite (see [5, Prop. 3.8] for equivalent conditions of silting-discreteness). (As a convention, a triangulated category without silting object is also called silting-discrete.)

2. (2) For a silting object A of $\mathcal {T}$ , we call $\mathcal {T}$ $2_A$ -silting finite if $ \mathsf{2}_{A}\operatorname{-silt}\mathcal {T}$ is a finite set.

3. (3) A 2-silting finite triangulated category is defined to be $2_A$ -silting finite for every silting object A of $\mathcal {T}$ .

Remark 2.4.

1. (1) The notion of $2_A$ -silting finiteness is nothing but that of $\tau $ -tilting finiteness for $\operatorname {End}_{\mathcal {T}}(A)$ [25, Cor. 2.8]. For $\tau $ -tilting theory, we refer to [2].

2. (2) The silting-discreteness is preserved under triangle equivalences, but the $2_A$ -silting finiteness is not always so; it depends on the choice of silting objects A.

Let us recall several results in [5], [10].

Proposition 2.5. Let A be a silting object, and let T be a presilting object of $\mathcal {T}$ .

1. (1) [5, Prop. 2.16] If $A\geq T\geq A[1]$ , then T is a direct summand of a silting object lying in the interval between A and $A[1]$ .

2. (2) [10, Prop. 2.14] Assume that $\mathcal {T}$ is $2_A$ -silting finite. If $A\geq T\geq A[2]$ , then there exists a silting object B of $\mathcal {T}$ satisfying $A\geq B\geq T\geq B[1]\geq A[2]$ .

3. (3) [10, Prop. 2.14] Assume that $\mathcal {T}$ is silting-discrete. Then there is a silting object P of $\mathcal {T}$ with $P\geq T\geq P[1]$ .

4. (4) [10, Th. 2.4] $\mathcal {T}$ is silting-discrete if and only if it is 2-silting finite.

In the following, we investigate the silting-discreteness of triangulated categories:

• • Full triangulated subcategories inherit the silting-discreteness (2.1);

• • Recollements make an influence on silting-discreteness (2.2).

Moreover, we will also consider silting-discrete algebras as these corollaries.

Let $\Lambda $ be a finite-dimensional K-algebra which is basic and ring-indecomposable, and modules are always finite-dimensional and right unless otherwise noted. The perfect derived category of $\Lambda $ is denoted by $\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )$ .

We say that $\Lambda $ is silting-discrete if so is $\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )$ . Here is an example of silting-discrete algebras which we should first observe; one frequently uses it in the rest of this paper.

Example 2.6. A piecewise hereditary algebra of type $\mathcal {H}$ , that is, the algebra is derived equivalent to a hereditary abelian category $\mathcal {H}$ with a tilting object, is silting-discrete if and only if $\mathcal {H}$ is of Dynkin type.

Let Q be a (finite) quiver. For an arrow $\alpha : i\to j$ of Q, we consider the formal inverse $\alpha ^{-1}:j\to i$ and define the virtual degree by $\deg (\alpha ^\pm )=\pm 1$ (double-sign corresponds). The virtual degree of a walk (a sequence of successive arrows and formal inverses) is defined by $\log $ -rule: $\deg (\alpha \beta )=\deg (\alpha )+\deg (\beta )$ . We say that Q is gradable if any closed walk has virtual degree 0. Here, a closed walk is a walk whose start and goal coincide. For instance, every tree quiver (i.e., the underlying graph is tree) is gradable. Note that a gradable quiver admits no oriented cycle; but the converse does not necessarily hold.

We observe silting-discrete algebras with radical square zero.

Proposition 2.7. Let $\Lambda $ be the radical-square-zero algebra given by a gradable quiver Q. Then $\Lambda $ is silting-discrete if and only if Q is Dynkin.

Proof. It follows from [55, Th. 4.4] that $\Lambda $ is derived equivalent to $KQ$ .

Note that there is a silting-discrete radical-square-zero algebra presented by an ungradable non-Dynkin acyclic quiver (Example 2.19).

We say that $\Lambda $ is gentle if it is presented by a quiver Q with ideal I such that:

1. (1) for any vertex of Q, there exist at most two incoming and at most two outgoing arrows;

2. (2) for each arrow $x: i\to j$ of Q, the number of arrows y stopping at i (starting from j) with $yx\not \in I$ ( $xy\not \in I$ ) is at most 1;

3. (3) for each arrow $x: i\to j$ of Q, the number of arrows y stopping at i (starting from j) with $yx\in I$ ( $xy\in I$ ) is at most 1;

4. (4) all relations in I are paths of length 2.

A quiver Q is called one-cycle provided its underlying graph contains exactly one cycle. A gentle one-cycle algebra is defined to be a gentle algebra given by a one-cycle quiver. We say that a gentle one-cycle algebra satisfies the clock condition if the numbers of clockwise- and counterclockwise-oriented relations on the cycle coincide.

The following proposition is also useful.

Proposition 2.8. A gentle one-cycle algebra $\Lambda $ is silting-discrete if and only if it does not satisfy the clock condition.

Proof. Assume that $\Lambda $ satisfies the clock condition. By [15, Th. (A)], we see that $\Lambda $ is a piecewise hereditary algebra of type $\widetilde {A}$ , which is not silting-discrete.

Assume that $\Lambda $ does not satisfy the clock condition. It follows from [53, Theorem] that $\Lambda $ is derived-discrete, and so it is silting-discrete by [56, Exam. 3.9 and Cor. 4.2].

Let us observe the silting-discreteness of type $\widetilde {A}$ with commutative relation.

Proposition 2.9. Let $\Lambda $ be the algebra given by the quiver of type $\widetilde {A_{p,q}}$ with commutative relation; suppose that $1<p\leq q$ . Then $\Lambda $ is silting-discrete if and only if $p=2$ and any q or $p=3$ and $q=3, 4, 5$ .

Proof. Let Q be the quiver . Since $\Lambda $ is the one-point extension of the path algebra $KQ$ by the indecomposable injective module corresponding to the vertex $\bigstar $ , we see that it is derived equivalent to the one-point extension of $KQ$ by the indecomposable projective module corresponding to the vertex $\bigstar $ [18, Th. 1], which is the path algebra of the quiver $\widetilde {Q}$

Hence, it turns out that $\Lambda $ is silting-discrete if and only if so is $K\widetilde {Q}$ if and only if one of the desired conditions holds.

2.1. Full triangulated subcategories

The following is the first main theorem.

Theorem 2.10. Let $\mathcal {U}$ be a full triangulated subcategory of $\mathcal {T}$ . If $\mathcal {T}$ is silting-discrete, then so is $\mathcal {U}$ .

Proof. We show that $\mathcal {U}$ is 2-silting finite; hence, it is silting-discrete by Proposition 2.5(4). Let T be a silting object in $\mathcal {U}$ . As T is presilting in $\mathcal {T}$ , we obtain a silting object A satisfying $A\geq T\geq A[1]$ by Proposition 2.5(3). Let U be a silting object of $\mathcal {U}$ with $T\geq U\geq T[1]$ . Since there is a triangle $T_1\to T_0\to U\to T_1[1]$ with $T_0, T_1\in \operatorname {\mathsf {add}} T$ in $\mathcal {U}$ [9, Prop. 2.24], we have inequalities $A\geq U\geq A[2]$ . Proposition 2.5(1) and (2) lead to the fact that U is a direct summand of a silting object lying in the interval between A and $A[2]$ . As the interval in $\operatorname {\mathsf {silt}}\mathcal {T}$ is a finite set, we deduce that $\{U\in \operatorname {\mathsf {silt}}\mathcal {U}\ |\ T\geq U\geq T[1] \}$ is also finite.

A typical example of fully faithful functors is the triangle functor $-\otimes _{e\Lambda e}e\Lambda :\mathsf {K^b}(\operatorname {\mathsf {proj}} e\Lambda e)\to \mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )$ for an idempotent e of $\Lambda $ . This yields a corollary of Theorem 2.10, which says that taking idempotent truncations brings the silting-discreteness from the original.

Corollary 2.11. Let e be an idempotent of $\Lambda $ . If $\Lambda $ is silting-discrete, then so is $e\Lambda e$ .

2.2. Recollements

In this subsection, we investigate an impact of recollements on silting-discreteness.

Let us begin with the study of silting reductions [31], which enable us to realize the Verdier quotient of $\mathcal {T}$ by the thick closure of a presilting object as a certain subfactor category of $\mathcal {T}$ and which make isomorphisms of silting posets.

Let S be a presilting object of $\mathcal {T}$ and put $\mathcal {S}:=\operatorname {\mathsf {thick}} S$ . The Verdier quotient of $\mathcal {T}$ by $\mathcal {S}$ is denoted by $\mathcal {T}/\mathcal {S}$ . Take a full subcategory $\mathcal {Z}$ of $\mathcal {T}$ as follows:

$$\begin{align*}\mathcal{Z}:=\{X\in\mathcal{T}\ |\ \operatorname{Hom}_{\mathcal{T}}(X,S[i])=0=\operatorname{Hom}_{\mathcal{T}}(S, X[i])\ \mbox{for any } i>0 \}.\end{align*}$$

We denote by $\frac {\mathcal {Z}}{[S]}$ the additive quotient of $\mathcal {Z}$ modulo the ideal $[S]$ , which admits a triangle structure [31, Th. 3.6]. Then, the silting reduction [31, Th. 3.6] shows that there is a triangle equivalence between $\mathcal {T}/\mathcal {S}$ and $\frac {\mathcal {Z}}{[S]}$ . Since $\mathcal {T}$ is Hom-finite and Krull–Schmidt, so is $\mathcal {T}/\mathcal {S}$ .

Now, we prepare a lemma for the main theorem of this subsection.

Lemma 2.12. Keep the notations above. If $\mathcal {T}$ is silting-discrete, then so is $\mathcal {T}/\mathcal {S}$ .

Proof. By the silting reduction [31, Th. 3.7 and Cor. 3.8], the natural functor $\mathcal {T}\to \mathcal {T}/\mathcal {S}$ induces an isomorphism $\operatorname {\mathsf {silt}}_S\mathcal {T}\simeq \operatorname {\mathsf {silt}}\mathcal {T}/\mathcal {S}$ of posets. Thus, it is not hard to deduce that $\mathcal {T}/\mathcal {S}$ is silting-discrete.

Remark 2.13. We are assuming that $\operatorname {\mathsf {silt}}\mathcal {T}\neq \emptyset $ to apply the silting reduction, while $\operatorname {\mathsf {silt}}\mathcal {T}/\mathcal {S}=\emptyset $ may occur (see §B.2).

To realize the goal of this subsection, we need (nonpositive) dg algebras.

Let $\mathcal {A}$ be a dg K-algebra and $\operatorname {\mathsf {per}}(\mathcal {A})$ denote the perfect derived category of $\mathcal {A}$ . If $\mathcal {A}$ is nonpositive with finite-dimensional cohomologies, then $\operatorname {\mathsf {per}}(\mathcal {A})$ is Hom-finite and Krull–Schmidt (see, e.g., [4, Prop. 6.12]).

We give a remarkable example [55, Exam. 5.3], which mentions that the zeroth cohomology $H^0(\mathcal {A})$ of $\mathcal {A}$ does not necessarily inherit the silting-discreteness from $\mathcal {A}$ ; moreover, the endomorphism algebra of a silting object does not always inherit the property either.

Example 2.14. Let $\Lambda $ be the algebra presented by the quiver Q

with relation $\alpha \beta \gamma =0$ . Then, we easily observe that $\Lambda $ is piecewise hereditary of type $E_6$ ; so, it is silting-discrete. Take $T:=S_1\oplus S_2[1]\oplus S_3[2]\oplus S_4[3]\oplus S_5[2]\oplus S_6[1]$ , where $S_i$ is the simple module corresponding to the vertex i of Q; it is in $\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )$ because $\Lambda $ has finite global dimension. We see that T is a silting object whose dg endomorphism algebra $\mathcal {A}$ is nonpositive with finite-dimensional cohomologies; so, $\operatorname {\mathsf {per}}(\mathcal {A})$ is Krull–Schmidt. By Keller–Rickard’s theorem [36], [48], we have a triangle equivalence $\operatorname {\mathsf {per}}(\mathcal {A})\simeq \mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )$ , whence $\operatorname {\mathsf {per}}(\mathcal {A})$ is silting-discrete. However, the zeroth cohomology $H^0(\mathcal {A})$ of $\mathcal {A}$ , which is isomorphic to $\operatorname {End}_{\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )}(T)$ , is the radical-square-zero algebra of $Q^{\mathrm {op}}$ , and it is derived equivalent to a hereditary algebra of type $\widetilde {D_5}$ . Thus, $H^0(\mathcal {A})$ is not silting-discrete.

This example also says that although a derived preprojective algebra $\mathcal {A}$ of Dynkin type (it is a nonpositive dg algebra) admits a silting-discrete perfect derived category [4, Cor. 8.6], it is still open whether a preprojective algebra of Dynkin type except $A_2, D_{2n}, E_7$ , and $E_8$ , which appears as the zeroth cohomology of $\mathcal {A}$ , is silting-discrete or not.

We aim at our goal of this subsection.

Recall that a recollement $\mathcal {R}$ consists of the following data of triangulated categories $\mathcal {D}, \mathcal {X}$ , and $\mathcal {Y}$ , and triangle functors between them:

such that:

1. (1) $(i^{*}, i_*),\ (i_!, i^!),\ (j_!, j^!)$ , and $(j^{*}, j_*)$ are adjoint pairs;

2. (2) $i_*\ (=i_!),\ j_!$ , and $j_*$ are fully faithful;

3. (3) any object D of $\mathcal {D}$ admits two triangles

   $$\begin{align*}\begin{array}{c@{\ \mbox{and }}c} i_!i^!(D)\to D\to j_*j^{*}(D)\to i_!i^!(D)[1] & j_!j^!(D)\to D\to i_*i^{*}(D)\to j_!j^!(D)[1], \end{array}\end{align*}$$
   where the morphisms around D are given by adjunctions.

Omitting the triangle functors, we write such a recollement as $\mathcal {R}=(\mathcal {Y}, \mathcal {D}, \mathcal {X})$ .

We denote by $\operatorname {\mathsf {D}}(-)$ the (unbounded) derived category, and for an object X of $\operatorname {\mathsf {D}}(\Lambda )$ , $\operatorname {\mathsf {Tria}}(X)$ stands for the smallest full triangulated subcategory of $\operatorname {\mathsf {D}}(\Lambda )$ containing X which is closed under all direct sums.

From now, we consider either of the following situations.

• • Start from a given recollement $(\mathcal {Y}, \operatorname {\mathsf {D}}(\Lambda ),\mathcal {X})$ with $\mathcal {X}$ compactly generated: Then, by [12, Th. 4.5], $\mathcal {X}$ can be written as $\operatorname {\mathsf {Tria}}(S)$ for some presilting object S of $\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )$ .

• • Start from a given presilting object S of $\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )$ : Since $\operatorname {\mathsf {Tria}}(S)$ is a smashing subcategory of $\operatorname {\mathsf {D}}(\Lambda )$ [12, Prop. 3.5], there exists a full triangulated subcategory $\mathcal {Y}$ of $\operatorname {\mathsf {D}}(\Lambda )$ such that $(\mathcal {Y}, \operatorname {\mathsf {D}}(\Lambda ), \operatorname {\mathsf {Tria}}(S))$ forms a recollement; actually, $\mathcal {Y}$ is right perpendicular to $\operatorname {\mathsf {Tria}}(S)$ [44, Cor. 2.4].

In any case, we have a recollement $(\mathcal {Y},\operatorname {\mathsf {D}}(\Lambda ),\operatorname {\mathsf {Tria}}(S))$ for a presilting object S of $\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )$ . Furthermore, thanks to Nicolas–Saorin’s result [44, §4], there is a dg K-algebra $\mathcal {A}$ and a morphism $\varphi :\Lambda \to \mathcal {A}$ of dg algebras such that $i_!=-\otimes ^{\mathsf {L}}_{\mathcal {A}}\mathcal {A}$ induces a recollement $(\operatorname {\mathsf {D}}(\mathcal {A}), \operatorname {\mathsf {D}}(\Lambda ),\operatorname {\mathsf {Tria}}(S))$ . Here, $\varphi $ is embedded into the triangle $X:=\operatorname {\mathsf {RHom}}_\Lambda (S, \Lambda )\otimes ^{\mathsf {L}}_{\mathcal {B}} S\to \Lambda \xrightarrow {\varphi }\mathcal {A}\to X[1]$ in $\operatorname {\mathsf {D}}(\Lambda ^{\mathrm {op}}\otimes _K\Lambda )$ , where $\mathcal {B}$ is the dg endomorphism algebra of S. We evidently observe that $H^i(\mathcal {A})$ is finite-dimensional for any i. One can also describe $\mathcal {A}$ as the dg endomorphism algebra of $\Lambda $ in a dg enhancement of $\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )/\operatorname {\mathsf {thick}}(S)$ . Such a morphism $\varphi $ is called a homological epimorphism (see [46]).

Now, we state the second main theorem of this paper.

Theorem 2.15. Keep the notations above. If $\Lambda $ is silting-discrete, then so is $\operatorname {\mathsf {per}}(\mathcal {A})$ .

A key point is an analog of the proof of [33, Cor. 2.12(a)].

Proof. The recollement $(\operatorname {\mathsf {D}}(\mathcal {A}), \operatorname {\mathsf {D}}(\Lambda ), \operatorname {\mathsf {Tria}}(S))$ as in before Theorem 2.15 gives rise to an exact sequence $0\to \operatorname {\mathsf {Tria}}(S)\to \operatorname {\mathsf {D}}(\Lambda )\xrightarrow {i^{*}}\operatorname {\mathsf {D}}(\mathcal {A})\to 0$ of triangulated categories. As S is presilting in $\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )$ , we derive from the Thomason–Trobaugh–Yao localization theorem [43, Th. 2.1] that $i^{*}$ induces a triangle equivalence $\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )/\operatorname {\mathsf {thick}}(S)\simeq \operatorname {\mathsf {per}}(\mathcal {A})$ , whence $\operatorname {\mathsf {per}}(\mathcal {A})$ is silting-discrete by Lemma 2.12.

Remark 2.16. Theorem 2.15 holds enough for a nonpositive dg K-algebra $\Lambda $ .

The proof of Theorem 2.15 spins off the following result, which mentions a Bongartz-type lemma for a given presilting object S; if $\Lambda \geq S\geq \Lambda [1]$ , then $\mathcal {A}$ is nonpositive because $H^i(\mathcal {A})\simeq H^{i+1}(X)$ , so the result gives a generalization of Proposition 2.5(1).

Corollary 2.17. Keep the notations as in before Theorem 2.15. Assume that $\mathcal {A}$ is nonpositive. Then S is partial silting.

Proof. In the proof of Theorem 2.15, we got a triangle equivalence $\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )/\operatorname {\mathsf {thick}} S\simeq \operatorname {\mathsf {per}}(\mathcal {A})$ . The silting reduction (Lemma 2.12) yields isomorphisms $\operatorname {\mathsf {silt}}_S(\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda ))\simeq \operatorname {\mathsf {silt}}(\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )/\operatorname {\mathsf {thick}} S)\simeq \operatorname {\mathsf {silt}}(\operatorname {\mathsf {per}}(\mathcal {A}))$ of posets. Since $\mathcal {A}$ is nonpositive, it is silting in $\operatorname {\mathsf {per}}(\mathcal {A})$ , whence $\operatorname {\mathsf {silt}}(\operatorname {\mathsf {per}}(\mathcal {A}))$ is nonempty; so is $\operatorname {\mathsf {silt}}_S(\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda ))$ . Thus, it turns out that S is partial silting.

Let e be an idempotent of $\Lambda $ . Following [33, Prop. 2.10] and its proof, we can construct a nonpositive dg K-algebra $\mathcal {A}_e$ and a homological epimorphism $\varphi :\Lambda \to \mathcal {A}_e$ of dg algebras; embed the canonical morphism $\Lambda e\otimes ^{\mathsf {L}}_{e\Lambda e}e\Lambda \to \Lambda $ into the triangle

$$\begin{align*}\Lambda e\otimes^{\mathsf{L}}_{e\Lambda e}e\Lambda\to\Lambda\xrightarrow{\varphi'}\mathcal{A}'\to\Lambda e\otimes_{e\Lambda e}^{\mathsf{L}}e\Lambda[1],\end{align*}$$

and take the standard truncations $\mathcal {A}_e:=\sigma ^{\leq 0}(\mathcal {A}')$ and $\varphi :=\sigma ^{\leq 0}(\varphi ')$ , where for a complex $(X^i, d^i)$ , the standard truncation $\sigma ^{\leq 0}(X)$ is the complex $\cdots \to X^{-1}\to \operatorname {Ker} d^0\to 0$ . Then, $\mathcal {A}_e$ and $\varphi $ are the desired dg algebra and homological epimorphism, since we have isomorphisms

(2.17.1) $$ \begin{align} H^i(\mathcal{A}_e)\simeq\begin{cases} \ 0, & (i>0), \\ \ \Lambda/\Lambda e\Lambda, & (i=0), \\ \ \operatorname{Ker}(\Lambda e\otimes_{e\Lambda e}e\Lambda\to\Lambda e\Lambda), & (i=-1), \\ \ \operatorname{Tor}^{e\Lambda e}_{-i-1}(\Lambda e, e\Lambda), & (i<-1). \end{cases} \end{align} $$

Here is a corollary of Theorem 2.15 and Corollary 2.17.

Corollary 2.18. Let e be an idempotent of $\Lambda $ . Then we have the following.

1. (1) If $\Lambda $ is silting-discrete, then so is $\operatorname {\mathsf {per}}(\mathcal {A}_e)$ .

2. (2) A presilting object of $\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )$ generating $\operatorname {\mathsf {thick}}(e\Lambda )$ is partial silting.

Proof. As above, we obtain a recollement $(\operatorname {\mathsf {D}}(\mathcal {A}_e), \operatorname {\mathsf {D}}(\Lambda ), \operatorname {\mathsf {Tria}}(e\Lambda ))$ and know that $\mathcal {A}_e$ is nonpositive. So, the first assertion directly follows from Theorem 2.15. If S is a presilting object of $\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )$ generating $\operatorname {\mathsf {thick}}(e\Lambda )$ , then we have $\operatorname {\mathsf {Tria}}(S)=\operatorname {\mathsf {Tria}}(e\Lambda )$ , whence the last assertion is derived from Corollary 2.17.

We remark that Corollary 2.18(2) can be shown more directly by the silting reduction. That is, let $\mathcal {S}:=\operatorname {\mathsf {thick}} S$ , then $\mathcal {S}=\operatorname {\mathsf {thick}} e\Lambda $ by assumption, and so the silting reduction yields isomorphisms $\operatorname {\mathsf {silt}}_S\Lambda \simeq \operatorname {\mathsf {silt}}\mathsf {K^b}(\operatorname {\mathsf {proj}}\Lambda )/\mathcal {S}\simeq \operatorname {\mathsf {silt}}_{e\Lambda }\Lambda $ of posets. As the last set is nonempty (because $\Lambda $ is the member), we have $\operatorname {\mathsf {silt}}_S\Lambda \neq \emptyset $ , whence S is partial silting.

In the rest of this subsection, we give an explicit description of $\mathcal {A}_e$ and its example thanks to [35].

We first construct a nonpositive dg algebra $\widetilde {\Lambda }$ quasi-isomorphic to $\Lambda $ (i.e., $H^i(\widetilde {\Lambda })\simeq H^i(\Lambda )$ ; see [45, Construction 2.6]). Let $\Lambda $ be presented by a (finite) quiver Q with admissible ideal I; write $Q^{(0)}:=Q$ . We make a graded quiver $Q^{(1)}$ by adding to $Q^{(0)}$ arrows in degree $-1$ which correspond to each element of I and consider a differential which sends new arrows to their corresponding relations; then $H^0(KQ^{(1)})=\Lambda $ . Picking out a generating set of $H^{-1}(KQ^{(1)})$ , one produces a graded quiver $Q^{(2)}$ by adding to $Q^{(1)}$ arrows in degree $-2$ which correspond to each element of the generating set and consider a differential as well. Iterating this, we get a graded quiver $Q^{(\infty )}:=\bigcup _{i=0}^\infty Q^{(i)}$ such that the dg quiver algebra $\widetilde {\Lambda }:=(KQ^{(\infty )}, d)$ is quasi-isomorphic to $\Lambda $ .

Now, we apply Theorem 7.1 of [35] to obtain that $\mathcal {A}_e$ is quasi-isomorphic to $\widetilde {\Lambda }/\widetilde {\Lambda }\widetilde {e}\widetilde {\Lambda }$ , where $\widetilde {e}$ is the idempotent of $\widetilde {\Lambda }$ corresponding to e.

Example 2.19. Let $\Lambda $ be the algebra presented by the quiver

with $\beta \gamma =0$ ; it is silting-discrete by Proposition 2.8. Then, $\widetilde {\Lambda }$ is the dg quiver algebra of

with $\deg (\delta )=-1$ , the other arrows are of degree 0 and with the trivial differential.

Let e be the primitive idempotent of $\Lambda $ corresponding to the vertex 3. We see that $\mathcal {A}_e$ is the dg quiver algebra of the graded Kronecker quiver with arrows of degrees 0 and $-1$ , and the trivial differential. This is silting-discrete by Corollary 2.18(1).

Note that the dg quiver algebra of the graded Kronecker quiver with arrows of the same degree $n\leq 0$ and the trivial differential is not silting-discrete; indeed, it is derived equivalent to the (ordinary) Kronecker algebra . Here, $P_1$ and $P_2$ are the indecomposable projective modules corresponding to the vertices 1 and 2, respectively.

This observation gives an example of silting-indiscrete algebras.

Example 2.20. Let Q be the quiver of type $\widetilde {A_{p,q}}$ ( $p,q\geq 2$ ); assign to the arrows of two paths (resp. length p and q) from the source to the sink $\alpha $ and $\beta $ , respectively. Then $\Lambda :=KQ/(\alpha ^p, \beta ^q)$ is silting-indiscrete.

Proof. We see that $\widetilde {\Lambda }$ is given by the graded quiver

with the dashed arrows of degree $-1$ . Taking the sum e of the primitive idempotents of $\Lambda $ corresponding to all but the vertices 1 and 2, it is obtained that $\widetilde {\Lambda }/\widetilde {\Lambda }\widetilde {e}\widetilde {\Lambda }$ is the dg quiver algebra of the graded Kronecker quiver with arrows of degree $-1$ , which is silting-indiscrete, so is $\Lambda $ by Corollary 2.18(1).

2.3. Homological epimorphisms of algebras

In this subsection, we restrict the results in §2.2 to ordinary algebras; that is, $\mathcal {A}$ is just an algebra.

Let $\Gamma $ be a finite-dimensional K-algebra. Recall that a homomorphism $\varphi :\Lambda \to \Gamma $ of algebras is a homological epimorphism if the canonical morphism $\Gamma \otimes ^{\mathsf {L}}_\Lambda \Gamma \to \Gamma $ is an isomorphism in the derived category of $(\Gamma , \Gamma )$ -bimodules; that is, the multiplication map $\Gamma \otimes _\Lambda \Gamma \to \Gamma $ is isomorphic and $\operatorname {Tor}^\Lambda _i(\Gamma ,\Gamma )=0$ for all $i>0$ . Note that the former condition $\Gamma \otimes _\Lambda \Gamma \simeq \Gamma $ is satisfied if and only if $\varphi $ is an epimorphism in the category of rings; it is called a ring-epimorphism [51, Prop. 1.1]. A surjective homomorphism (i.e., an ordinary epimorphism) is a ring-epimorphism, but the converse is not always true.

As a homological epimorphism $\Lambda \to \Gamma $ of algebras induces a recollement $(\operatorname {\mathsf {D}}(\Gamma ), \operatorname {\mathsf {D}}(\Lambda ), \mathcal {X})$ by [44, §4], we immediately obtain the following result from Theorem 2.15 under a subtle condition.

Corollary 2.21. Let $\varphi :\Lambda \to \Gamma $ be a homological epimorphism and assume that $\mathcal {X}$ is compactly generated; for example, the projective dimension of the right $\Lambda $ -module $\Gamma _\Lambda $ is finite. If $\Lambda $ is silting-discrete, then so is $\Gamma $ .

We say that an idempotent e of $\Lambda $ (or the ideal $\Lambda e\Lambda $ ) is stratifying if the canonical homomorphism $\pi :\Lambda \to \Lambda /\Lambda e\Lambda $ is a homological epimorphism; equivalently, the dg algebra $\mathcal {A}_e$ as in Corollary 2.18 is just isomorphic to $\Lambda /\Lambda e\Lambda $ . For example, if $\Lambda $ is hereditary (i.e., $\Lambda =KQ$ for an acyclic quiver Q), then any idempotent e of $\Lambda $ is stratifying; notice that $\widetilde {\Lambda }$ as in the last of §2.2 is nothing but $(KQ, 0)=\Lambda $ . More generally, if $\Lambda e\Lambda $ is projective as a right $\Lambda $ -module, then it is stratifying. Here is a direct consequence of Corollaries 2.18/2.21.

Corollary 2.22. Let e be a stratifying idempotent of $\Lambda $ . If $\Lambda $ is silting-discrete, then so is $\Lambda /\Lambda e\Lambda $ .

A typical example of stratifying idempotents is a primitive idempotent corresponding to a source or a sink in the Gabriel quiver of $\Lambda $ . So, we have:

Example 2.23. Let e be a primitive idempotent corresponding to a source or a sink in the Gabriel quiver of $\Lambda $ . If $\Lambda $ is silting-discrete, then so is $\Lambda /\Lambda e\Lambda $ . In other words, if a one-point extension algebra of $\Lambda $ is silting-discrete, then so is $\Lambda $ .

We remark that this example can also be got from the fact that $\Lambda /\Lambda e\Lambda $ is isomorphic to $(1-e)\Lambda (1-e)$ for such an idempotent e.

Note that the converse of Example 2.23 does not necessarily hold; a one-point extension of a path algebra of Dynkin type is often of non-Dynkin type. This also means that the silting-discreteness of both sides in a recollement is not always passed to the middle in the recollement.

We apply Example 2.23 to tree quiver algebras.

Example 2.24. Let $\Lambda $ be a tree quiver algebra; that is, the underlying graph of its Gabriel quiver is a tree. If $\Lambda $ is silting-discrete, then so is $\Lambda /\Lambda e\Lambda $ for every idempotent e of  $\Lambda $ .

2.4. Remark on Kronecker algebras

As is well-known, the Kronecker algebras are a first and good example of $\tau $ -tilting infinite algebras. When we discuss silting-discreteness, one often needs graded Kronecker algebras, as we already observed in the preceding subsections. Now, we summarize it.

Let $\mathcal {K}$ be the dg quiver algebra of a graded n-Kronecker quiver with trivial differential. For $\lambda :=(n_i)_i\in \bigoplus _{i\in \mathbb {Z}}\mathbb {Z}_{\geq 0}$ , we write $\mathcal {K}:=\mathcal {K}(\lambda )$ , where $n_i$ stands for the number of arrows of degree i; we will omit the zero parts. Note that $n=\sum _i n_i$ . We remark that applying shifts, $\mathcal {K}((n_i)_i)$ is derived equivalent to $\mathcal {K}((n_{i+m})_i)$ for every $m\in \mathbb {Z}$ .

In case that $n_i\neq 0$ for at most two components i, we get the following result.

Proposition 2.25. Let $i<0$ . Then $\mathcal {K}=\mathcal {K}(n_0, n_i)$ is silting-discrete if and only if $n_0, n_i\leq 1$ .

Proof. If $n_0\geq 2$ , then $H^0(\mathcal {K})$ is the ordinary (2-)Kronecker algebra; so, it is not $\tau $ -tilting finite. Therefore, $\mathcal {K}$ is not silting-discrete. Let $n_i\geq 2$ and $S:=H^0(\mathcal {K})/\operatorname {rad} H^0(\mathcal {K})$ . Then the Koszul dual $\mathcal {K}^!:=\operatorname {Ext}_{\mathcal {K}}^{*}(S, S)$ of $\mathcal {K}$ is quasi-isomorphic to $\mathcal {K}(n^{\prime }_{1-i}, n^{\prime }_1)$ , which is derived equivalent to $\mathcal {K}(n^{\prime \prime }_0, n^{\prime \prime }_i)$ . Here, $n^{\prime \prime }_0=n^{\prime }_{1-i}=n_i$ and $n^{\prime \prime }_i=n^{\prime }_1=n_0$ . Since $\mathcal {K}$ and $\mathcal {K}^!$ are derived equivalent [52, §5], we obtain that $\mathcal {K}$ is not silting-discrete.

Let us show the if part; we only handle the case that $n_0=1=n_i$ . Let $\Lambda $ be the radical-square-zero algebra given by the quiver

It follows from Proposition 2.8 that $\Lambda $ is silting-discrete. Let $e_v$ be the primitive idempotent of $\Lambda $ corresponding to the vertex v. Taking the idempotent $e:=e_1+\cdots +e_{-i}$ , we see that $\mathcal {K}$ is quasi-isomorphic to $\mathcal {A}_e$ , whence it is silting-discrete by Corollary 2.18.

Remark 2.26. We know from [41, Cor. 3.2] that if $\mathcal {K}$ is derived equivalent to an ordinary algebra, then $\mathcal {K}=\mathcal {K}(n_0, n_{-1})$ up to shift. Moreover, by [41, Cor. 3.10], it is seen that $\mathcal {K}(n_0, n_{-1})$ is derived equivalent to the algebra presented by the quiver consisting of vertices 1 and 2 with $n_0$ arrows from 1 to 2 (say x) and $n_{-1}$ arrows from 2 to 1 (say y), and with all relations $yx=0$ . This is just a quasi-hereditary algebra with two simples.

Remark 2.27. We observe that $\mathcal {K}(1,1)$ for an arbitrary degree is derived equivalent to a graded gentle one-cycle algebra, which behaves like (ordinary) derived-discrete algebras. We hope that graded gentle one-cycle algebras which are not derived equivalent to $K\widetilde {A}$ are silting-discrete (see [34]).

2.5. Remark on quotients

We give a remark on the inheritance of silting-discreteness by (idempotent) quotients, comparing with $\tau $ -tilting finiteness.

First, note that algebra (ordinary) epimorphisms cannot necessarily transmit the silting-discreteness (see Proposition 2.9 and Example 2.20).

Next, let us observe that idempotent quotients cannot always inherit the silting-discreteness, either. Let $\Gamma $ be the algebra given by the quiver

with relations $\beta \alpha =\delta \alpha =\gamma \varepsilon =0$ and $\gamma \beta =yx$ . To check that $\Gamma $ is a piecewise hereditary algebra of type $E_7$ , we first show the following claim.

Claim. Let $\Lambda $ be a piecewise hereditary algebra of Dynkin type and M an indecomposable right $\Lambda $ -module. Then the one-point extension algebra $\begin {pmatrix} K & M \\ 0 & \Lambda \end {pmatrix}$ is derived equivalent to a hereditary algebra.

Proof. Let A be a hereditary algebra of Dynkin type which is derived equivalent to $\Lambda $ . Applying shift functors if needed, we may assume that M corresponds to an indecomposable A-module N under a derived equivalence between $\Lambda $ and A. Since A is representation-finite, there is an integer $\ell \geq 0$ with $P:=\tau ^\ell (N)$ projective. Here, $\tau $ denotes the Auslander–Reiten translation. As it induces a derived autoequivalence of A, we can take a derived equivalence between $\Lambda $ and A which sends M to P. By [18, Th. 1], we obtain that $\begin {pmatrix} K & M \\ 0 & \Lambda \end {pmatrix}$ is derived equivalent to $\begin {pmatrix} K & P \\ 0 & A \end {pmatrix}$ , which is hereditary.

We recall the definitions of the Coxeter matrix, Coxeter polynomial, and spectral radius of a hereditary algebra $\Lambda $ . Let C be the Cartan matrix of $\Lambda $ . Then the matrix $-C^{-1}C^t$ and its characteristic polynomial are called the Coxeter matrix and the Coxeter polynomial of $\Lambda $ , respectively. Here, $C^t$ denotes the transpose of C. Let $\alpha _1, \ldots , \alpha _n$ be the eigenvalues of the Coxeter matrix; that is, the complex roots of the Coxeter polynomial. The spectral radius of $\Lambda $ is defined to be the maximum of the absolute values of $\alpha _i$ ’s.

Let us return our attention to $\Gamma $ . Since the full subquiver with the vertices 2, 3, 4, and 7 makes a piecewise hereditary algebra of type $D_4$ , we observe by the claim that the one-point extension at the vertex 5 is derived equivalent to a hereditary algebra whose Coxeter polynomial is $(x+1)(x^4+1)$ , which is of type $D_5$ . Therefore, it turns out that the algebra presented by the full subquiver with the vertices 2, 3, 4, 5, and 7 is piecewise hereditary of type $D_5$ . As a similar argument (use the dual of the claim), we see that $\Gamma $ is derived equivalent to a hereditary algebra with Coxeter polynomial $(x+1)(x^6-x^3+1)$ , which is of type $E_7$ . Thus, we figure out that $\Gamma $ is silting-discrete.

However, it is obtained by Example 2.14 that the idempotent quotient $\Gamma /\Gamma e_7\Gamma $ is not silting-discrete, where $e_7$ denotes the primitive idempotent corresponding to the vertex 7.

Remark 2.28. Although Coxeter polynomials are a derived invariant, we cannot judge the silting-discreteness of a given algebra only by them, in general. Actually, the extended canonical algebra of type $\langle 2,4,6 \rangle $ , which is given by the quiver

with relation $z^6=x^2-y^4$ , has the same Coxeter polynomial as a (piecewise) hereditary algebra of type $D_{12}$ [39, Prop. 5.9], but it is not silting-discrete. Note that the algebra is never derived equivalent to a hereditary algebra; for a hereditary algebra, if the spectral radius ( $=1$ ) is not a root of the Coxeter polynomial, then the algebra is representation-finite; if the spectral radius ( $=1$ ) is a root of the Coxeter polynomial, then the algebra is representation-tame; if the spectral radius is strictly greater than 1, then the algebra is representation-wild.

3. Applications

In this section, we investigate the silting-discreteness of finite-dimensional algebras; in particular, we focus on

• • simply connected tensor algebras (3.1) and

• • self-injective Nakayama algebras (3.2).

Let $\Lambda $ be a finite-dimensional K-algebra which is basic and ring-indecomposable. Let $\overrightarrow {A_n}$ be the quiver . For two successive arrows in the Gabriel quiver of $\Lambda $ , we draw if $\alpha \beta =0$ .

3.1. Simply connected tensor algebras

In this subsection, we completely classify silting-discrete simply connected tensor algebras. Since we already know that the tensor algebra of three nonlocal algebras is nothing but not silting-discrete [7, Prop. 4.1], one turns the interest to the classification of silting-discrete simply connected tensor algebras of two algebras. Here is the first case; one of the two is K.

Proposition 3.1. A simply connected algebra is silting-discrete if and only if it is piecewise hereditary of Dynkin type.

Proof. The if part is trivial. We show the only if part. Let $\Lambda $ be a silting-discrete simply connected algebra. Note that the silting-discreteness implies the $2_\Lambda $ -silting finiteness (the $\tau $ -tilting finiteness for $\Lambda $ ). We see by [54, Th. 3.4] that $\Lambda $ is representation-finite. Since $\Lambda $ is silting-discrete, any reflection of $\Lambda $ is $\tau $ -tilting finite. Moreover, it follows from [14, Lem. 3.1] that every reflection of $\Lambda $ is simply connected, so it is representation-finite. Then, we apply Proposition 3.3 of [14] to deduce that the repetitive algebra of $\Lambda $ is locally representation-finite, which implies that it is piecewise hereditary of Dynkin type by [28].

In the rest of this subsection, let A and B be triangular algebras which are basic and ring-indecomposable, and put $\Lambda :=A\otimes _KB$ . Here, a triangular algebra is a finite-dimensional K-algebra whose Gabriel quiver is acyclic.

Our goal of this subsection is to give a complete classification of silting-discrete simply connected tensor algebras. We know that $\Lambda $ is silting-discrete if A is local and B is silting-discrete [7, Th. 2.1]. Let us give an easy observation.

Lemma 3.2. If $\Lambda $ is silting-discrete, then so are both A and B.

Proof. Let e be a primitive idempotent of A. As A is triangular, we have an isomorphism $eAe\simeq K$ . Apply Corollary 2.11 to deduce that $B\simeq (e\otimes 1)\Lambda (e\otimes 1)$ is silting-discrete.

We determine the algebra structure of one of the components when $\Lambda $ is silting-discrete.

Lemma 3.3. If $\Lambda $ is silting-discrete, then at least one of A and B has at most two nonisomorphic simple modules. In particular, it is isomorphic to K or $K\overrightarrow {A_2}$ .

Proof. By Lemma 3.2, it is observed that A and B are silting-discrete, and so they have no multiple arrow in their Gabriel quivers [26, Th. 5.12(d)]. Note that for every idempotent e of A, $eAe$ is a silting-discrete triangular algebra.

Now, assume that A has at least three nonisomorphic simple modules. As above, there exists an idempotent e of A such that $eAe$ is isomorphic to one of the following:

From Corollary 2.11, we obtain that $(e\otimes 1)\Lambda (e\otimes 1)=(eAe)\otimes _KB$ is silting-discrete. Note that the first two of the three algebras above are derived equivalent to the path algebra $K\overrightarrow {A_3}$ . If $eAe$ is one of the first two, then $(eAe)\otimes _KB$ is derived equivalent to $K\overrightarrow {A_3}\otimes _KB$ , which must be silting-discrete. This yields by [7, Th. 4.4] that B is a Nakayama algebra with radical square zero, whence it is isomorphic to $K\overrightarrow {A_r}/\operatorname {rad}^2K\overrightarrow {A_r}$ for some $r>0$ ; because B is triangular. We see that $(eAe)\otimes _KB$ is derived equivalent to $K\overrightarrow {A_3}\otimes _KK\overrightarrow {A_r}$ , which is also silting-discrete. By [7, Th. 4.4] again, we have $r\leq 2$ .

Finally, let us suppose that $eAe$ is the last of the three algebras above, which is derived equivalent to the algebra C given by the quiver

with $\gamma \beta =0$ . So, we observe that $(eAe)\otimes _KB$ is derived equivalent to $C\otimes _KB$ , which is silting-discrete. Since there is an epimorphism $C\otimes _KB\to K\overrightarrow {A_3}\otimes _KB$ , it turns out that $K\overrightarrow {A_3}\otimes _KB$ is $\tau $ -tilting finite [26]. As a similar argument above, we conclude that B has at most two nonisomorphic simple modules.

We often call the algebra $K\overrightarrow {A_2}\otimes _KK\overrightarrow {A_r}$ the commutative ladder of degree r, which is presented by the quiver with relations as follows:

Now, we realize the goal of this subsection, which gives a complete classification of silting-discrete simply connected tensor algebras.

Theorem 3.4. Assume that A and B are nonlocal and simply connected. Then the following are equivalent:

1. (1) $\Lambda $ is silting-discrete.

2. (2) It is a piecewise hereditary algebra of type $D_4, E_6$ , or $E_8$ .

3. (3) It is derived equivalent to the commutative ladder of degree $\leq 4$ .

Note that the case of $\Lambda $ being a piecewise hereditary algebra of type $A_2$ cannot occur since if so, then either A or B is local.

Proof. The equivalence (2) $\Leftrightarrow $ (3) is due to [38]. We know the implication (2) $\Rightarrow $ (1) holds. Let us show the implication (1) $\Rightarrow $ (3) holds true. Applying Lemma 3.3, we may suppose that $A\simeq K\overrightarrow {A_2}$ . By Lemma 3.2, one sees that B is silting-discrete. Therefore, it follows from Proposition 3.1 that B is piecewise hereditary of Dynkin type $\Delta $ , whence $\Lambda $ is derived equivalent to $K\overrightarrow {A_2}\otimes _KK\overrightarrow {\Delta }$ . Since this is silting-discrete, we observe that $\Delta $ must be of type $A_r$ by [7, Th. 3.2]; that is, $\Lambda $ is derived equivalent to the commutative ladder of degree r. By [7, Exam. 3.3], we get $r\leq 4$ .

Remark 3.5. Lemma 3.3 also says that if $\Lambda $ is silting-discrete, then at least one of A and B is automatically simply connected. Then, we ask whether they both are simply connected or not. In fact, it seems to be unknown that $K\overrightarrow {A_2}\otimes _KC$ is silting-discrete, where C is the algebra presented by the quiver with relation:

(It is $\tau $ -tilting finite, thanks to Aoki’s QPA program.) This is also one reason why we cannot drop the assumption of A and B being simply connected in Theorem 3.4.

3.2. Self-injective Nakayama algebras

Ten years ago, the first-named author of this paper showed that any representation-finite symmetric algebra is silting-discrete [5]. Then, we might naturally hope that every representation-finite self-injective algebra is also silting-discrete. However, we here give the following surprising result, which says that the guess does not necessarily hold.

Theorem 3.6. Let $\Lambda $ be a (nonlocal) self-injective Nakayama algebra; that is, it is presented by the cyclic quiver with n vertices, and the rth radical is zero for some $n, r>1$ . Then $\Lambda $ is not silting-discrete if (i) $r=3, 4$ and $n\geq 11$ ; (ii) $r=5,6$ and $n\geq r+8$ ; or (iii) $r\geq 7$ and $n\geq 2r+1$ .

Let $n,r>1$ . We denote by $N_{n,r}$ the algebra given by the quiver

with $x^r=0$ . The primitive idempotent corresponding to the vertex i of the quiver is denoted by $e_i$ . As is well-known, an algebra is a nonlocal self-injective (symmetric) Nakayama algebra if and only if it is isomorphic to $N_{n, r}$ for some $n,r>1$ ( $r\equiv 1\pmod n$ ).

We give an example of silting-discrete self-injective Nakayama algebras.

Proposition 3.7. If $r=2$ or $r\equiv 1\,\pmod n$ , then $N_{n,r}$ is silting-discrete.

Proof. By Proposition 2.8 for $r=2$ and [5, Th. 5.6] for $r\equiv 1\,\pmod n$ .

Put $A(n,r):=K\overrightarrow {A_n}/\operatorname {rad}^r K\overrightarrow {A_n}$ . Under a suitable condition, we can get $A(n,r)$ by an idempotent truncation of a self-injective Nakayama algebra.

Lemma 3.8. Let $1\leq s\leq n$ and put $e:=e_1+\cdots +e_s$ . If $s+r\leq n+1$ , then $eN_{n,r}e$ is isomorphic to $A(s,r)$ .

Proof. Straightforward.

Thanks to the list of [29] (see also [39]), we give a complete classification.

Lemma 3.9.

1. (1) If $A(s,r)$ is not silting-discrete, then neither is $A(s+1, r)$ .

2. (2) $A(s,r)$ is silting-discrete if and only if one of the following cases occurs:

   1. (i) $r=2$ ;

   2. (ii) $r=3, 5, 6$ and $s\leq 8$ ;

   3. (iii) $r=4$ and $s\leq 7$ ;

   4. (iv) $r\geq 7$ and $s=r+1$ .

      Table 1 The list of Happel–Seidel (the entry describes the type of $A(s,r)$ if it is piecewise hereditary, otherwise the hypen “–” is put)

      

Proof. (1) Apply Corollary 2.11 to $A(s, r)\simeq (1-e_{s+1})A(s+1, r)(1-e_{s+1})$ .

(2) For each r, the list of Happel–Seidel (Table 1) describes a border of s for which $A(s, r)$ is derived equivalent to a hereditary algebra of Dynkin type or not. Then, by (1), we completely understand whether $A(s, r)$ is silting-discrete or not. For instance, when $r\geq 7$ , $A(r+2, r)$ is derived equivalent to a path algebra of type $T_{23(r-1)}$ ( $\widetilde {E_8}=T_{236}$ ), which is not Dynkin; so, it is silting-indiscrete.

Now, we show the main theorem of this subsection.

Proof of Theorem 3.6

Let $\Lambda :=N_{n,r}$ for $n,r>1$ . We put s in each case as follows:

1. (i) $r=3,4$ and $n\geq 11\rightsquigarrow s=9\ (r=3)$ or $s=8\ (r=4)$ .

2. (ii) $r=5,6$ and $n\geq r+8\rightsquigarrow s=9$ .

3. (iii) $r\geq 7$ and $n\geq 2r+1\rightsquigarrow s=r+2$ .

In all the cases, the assumption of Lemma 3.8 is satisfied; hence, $e\Lambda e\simeq A(s,r)$ .

By Lemma 3.9, we see that $A(s,r)$ is not silting-discrete for these pairs $(s,r)$ , whence $\Lambda $ is not silting-discrete by Corollary 2.11.

Remark 3.10. As an analog of silting-discreteness, the notion of the tilting-discreteness for (self-injective) algebras was introduced by Chan, Koenig, and Liu. It is known that any representation-finite self-injective algebra (so, every self-injective Nakayama algebra) is tilting-discrete [10], [22].