Silting-Discreteness in Triangulated Categories

• Silting-discreteness is a finiteness property where every interval of silting objects in a triangulated category is finite, controlling mutation behavior.
• It connects τ-tilting, t-structures, and stability conditions, underpinning significant classification results in algebraic geometry and representation theory.
• This property is preserved under operations like full triangulated subcategories and homological epimorphisms, enabling robust reduction techniques in complex categorical settings.

Silting-discreteness is a structural finiteness property for triangulated categories, particularly those of algebraic or geometric origin, that governs the distribution and mutation behavior of silting objects within the category. A triangulated category is silting-discrete if the partially ordered set of basic silting objects is "locally finite," meaning that every interval in this poset contains only finitely many silting objects. This concept plays a pivotal role in the interaction between silting theory, τ-tilting theory, t-structure theory, and the study of stability conditions, providing robust finiteness criteria and classification results across representation theory, algebraic geometry, and related fields (Hügel, 2018, Aihara et al., 2023, Chang et al., 30 Dec 2025, Adachi et al., 2017, Aihara et al., 2020, Kimura et al., 2024).

1. Definitions and Characterizations

Let $\mathcal{T}$ be a $K$-linear, Hom-finite, Krull–Schmidt triangulated category such as $K^b(\mathrm{proj}\,A)$ for a finite-dimensional algebra $A$. An object $T \in \mathcal{T}$ is called:

• Presilting: $\operatorname{Hom}_{\mathcal{T}}(T, T[i]) = 0$ for all $i>0$.
• Silting: $T$ is presilting and $\operatorname{thick}(T) = \mathcal{T}$.
• Partial silting: A direct summand of a silting object.

The set $\operatorname{silt}\,\mathcal{T}$ of isomorphism classes of basic silting objects carries a natural partial order: $T \geq U$ if $\operatorname{Hom}_{\mathcal{T}}(T, U[i])=0$ for all $i>0$.

Silting-discrete: $\mathcal{T}$ is silting-discrete if for any silting object $A$ and $d>0$, the interval

$${}^{d}_{A}\operatorname{silt} \mathcal{T} = \{ T \in \operatorname{silt}\,\mathcal{T} \mid A \geq T \geq A[d-1] \}$$

is finite. Equivalently, one can require only $d=2$ (“2-silting finiteness”) (Aihara et al., 2023, Hügel, 2018, Adachi et al., 2017).

Further characterizations include:

Characterization	Description
(C1) Local finiteness	All intervals ${}^{d}_{A}\operatorname{silt} \mathcal{T}$ are finite for each silting $A$ and $d>0$.
(C2) Compactness	Every silting object in $D(A)$ is equivalent (up to shift) to one in $K^b(\mathrm{proj}\,A)$.
(C3) τ-tilting finiteness	$\operatorname{End}_{\mathcal{T}}(A)$ is τ-tilting finite for all basic silting $A$.

Equivalence of these conditions is established in (Hügel, 2018, Adachi et al., 2017, Aihara et al., 2020, Chang et al., 30 Dec 2025).

2. Structural Theory and Inheritance Properties

Silting-discreteness enjoys strong inheritance and reduction properties:

• Full Triangulated Subcategories: If $\mathcal{T}$ is silting-discrete, any full triangulated subcategory $\mathcal{U}$ is also silting-discrete (Aihara et al., 2023).
• Idempotent Truncations: If $A$ is silting-discrete and $e$ is any idempotent, then $eAe$ is silting-discrete (Aihara et al., 2023).
• Homological Epimorphisms: If $\varphi: A \to B$ is a homological epimorphism with certain compactness assumptions, then $\mathrm{per}(A)$ silting-discrete implies $\mathrm{per}(B)$ silting-discrete (Aihara et al., 2023).
• Skew Group Algebras: Under mild hypotheses on a finite group $G$ acting on $\Lambda$, if $\Lambda$ is $G$-stable silting-discrete, then the skew group algebra $A = \Lambda*G$ is silting-discrete (Kimura et al., 2024).

This transmission via functors and reductions underscores the robust nature of silting-discreteness in derived and module categories.

3. Relation to τ-Tilting and t-Structures

Silting-discreteness is intimately connected with other finiteness properties:

• For any algebra $A$ and basic silting $M$, $A$ is silting-discrete if and only if $\operatorname{End}_{D(A)}(M)$ is τ-tilting finite (Hügel, 2018, Chang et al., 30 Dec 2025). In the context of gentle and skew-gentle algebras, this translates to representation-finiteness characterizations via surface geometry (Chang et al., 30 Dec 2025).
• In the context of the Adachi–Mizuno–Yang theorem, silting-discreteness for $K^b(\mathrm{proj} A)$ is equivalent to $t$-discreteness for $D^b(\mathrm{mod}\,A)$, and the stability space $\operatorname{Stab}(D^b(\mathrm{mod}\,A))$ is contractible (Adachi et al., 2017).
• Silting-discrete algebras possess a Bongartz-type property: every presilting object admits a silting complement (completion) (Aihara et al., 2023, Hügel, 2018).

4. Classification Results and Key Examples

Classification theorems for silting-discrete algebras and categories are known in several cases:

• Local and Representation-Finite Hereditary Algebras: Always silting-discrete (Aihara et al., 2023, Hügel, 2018).
• Piecewise Hereditary Algebras (Dynkin type): Silting-discrete if and only if of Dynkin type (Aihara et al., 2023).
• Simply-Connected Tensor Algebras: The tensor algebra $A \otimes B$ of non-local simply-connected triangular algebras is silting-discrete if and only if it is derived-equivalent to a commutative ladder of degree at most 4 (types $D_4, E_6, E_7$) (Aihara et al., 2023).
• Gentle and Skew-Gentle Algebras: For homologically smooth proper graded gentle algebras, silting-discreteness of $\mathrm{per}(A)$ is equivalent to the associated surface having genus zero and all winding numbers nonzero for simple closed curves. For skew-gentle, additional orbifold restrictions apply (Chang et al., 30 Dec 2025).
• Selfinjective Nakayama Algebras: Certain parameter ranges yield silting-indiscreteness; extreme cases such as $r=2$ or $r \equiv 1$ (mod $n$) are silting-discrete (Aihara et al., 2023).
• Weakly Symmetric Tubular and Related Algebras: Families such as weakly symmetric tubular algebras with nonsingular Cartan are tilting-discrete (Aihara et al., 2020).

These results are summarized in the following table:

Family	Silting-discrete Criterion
Local, rep-finite hereditary, symmetric	Always silting-discrete
Piecewise hereditary	If and only if Dynkin type
Tensor of simply-connected triangular	Derived-equivalent to commutative ladder of degree ≤4
Gentle/skew-gentle (surface model)	Genus zero, all (orbifold-adjusted) winding numbers nonzero
Weakly symmetric tubular	Nonsingular Cartan matrix

5. Applications and Consequences

Silting-discreteness exerts deep influence on the structure and mutation theory of derived categories:

• Mutation Graph Connectedness: The mutation (Hasse) quiver on silting objects is connected; any silting object can be reached from the trivial silting via finite mutations (Hügel, 2018).
• Torsion Classes and t-Structures: There are only finitely many intermediate t-structures and functorially finite torsion classes between any two given ones (Hügel, 2018, Adachi et al., 2017).
• Reduction Theorems: Many reduction techniques preserve silting-discreteness, facilitating classification and explicit computations (Aihara et al., 2023, Kimura et al., 2024).
• Stability Spaces: When silting-discreteness holds, the stability manifold of the associated derived category is contractible, with implications for geometric and categorical stability (Adachi et al., 2017).
• Classification of thick subcategories: In asotic cases with Bongartz property, every thick subcategory generated by a silting object is equivalent to the homotopy category of projectives for an idempotent truncation of an algebra derived-equivalent to $A$ (Aihara et al., 2023).

6. Geometric and Combinatorial Realizations

For gentle and skew-gentle algebras, silting-discreteness can be characterized in terms of surface topology and line field invariants:

• Surface Model: The algebra corresponds to a surface $S$ (possibly an orbifold), a line field $\eta$, and a polygonal dissection, with silting-discreteness equivalent to $g=0$ and nonzero winding numbers $\omega_\eta(\ell)$ for all essential closed curves (Chang et al., 30 Dec 2025).
• Combinatorial Invariants: For certain preprojective and mesh algebras, the poset of silting complexes matches the weak order on the Weyl group or the braid group of associated Coxeter type (Kimura et al., 2024).

This geometric viewpoint enables explicit constructions and counterexamples, linking representation theory, topology, and category theory.

7. Open Problems and Limitations

While silting-discreteness is robust under many categorical operations, limitations arise in skew-gentle cases where reduction strategies face obstacles because the endomorphism algebras of silting objects may exit the class of skew-gentle algebras (Chang et al., 30 Dec 2025). Moreover, in certain cases (e.g., higher genus surfaces or parameter ranges for selfinjective Nakayama algebras), silting-discreteness fails, with infinitely many silting objects in intervals between mutations (Aihara et al., 2023).

A plausible direction is the further development of invariants and derived invariants describing silting-discrete classes, particularly in dg, orbifold, or higher Calabi–Yau contexts. The relationship between silting-discreteness and the contractibility of stability conditions spaces continues to motivate new work in the interplay of homological algebra, geometry, and combinatorics.

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References:

(Aihara et al., 2023, Chang et al., 30 Dec 2025, Hügel, 2018, Adachi et al., 2017, Aihara et al., 2020, Kimura et al., 2024)