2-Term Silting Complexes

• 2-term silting complexes are objects in triangulated categories concentrated in degrees -1 and 0 that define intermediate co-t-structures and link to τ-tilting theory.
• They establish a framework for classifying torsion classes and silting modules via bijections with support τ-tilting pairs, aiding explicit computations in representation theory.
• Their mutation, reduction, and extension properties enable deep insights into endomorphism algebras and global dimension bounds in various algebraic settings.

A 2-term silting complex is, in the classical algebraic setting, a silting object in $K^{b}(\mathrm{proj}\,\Lambda)$ concentrated in degrees $-1$ and $0$; in the intrinsic triangulated formulation, relative to a bounded co-t-structure with co-heart $\mathcal S$, it is a silting subcategory contained in $\mathcal S * \Sigma \mathcal S$, where every object fits into a triangle $s_1\to x\to s_0\to \Sigma s_1$ with $s_0,s_1\in\mathcal S$ (Iyama et al., 2013). The notion occupies a central position in modern representation theory because it identifies the 2-step part of silting theory that simultaneously controls intermediate co-t-structures, support $\tau$-tilting theory, torsion classes, and, in many settings, hearts of bounded t-structures.

1. Basic definition and ambient framework

Let $\mathcal T$ be a triangulated category with suspension functor $\Sigma$. A full subcategory $-1$0 is presilting if

$-1$1

and silting if it is presilting and generates $-1$2 as a thick subcategory: $-1$3 An object $-1$4 is a silting object when $-1$5 is a silting subcategory (Iyama et al., 2013).

Fix a bounded co-t-structure $-1$6 on $-1$7, with co-heart

$-1$8

The subcategory

$-1$9

is equal to $0$0, and is idempotent complete and extension-closed (Iyama et al., 2013). A two-term silting subcategory relative to $0$1 is then precisely a silting subcategory $0$2. In the standard algebraic case $0$3 and $0$4 embedded as stalk complexes in degree $0$5, $0$6 consists exactly of complexes concentrated in degrees $0$7, so the abstract and classical definitions coincide (Iyama et al., 2013).

A recurring source of confusion is that “two-term” is not merely a statement about literal cochain degrees in an arbitrary triangulated category. In the intrinsic formulation it is relative to a chosen silting co-heart. The degree description is recovered only after fixing a standard silting subcategory such as $0$8 in $0$9 (Iyama et al., 2013).

2. Intermediate co-t-structures and intrinsic characterization

The decisive structural statement is that two-term silting is the silting-theoretic realization of intermediate co-t-structures. If $\mathcal S$0 and $\mathcal S$1 are bounded co-t-structures on $\mathcal S$2, then $\mathcal S$3 is called intermediate with respect to $\mathcal S$4 when

$\mathcal S$5

Theorem 2.3 identifies these data with two-term silting subcategories via

$\mathcal S$6

yielding a bijection between intermediate bounded co-t-structures and silting subcategories contained in $\mathcal S$7 (Iyama et al., 2013).

This result rests on the broader correspondence between bounded co-t-structures and silting subcategories. In the form quoted from Mendoza–Sáenz–Santiago–Souto Salorio, the assignment

$\mathcal S$8

is a bijection between bounded co-t-structures on an essentially small idempotent complete triangulated category and silting subcategories (Iyama et al., 2013). The two-term theory is therefore the interval

$\mathcal S$9

inside the poset of co-t-structures.

Over arbitrary rings, the same picture reappears in derived form. There are bijections between equivalence classes of 2-silting complexes, equivalence classes of silting modules, 2-silting t-structures in $\mathcal S * \Sigma \mathcal S$0, and co-t-structures $\mathcal S * \Sigma \mathcal S$1 in $\mathcal S * \Sigma \mathcal S$2 with

$\mathcal S * \Sigma \mathcal S$3

and $\mathcal S * \Sigma \mathcal S$4 closed under coproducts (Hügel et al., 2014). In that formulation, 2-term silting complexes are the boundary case where the silting–t-structure correspondence still has a direct module-theoretic incarnation.

3. $\mathcal S * \Sigma \mathcal S$5-tilting, silting modules, and torsion classes

The second foundational axis is the passage from two-term silting to support $\mathcal S * \Sigma \mathcal S$6-tilting. For a silting subcategory $\mathcal S * \Sigma \mathcal S$7, the restricted Yoneda functor

$\mathcal S * \Sigma \mathcal S$8

induces an equivalence

$\mathcal S * \Sigma \mathcal S$9

Within this quotient, presilting subcategories correspond to $s_1\to x\to s_0\to \Sigma s_1$0-rigid pairs, and silting subcategories correspond to support $s_1\to x\to s_0\to \Sigma s_1$1-tilting pairs; under a Krull–Schmidt uniqueness condition on decompositions in $s_1\to x\to s_0\to \Sigma s_1$2, the correspondence is bijective on both presilting and silting subcategories (Iyama et al., 2013).

Under the standard algebraic hypotheses that $s_1\to x\to s_0\to \Sigma s_1$3 is Krull–Schmidt, $s_1\to x\to s_0\to \Sigma s_1$4-linear, Hom-finite, and $s_1\to x\to s_0\to \Sigma s_1$5 is a basic silting object with $s_1\to x\to s_0\to \Sigma s_1$6, one obtains an algebra $s_1\to x\to s_0\to \Sigma s_1$7 and an equivalence $s_1\to x\to s_0\to \Sigma s_1$8. The functor $s_1\to x\to s_0\to \Sigma s_1$9 then induces a bijection between basic silting objects of $s_0,s_1\in\mathcal S$0 lying in $s_0,s_1\in\mathcal S$1, that is, two-term silting objects, and basic support $s_0,s_1\in\mathcal S$2-tilting $s_0,s_1\in\mathcal S$3-modules (Iyama et al., 2013). In the classical case $s_0,s_1\in\mathcal S$4, this recovers the Adachi–Iyama–Reiten correspondence between basic 2-term silting complexes and basic support $s_0,s_1\in\mathcal S$5-tilting $s_0,s_1\in\mathcal S$6-modules (Iyama et al., 2013).

The torsion-theoretic side is equally rigid. For an essentially small additive category $s_0,s_1\in\mathcal S$7, a torsion class in $s_0,s_1\in\mathcal S$8 is a full subcategory closed under factor modules and extensions, and it is finitely generated if it has the form $s_0,s_1\in\mathcal S$9 for some $\tau$0. Theorem 5.1 identifies support $\tau$1-tilting pairs $\tau$2 with finitely generated torsion classes $\tau$3 such that every finitely generated projective $\tau$4-module has a left $\tau$5-approximation, and moreover

$\tau$6

Composed with the two-term silting correspondence, this yields the chain

$\tau$7

in the appropriate module category (Iyama et al., 2013).

For arbitrary rings, silting modules make the same bridge explicit. A silting module $\tau$8 is one for which there exists a projective presentation $\tau$9 with $\mathcal T$0, and equivalence classes of 2-silting complexes correspond bijectively to equivalence classes of silting modules via $\mathcal T$1 (Hügel et al., 2014). Over finite-dimensional algebras, partial silting coincides with $\mathcal T$2-rigidity and silting coincides with support $\mathcal T$3-tilting (Hügel et al., 2014).

4. Mutation, completions, reduction, and finiteness

Two-term silting is unusually well adapted to mutation. For a module-finite algebra $\mathcal T$4 over a commutative noetherian ring, any 2-term presilting complex admits both Bongartz and co-Bongartz completions, because $\mathcal T$5 is functorially finite in $\mathcal T$6. If $\mathcal T$7 is Krull–Schmidt and $\mathcal T$8 is almost complete, then the co-Bongartz completion is an irreducible left mutation of the Bongartz completion, and an almost complete 2-term presilting complex has at most these two completions (Kimura, 2020). This extends the finite-dimensional mutation picture to noetherian algebras.

Reduction results are equally strong. If $\mathcal T$9 is complete local noetherian, $\Sigma$0 is module-finite over $\Sigma$1, and $\Sigma$2 is a two-sided ideal, then the reduction map $\Sigma$3 induces isomorphisms of posets

$\Sigma$4

compatible with $\Sigma$5 and $\Sigma$6 (Kimura, 2020). This makes the 2-term silting theory of many noetherian algebras accessible through finite-dimensional quotients.

Finiteness questions are naturally expressed through $\Sigma$7-tilting finiteness. An algebra $\Sigma$8 is $\Sigma$9-tilting-finite exactly when it has only finitely many 2-term silting objects in $-1$00, up to isomorphism (Aihara et al., 2020). The cited survey records large families where this finiteness is established, including weakly symmetric algebras of tubular type with non-singular Cartan matrix and non-standard selfinjective algebras socle-equivalent to tubular type; in these cases the paper gives explicit counts of support $-1$01-tilting modules, hence of 2-term silting complexes (Aihara et al., 2020). It also records that representation-finite algebras are $-1$02-tilting-finite, while the converse fails in general, so finiteness of two-term silting is strictly weaker than representation-finiteness globally (Aihara et al., 2020).

A second misconception is therefore excluded by the current literature: finite 2-term silting behavior does not generally force classical representation-finiteness, although it does so in specific classes such as quasitilted algebras, tree quiver algebras satisfying separation conditions, radical-square-zero tree algebras, and locally hereditary algebras (Aihara et al., 2020).

5. Relative, dg, and Gorenstein extensions

The 2-term formalism extends beyond ordinary module categories in two distinct directions. First, it admits a relative triangulated formulation. If $-1$03 is a Hom-finite Krull–Schmidt triangulated category and $-1$04 is rigid, a subcategory is called two-term with respect to $-1$05 when it lies in $-1$06. Two-term $-1$07-rigid subcategories correspond bijectively to $-1$08-rigid subcategories of $-1$09, and two-term weak $-1$10-cluster tilting subcategories correspond bijectively to support $-1$11-tilting subcategories of $-1$12. When $-1$13 is silting, the two-term weak $-1$14-cluster tilting subcategories are precisely the two-term silting subcategories of Iyama–Jørgensen–Yang (Zhou et al., 2018).

Second, the theory persists for non-positive dg algebras. If $-1$15 is a non-positive dg algebra with finite-dimensional total cohomology, then $-1$16 is Krull–Schmidt with silting object $-1$17, and the $-1$18 case of the extended-heart framework yields poset isomorphisms

$-1$19

where $-1$20 is the heart of the standard t-structure (Huang et al., 11 Jun 2026). In the same dg setting, 2-term silting objects in $-1$21 correspond to basic $-1$22-tilting pairs in $-1$23, and $-1$24-cluster morphism categories can be constructed from 2-term presilting objects via silting reduction (Børve, 2021).

The Gorenstein analogue replaces projectives by Gorenstein-projectives and $-1$25 by the Gorenstein derived category $-1$26. A 2-term complex

$-1$27

with $-1$28 is 2-term Gorenstein silting if it is rigid in $-1$29 and generates $-1$30. For a finite-dimensional Gorenstein algebra of finite CM-type, such complexes are equivalent to Gorenstein silting modules via $-1$31, partial 2-term Gorenstein silting corresponds to $-1$32-rigidity, and the complex induces both a torsion pair in $-1$33 and a t-structure on $-1$34 whose heart is equivalent to $-1$35 (Gao et al., 2022).

These extensions show that 2-term silting is not confined to the classical finite-dimensional setup. It remains meaningful whenever a suitable “degree two window” can be identified, whether by a silting subcategory, a dg algebra, or a Gorenstein-projective replacement.

6. Classification results, endomorphism algebras, and representation-theoretic invariants

In several important classes, 2-term silting complexes admit explicit classification. For complete preprojective algebras $-1$36 of non-Dynkin type, two canonical families of two-term tilting complexes, $-1$37 and $-1$38, are constructed from the ideals $-1$39 indexed by the Coxeter group $-1$40. In affine type, every two-term silting complex is actually tilting, and every two-term tilting complex is either $-1$41 or $-1$42; moreover the cones $-1$43 are the closures of the Weyl chambers, while $-1$44 (Kimura et al., 2019). This gives a Coxeter-theoretic and geometric classification of the 2-term silting fan.

Over hereditary algebras, endomorphism algebras of 2-term silting complexes form the class of silted algebras. These are exactly the tilted algebras and the strictly shod algebras; more generally, the endomorphism algebras of 2-term silting complexes over Ext-finite hereditary abelian categories are exactly the shod algebras (Buan et al., 2015). In Dynkin type, this can be made algorithmic: for a hereditary path algebra $-1$45, every basic 2-term silting complex has the form $-1$46, where $-1$47 for an idempotent $-1$48 and $-1$49 is a basic tilting $-1$50-module, and this yields explicit computations of all basic 2-term silting complexes and their endomorphism algebras in small $-1$51 and $-1$52 cases (Xing, 2021).

The homological invariants of endomorphism algebras of 2-term silting complexes are more delicate than in classical tilting. If $-1$53 and $-1$54 is a 2-term silting complex, then

$-1$55

For each $-1$56, however, there exists an algebra $-1$57 with $-1$58 admitting a 2-term silting complex $-1$59 such that $-1$60 is infinite (Buan et al., 2016). Under the additional hypothesis $-1$61, one has the more uniform bound

$-1$62

(Buan et al., 2016). Thus the endomorphism algebra of a 2-term silting complex behaves like a tilted algebra only under extra constraints.

Representation dimension exhibits a similar conditional stability. If $-1$63 is a separating 2-term silting complex with $-1$64 for each $-1$65, then

$-1$66

If $-1$67 is both separating and splitting, then $-1$68 is a splitting and separating tilting $-1$69-module, and

$-1$70

These equalities generalize compare theorems for classical tilting modules to the 2-term silting context (2002.04582).

Taken together, these results position 2-term silting complexes as a sharply delimited yet unusually rich region of silting theory. They are rigid enough to admit explicit classification in important examples, to control torsion-theoretic and co-t-structural data, and to support strong reduction theorems, but flexible enough to exhibit phenomena—such as strictly shod endomorphism algebras or infinite global dimension of endomorphism rings—that lie beyond classical tilting.