---
title: 2-Term Silting Complexes
url: https://www.emergentmind.com/topics/2-term-silting-complexes
type: topic
---

# 2-Term Silting Complexes

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 \(s_0,s_1\in\mathcal S\) [1311.4891]. 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 \(\mathcal S\subseteq \mathcal T\) is presilting if
\[
\operatorname{Hom}_{\mathcal T}(s,\Sigma^{i}s')=0
\quad\text{for all } s,s'\in\mathcal S \text{ and all } i>0,
\]
and silting if it is presilting and generates \(\mathcal T\) as a thick subcategory:
\[
\operatorname{thick}(\mathcal S)=\mathcal T.
\]
An object \(u\in\mathcal T\) is a silting object when \(\operatorname{add}(u)\) is a silting subcategory [1311.4891].

Fix a bounded co-t-structure \((\mathcal A,\mathcal B)\) on \(\mathcal T\), with co-heart
\[
\mathcal S=\mathcal A\cap \Sigma^{-1}\mathcal B.
\]
The subcategory
\[
\mathcal S * \Sigma\mathcal S
=
\Bigl\{ t\in\mathcal T \,\Bigm|\, \exists\ \text{triangle } s_1\to t\to s_0\to \Sigma s_1 \text{ with } s_0,s_1\in\mathcal S \Bigr\}
\]
is equal to \(\Sigma\mathcal A\cap \Sigma^{-1}\mathcal B\), and is idempotent complete and extension-closed [1311.4891]. A two-term silting subcategory relative to \(\mathcal S\) is then precisely a silting subcategory \(\mathcal S'\subseteq \mathcal S * \Sigma\mathcal S\). In the standard algebraic case \(\mathcal T=K^{b}(\mathrm{proj}\,\Lambda)\) and \(\mathcal S=\mathrm{proj}\,\Lambda\) embedded as stalk complexes in degree \(0\), \(\mathcal S * \Sigma\mathcal S\) consists exactly of complexes concentrated in degrees \(-1,0\), so the abstract and classical definitions coincide [1311.4891].

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 \(\mathrm{proj}\,\Lambda\) in \(K^{b}(\mathrm{proj}\,\Lambda)\) [1311.4891].

## 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 \((\mathcal A,\mathcal B)\) and \((\mathcal A',\mathcal B')\) are bounded co-t-structures on \(\mathcal T\), then \((\mathcal A',\mathcal B')\) is called intermediate with respect to \((\mathcal A,\mathcal B)\) when
\[
\mathcal A \subseteq \mathcal A' \subseteq \Sigma\mathcal A.
\]
Theorem 2.3 identifies these data with two-term silting subcategories via
\[
(\mathcal A',\mathcal B') \longmapsto \mathcal A' \cap \Sigma^{-1}\mathcal B',
\]
yielding a bijection between intermediate bounded co-t-structures and silting subcategories contained in \(\mathcal S * \Sigma\mathcal S\) [1311.4891].

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
\[
(\mathcal A,\mathcal B)\longmapsto \mathcal A\cap \Sigma^{-1}\mathcal B
\]
is a bijection between bounded co-t-structures on an essentially small idempotent complete triangulated category and silting subcategories [1311.4891]. The two-term theory is therefore the interval
\[
\mathcal A \subseteq \mathcal A' \subseteq \Sigma\mathcal A
\]
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 \(\mathbf D(A)\), and co-t-structures \((\mathcal U_{\ge 0},\mathcal U_{\le 0})\) in \(\mathbf D(A)\) with
\[
\mathbf D^{\ge -1}\subseteq \mathcal U_{\le 0}\subseteq \mathbf D^{\le 0}
\]
and \(\mathcal U_{\le 0}\) closed under coproducts [1405.2531]. 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. \(\tau\)-tilting, silting modules, and torsion classes

The second foundational axis is the passage from two-term silting to support \(\tau\)-tilting. For a silting subcategory \(\mathcal S\subseteq\mathcal T\), the restricted Yoneda functor
\[
F:\mathcal T\longrightarrow \mathrm{Mod}\,\mathcal S,\qquad t\longmapsto \mathcal T(-,t)|_{\mathcal S}
\]
induces an equivalence
\[
(\mathcal S * \Sigma\mathcal S)/[\Sigma\mathcal S] \xrightarrow{\sim} \mathrm{mod}\,\mathcal S.
\]
Within this quotient, presilting subcategories correspond to \(\tau\)-rigid pairs, and silting subcategories correspond to support \(\tau\)-tilting pairs; under a Krull–Schmidt uniqueness condition on decompositions in \(\mathcal S * \Sigma\mathcal S\), the correspondence is bijective on both presilting and silting subcategories [1311.4891].

Under the standard algebraic hypotheses that \(\mathcal T\) is Krull–Schmidt, \(k\)-linear, Hom-finite, and \(s\) is a basic silting object with \(\mathcal S=\operatorname{add}(s)\), one obtains an algebra \(E=\operatorname{End}_{\mathcal T}(s)\) and an equivalence \(\mathrm{mod}\,\mathcal S\simeq \mathrm{mod}\,E\). The functor \(\mathcal T(s,-)\) then induces a bijection between basic silting objects of \(\mathcal T\) lying in \(\mathcal S * \Sigma\mathcal S\), that is, two-term silting objects, and basic support \(\tau\)-tilting \(E\)-modules [1311.4891]. In the classical case \(\mathcal T=K^b(\mathrm{proj}\,\Lambda)\), this recovers the Adachi–Iyama–Reiten correspondence between basic 2-term silting complexes and basic support \(\tau\)-tilting \(\Lambda\)-modules [1311.4891].

The torsion-theoretic side is equally rigid. For an essentially small additive category \(\mathcal C\), a torsion class in \(\mathrm{Mod}\,\mathcal C\) is a full subcategory closed under factor modules and extensions, and it is finitely generated if it has the form \(\mathrm{Fac}\,\mathcal M\) for some \(\mathcal M\subseteq \mathrm{mod}\,\mathcal C\). Theorem 5.1 identifies support \(\tau\)-tilting pairs \((\mathcal M,\mathcal E)\) with finitely generated torsion classes \(\mathcal T\subseteq \mathrm{Mod}\,\mathcal C\) such that every finitely generated projective \(\mathcal C\)-module has a left \(\mathcal P(\mathcal T)\)-approximation, and moreover
\[
\mathcal P(\mathrm{Fac}\,\mathcal M)=\mathcal M.
\]
Composed with the two-term silting correspondence, this yields the chain
\[
\text{two-term silting}
\longleftrightarrow
\text{support }\tau\text{-tilting}
\longleftrightarrow
\text{finitely generated torsion classes}
\]
in the appropriate module category [1311.4891].

For arbitrary rings, silting modules make the same bridge explicit. A silting module \(M\) is one for which there exists a projective presentation \(\sigma\) with \(\mathrm{Gen}(M)=\mathcal D_\sigma\), and equivalence classes of 2-silting complexes correspond bijectively to equivalence classes of silting modules via \(H^0\) [1405.2531]. Over finite-dimensional algebras, partial silting coincides with \(\tau\)-rigidity and silting coincides with support \(\tau\)-tilting [1405.2531].

## 4. Mutation, completions, reduction, and finiteness

Two-term silting is unusually well adapted to mutation. For a module-finite algebra \(A\) over a commutative noetherian ring, any 2-term presilting complex admits both Bongartz and co-Bongartz completions, because \(\mathrm{add}\,P\) is functorially finite in \(K^b(\mathrm{proj}\,A)\). If \(K^b(\mathrm{proj}\,A)\) is Krull–Schmidt and \(P\) 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 [2006.01677]. This extends the finite-dimensional mutation picture to noetherian algebras.

Reduction results are equally strong. If \((R,\mathfrak m)\) is complete local noetherian, \(A\) is module-finite over \(R\), and \(I\subseteq \mathfrak m A\) is a two-sided ideal, then the reduction map \(P\mapsto \overline P=(A/I)\otimes_A P\) induces isomorphisms of posets
\[
2\text{-}\mathrm{silt}\,A \xrightarrow{\sim} 2\text{-}\mathrm{silt}(A/I),
\qquad
\mathrm{m\!-\!silt}\,A \xrightarrow{\sim} \mathrm{m\!-\!silt}(A/I),
\qquad
\mathrm{f\!-\!tors}\,A \xrightarrow{\sim} \mathrm{f\!-\!tors}(A/I),
\]
compatible with \(H^0\) and \(\mathrm{Fac}\) [2006.01677]. This makes the 2-term silting theory of many noetherian algebras accessible through finite-dimensional quotients.

Finiteness questions are naturally expressed through \(\tau\)-tilting finiteness. An algebra \(A\) is \(\tau\)-tilting-finite exactly when it has only finitely many 2-term silting objects in \(K^{b}(\mathrm{proj}\,A)\), up to isomorphism [2002.08534]. 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 \(\tau\)-tilting modules, hence of 2-term silting complexes [2002.08534]. It also records that representation-finite algebras are \(\tau\)-tilting-finite, while the converse fails in general, so finiteness of two-term silting is strictly weaker than representation-finiteness globally [2002.08534].

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 [2002.08534].

## 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 \(\mathcal C\) is a Hom-finite Krull–Schmidt triangulated category and \(\mathcal R\subseteq \mathcal C\) is rigid, a subcategory is called two-term with respect to \(\mathcal R\) when it lies in \(\mathcal R * \mathcal R[1]\). Two-term \(\mathcal R[1]\)-rigid subcategories correspond bijectively to \(\tau\)-rigid subcategories of \(\mathrm{mod}\,\mathcal R\), and two-term weak \(\mathcal R[1]\)-cluster tilting subcategories correspond bijectively to support \(\tau\)-tilting subcategories of \(\mathrm{mod}\,\mathcal R\). When \(\mathcal R\) is silting, the two-term weak \(\mathcal R[1]\)-cluster tilting subcategories are precisely the two-term silting subcategories of Iyama–Jørgensen–Yang [1811.12588].

Second, the theory persists for non-positive dg algebras. If \(A\) is a non-positive dg algebra with finite-dimensional total cohomology, then \(\mathrm{per}(A)\) is Krull–Schmidt with silting object \(A\), and the \(d=1\) case of the extended-heart framework yields poset isomorphisms
\[
2\text{-}\mathrm{silt}\,A
\;\cong\;
\operatorname{h.cotors}\,\operatorname{add}(A)^{[0,1]}
\;\cong\;
\mathrm{f.}s\text{-}\operatorname{-tors}\,\mathcal H_A^1,
\]
where \(\mathcal H_A^1\simeq \mathrm{mod}\,H^0(A)\) is the heart of the standard t-structure [2606.13508]. In the same dg setting, 2-term silting objects in \(\mathrm{per}(A)\) correspond to basic \(\tau\)-tilting pairs in \(\mathrm{mod}\,H^0(A)\), and \(\tau\)-cluster morphism categories can be constructed from 2-term presilting objects via silting reduction [2110.03472].

The Gorenstein analogue replaces projectives by Gorenstein-projectives and \(K^b(\mathrm{proj}\,A)\) by the Gorenstein derived category \(D_{gp}^b(A)\). A 2-term complex
\[
G^\bullet: G_1 \xrightarrow{d^1} G_0
\]
with \(G_i\in \mathrm{Gproj}\,A\) is 2-term Gorenstein silting if it is rigid in \(D_{gp}^b(A)\) and generates \(K^b(\mathrm{Gproj}\,A)\). For a finite-dimensional Gorenstein algebra of finite CM-type, such complexes are equivalent to Gorenstein silting modules via \(H^0(G^\bullet)\), partial 2-term Gorenstein silting corresponds to \(\tau_G\)-rigidity, and the complex induces both a torsion pair in \(\mathrm{mod}\,A\) and a t-structure on \(D_{gp}^b(A)\) whose heart is equivalent to \(\mathrm{mod}\,\operatorname{End}(G^\bullet)^{\mathrm{op}}\) [2209.00520].

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 \(\Lambda_\Delta\) of non-Dynkin type, two canonical families of two-term tilting complexes, \(P_w\) and \(R_w\), are constructed from the ideals \(I_w\) indexed by the Coxeter group \(W_\Delta\). In affine type, every two-term silting complex is actually tilting, and every two-term tilting complex is either \(P_w\) or \(R_w\); moreover the cones \(C(P_w)\) are the closures of the Weyl chambers, while \(C(R_w)=-C(P_w)\) [1908.02424]. 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 [1506.03649]. In Dynkin type, this can be made algorithmic: for a hereditary path algebra \(A=KQ\), every basic 2-term silting complex has the form \(M\oplus P[1]\), where \(P=eA\) for an idempotent \(e\) and \(M\) is a basic tilting \(A/(e)\)-module, and this yields explicit computations of all basic 2-term silting complexes and their endomorphism algebras in small \(A_n\) and \(D_n\) cases [2106.00970].

The homological invariants of endomorphism algebras of 2-term silting complexes are more delicate than in classical tilting. If \(\mathrm{gl.dim}\,A\le 2\) and \(P\) is a 2-term silting complex, then
\[
\mathrm{gl.dim}\,\operatorname{End}_{D^b(A)}(P)\le 7.
\]
For each \(n>2\), however, there exists an algebra \(A\) with \(\mathrm{gl.dim}\,A=n\) admitting a 2-term silting complex \(P\) such that \(\mathrm{gl.dim}\,\operatorname{End}_{D^b(A)}(P)\) is infinite [1605.09255]. Under the additional hypothesis \(\mathrm{pd}_A H^0(P)\le 1\), one has the more uniform bound
\[
\mathrm{gl.dim}\,\operatorname{End}_{D^b(A)}(P)\le 2\,\mathrm{gl.dim}\,A+2
\]
[1605.09255]. 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 \(P\) is a separating 2-term silting complex with \(\mathrm{id}_A X\le 1\) for each \(X\in F(P)\), then
\[
\mathrm{rep.dim}\,\operatorname{End}_{D^b(A)}(P)=\mathrm{rep.dim}\,A.
\]
If \(P\) is both separating and splitting, then \(H^0(P)\) is a splitting and separating tilting \(A/\operatorname{ann}_A(P)\)-module, and
\[
\mathrm{rep.dim}\,\operatorname{End}_{A}(H^0(P))
=
\mathrm{rep.dim}\,A/\operatorname{ann}_A(P).
\]
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.

Source: https://www.emergentmind.com/topics/2-term-silting-complexes