Partial Silting Modules

• Partial silting modules are A-modules with a fixed projective presentation that satisfy homological conditions to form torsion classes.
• They generalize classical tilting modules by inducing torsion pairs and providing left approximations, often via Bongartz completion.
• They underpin universal localization and link with t-structures and silting mutations in derived categories, enhancing structural classification.

A partial silting module is an $A$-module $T$ endowed with a specific projective presentation and associated torsion-theoretic and approximation properties that generalize classical tilting theory to broader contexts. This concept is central in the structural paper of module categories, torsion classes, universal localization, $t$-structures in derived categories, and the combinatorics of tilting and silting mutations. The notion arises as both a weakening of silting modules—requiring only the vanishing of certain derived homomorphisms and the existence of left approximations—and as the key object in parametrizing and “lifting” torsion data in module and derived categories, with direct analogues and implications in the context of silting objects and subcategories in triangulated categories.

1. Definitions, Formulations, and Foundational Properties

Let $A$ be a (not necessarily commutative) ring and $T$ an $A$-module with a fixed projective presentation:

$\sigma: P_1 \to P_0 \to T \to 0.$

Define the class:

$$\mathcal{D}_\sigma = \{ X \in \operatorname{Mod}(A) \mid \operatorname{Hom}_A(\sigma, X) \text{ is surjective} \}.$$

$T$ is called a partial silting module (with respect to $\sigma$) if:

• (S1) $\mathcal{D}_\sigma$ is a torsion class;
• (S2) $T \in \mathcal{D}_\sigma$.

$T$ is called a silting module if $\mathcal{D}_\sigma = \operatorname{Gen}(T)$, i.e., the torsion class generated by $T$ coincides with $\mathcal{D}_\sigma$ (Hügel et al., 2014). If $\sigma$ is injective, the notions of partial silting and partial tilting coincide.

Partial silting modules generalize both tilting modules and support $\tau$-tilting modules and are, for any projective presentation, modules that satisfy enough self-orthogonality to induce torsion pairs and approximation sequences, yet may not generate the entire torsion class or derived category.

Key formulas:

• Silting: $\mathcal{D}_\sigma = \operatorname{Gen}(T)$.
• For two-term complexes, $T$ is silting if and only if the associated complex $\sigma$ is a two-term silting complex and $(\mathcal{D}_\sigma, T^\perp)$ is a torsion pair.
• Bongartz complement (see §3): Any partial silting $T$ is a direct summand of a silting module $T'$ with $\operatorname{Gen}(T') = \mathcal{D}_\sigma$.

2. Torsion Theory, Approximations, and the Bongartz Completion

Every partial silting module $T$ is associated with a torsion pair $(\operatorname{Gen}(T), T^\perp)$ in $\operatorname{Mod}(A)$, with $$\operatorname{Gen}(T) \subseteq \mathcal{D}_\sigma \subseteq T^\perp$$. If $T$ is silting, then $\mathcal{D}_\sigma$ is definable, closed under extensions, coproducts, direct limits, and pure quotients (Hügel, 2018). Left $\operatorname{Gen}(T)$-approximations are provided by surjective morphisms $A \to T_0$ factoring through $\sigma$.

Bongartz completion: Every partial silting module $T$ is a direct summand of a silting module $T' = T \oplus M$ such that $\operatorname{Gen}(T') = \mathcal{D}_\sigma$ [(Hügel et al., 2014), Theorem 3.15]. This is constructed via pushout diagrams starting from a $T$-preenvelope of $A$: $$\begin{array}{ccc} P_1 & \xrightarrow{\sigma} & P_0 \ \downarrow & & \downarrow \ A & \to & M \end{array}$$ with $T' = M \oplus \operatorname{Coker}(A \to M)$ silting and $\operatorname{Gen}(T') = \operatorname{Gen}(T'')$ for the torsion class generated by the partial silting $T$.

This completion links directly with the classical Bongartz complement for partial tilting modules and enables passage from partial objects to unique extensions with maximally defined torsion-theoretic properties.

3. Partial Silting Modules and Universal Localisation

Partial silting modules encode universal localisation data via their projective presentations. Any partial silting module $T$ (via $\sigma$) defines a bireflective subcategory:

$$\mathcal{Y}_\sigma = \{ X \mid \operatorname{Hom}_A(\sigma, X) \ \text{is bijective} \}$$

which is the essential image of a ring epimorphism $f: A \to B$ (a "silting epimorphism") (Hügel et al., 2015, Marks et al., 2016). For hereditary or perfect rings, all torsion classes generated by silting modules arise as such images, and for hereditary rings, all homological epimorphisms (universal localisations) are parametrized by minimal silting modules (Hügel et al., 2015).

In commutative rings, every universal localisation (inverting maps between countably generated projectives) arises from a partial silting module, and conversely each partial silting module yields a universal localisation (Marks et al., 2016, Šťovíček, 3 Oct 2025). It is shown that in the commutative context, every flat ring epimorphism is a silting epimorphism and vice versa.

Summary of relationships: | Partial Silting Module $T$ | Associated Epimorphism $f: A \to B$ | Universal Localisation | |---------------------------|--------------------------------------------|-------------------------------| | via $\sigma$: $P_1 \to P_0$ | $X \mapsto \operatorname{Hom}_A(\sigma, X)$ | inverts $\sigma$ in $\operatorname{Mod}(A)$ |

4. Role in Mutation, $t$-Structures, and Derived Categories

Partial silting modules correspond to partial silting objects (or almost complete silting objects) in triangulated categories. In the derived category setting, a partial silting object $M$ satisfies:

• $\operatorname{Hom}_\mathcal{T}(M, M[i]) = 0$ for $i > 0$ (semi-selforthogonality);
• $M$ is contained in the subcategory it generates.

Silting mutation (Aihara et al., 2010) is a process of replacing an indecomposable summand in a silting object via a triangle:

$M \to D \to N \to M[1]$

where $D$ is a covariantly finite subcategory of $M$, yielding a new silting object (or partial silting module) via $\mu^+(M;D) = \operatorname{add}(D \cup \{N\})$. For almost complete (partial) silting objects, iterated mutation enables reaching any completion, and the set of silting objects forms a connected mutation graph (silting quiver).

In compactly generated triangulated categories (including derived module categories with coproducts), every localizing subcategory is generated by a partial silting object and thus admits an induced silting $t$-structure (Hügel et al., 2019). The aisle of the $t$-structure associated to a partial silting set is the Milnor (homotopy) colimit of iterated cones with associated objects in $\operatorname{Add}(T)[n]$ (Nicolas et al., 2015).

Partial silting sets form a bridge to the theory of $t$-structures: equivalence classes of partial silting sets correspond to $t$-structures whose heart has a projective generator and is equivalent to a module category on some abelian category (Nicolas et al., 2015). When compactness is involved, partial silting objects correspond to nondegenerate $t$-structures with certain boundedness and finiteness conditions.

5. Relationships to Gorenstein and Wakamatsu Silting

Partial silting modules are generalized further in the theory of Gorenstein silting modules (Gao et al., 2021, Gao et al., 2022), where presentations are by Gorenstein-projective modules and the relevant torsion class is closed under G-exact sequences. For finite-dimensional algebras of finite CM-type, partial Gorenstein silting modules are in bijection with $\tau_G$-rigid modules.

In the derived category, the notion of Wakamatsu-silting complexes further encompasses partial silting modules. An object is Wakamatsu-silting if it is semi-selforthogonal and the regular module $R$ lies in the class generated by positive shifts. Every (partial) silting module yields a Wakamatsu-silting complex as a stalk complex (Wei, 2013).

6. Applications, Examples, and Structural Classifications

Partial silting modules form the basis for connecting varieties of algebraic and module-theoretic structures:

• Classification of torsion theories: Over Noetherian, hereditary, or perfect rings, definable torsion classes, enveloping torsion classes, and homological localising subcategories are each generated by a (minimal) silting module, and every partial silting module can be completed to a silting module giving rise to a torsion pair (Breaz et al., 2016, Hügel, 2018).
• Ring epimorphisms and localisations: Parametrisation of all homological ring epimorphisms and universal localisations is achieved via partial silting modules. In the commutative case, the classes of flat and silting epimorphisms coincide, tightly linking partial silting modules with Gabriel filters and the geometry of $\operatorname{Spec}(A)$ (Šťovíček, 3 Oct 2025).
• Gluing and recollement: Partial silting sets and modules can be glued along recollements to build silting modules or objects in the central term, controlling torsion theories and $t$-structures across extensions and quotient rings, as in constructions for upper-triangular matrix rings (Saorín et al., 2018, Gao et al., 2022).
• Examples: For the Kronecker algebra, the module $T = P_2/\operatorname{soc} P_2$ is silting but not tilting (Hügel et al., 2014). For a ring $R$ and idempotent ideal $I$, $R/I$ is silting if and only if $I$ is the trace of a projective module (at least for semiperfect $R$) (Argudin-Monroy et al., 26 Nov 2024).

7. Summary Table: Key Criteria and Relationships

Module Type	Criteria	Associated Structure
Partial silting	$P_1 \to P_0 \to T,\ \mathcal{D}_\sigma$ torsion, $T\in \mathcal{D}_\sigma$	Torsion pair, approximations
Silting	Partial silting + $\mathcal{D}_\sigma = \operatorname{Gen}(T)$	Silting class (definable, extension-closed)
Bongartz completion	Partial silting $T$ is direct summand in silting $T'$ with $\operatorname{Gen}(T') = \mathcal{D}_\sigma$	All approximations retained
Silting epimorphism	Partial silting $T\rightsquigarrow$ universal localisation (inverts $\sigma$)	Ring epimorphism, universal property
Gluing	Partial silting modules from recollement outer terms glued to central term	New silting in recollement central category

Partial silting modules thus function as fundamental building blocks in the theory of tilting, torsion pairs, localisations, and triangulated structures, with applications spanning classification, approximation theory, and modular representation theory. Their paper unifies and extends classical themes around generation, mutation, and the architecture of module and derived categories (Hügel et al., 2014, Buan et al., 2015, Marks et al., 2016, Hügel, 2018, Argudin-Monroy et al., 26 Nov 2024, Šťovíček, 3 Oct 2025).