Complete Tilted Systems

• Complete tilted systems are thoroughly classified structures in tilting theory that integrate universal localizations, Prüfer modules, and Lukas modules as building blocks.
• They underpin stability conditions in derived categories and moduli constructions, enabling precise analyses of wall-crossing phenomena.
• These systems extend to cluster tilting and gentle algebras, revealing explicit bounds on complements and demonstrating deep duality and classification properties.

A complete tilted system refers to a thoroughly classified structure arising in the context of tilted or tilting-theoretic methods across several mathematical domains, particularly in the representation theory of algebras and in the paper of stability conditions via abelian and derived categories. The terminology encompasses both classical tilting modules over certain algebras and their analogues in modern categorical settings, such as hearts of t-structures generated by tilting with respect to torsion pairs. The paradigm of "completeness" pertains both to the classification (no further indecomposable summands possible) and to the realization of these systems as building blocks for large or infinite-dimensional objects, especially over hereditary or gentle algebras, and in the theory of stability conditions on derived categories.

1. Tilting Theory and Complete Tilted Systems in Tame Hereditary Algebras

A finite-dimensional $k$-algebra $R$ is hereditary if $\operatorname{gl.dim} R \le 1$, and tame if, for each $d$, all but finitely many isomorphism classes of indecomposable $d$-dimensional modules occur in one-parameter families. The structure of $\operatorname{mod} R$ decomposes into preprojective, regular (as a union of “tubes”), and preinjective components. For tame hereditary algebras, the theory of tilting modules provides a complete classification of infinite-dimensional tilting modules.

A module $T$ is tilting if:

• $\operatorname{proj\,dim} T \leq 1$,
• $\operatorname{Ext}_R^1(T, T^{(\kappa)}) = 0$ for all cardinals $\kappa$,
• There exists an exact sequence $$0 \rightarrow R \rightarrow T_0 \rightarrow T_1 \rightarrow 0$$ with $T_0, T_1 \in \operatorname{Add}(T)$.

The central classification theorem specifies that every infinite-dimensional tilting module over a tame hereditary algebra $R$ is, up to equivalence, uniquely of the form

$T \cong R_\mathcal{U} \oplus (R_\mathcal{U} / R),$

where $\mathcal{U}$ is a union of tubes of the Auslander–Reiten quiver of $R$, $R_\mathcal{U}$ denotes the universal localization of $R$ at $\mathcal{U}$, and $R_\mathcal{U} / R$ is a direct sum of the corresponding Prüfer modules for tubes in $\mathcal{U}$. The tilting class $\operatorname{Gen} T$ is thus $\mathcal{U}^\perp$ in $\operatorname{Mod} R$.

In special cases such as the Kronecker algebra (the prototypical tame hereditary algebra), all infinite-dimensional tilting modules are constructed this way except for the Lukas tilting module $L$, characterized by the property that $\operatorname{Gen} L = \mathfrak{p}^\perp$ (all modules without indecomposable preprojective summands) (Hügel et al., 2010).

2. Structure of Complete Tilted Systems: Building Blocks and Decomposition

The principal structural components of complete tilted systems in the context of tame hereditary algebras are:

• Universal localizations $R_\mathcal{U}$: $R$-modules obtained by inverting maps associated to a set of quasi-simples $\mathcal{U}$ across chosen tubes.
• Prüfer modules $U[\infty]$: Direct limits along the rays in given tubes, constituting the $\mathcal{U}$-indexed summands in $R_\mathcal{U}/R$.
• Lukas module $L$: A countably generated, infinite-dimensional tilting module whose tilting class consists of all modules with no preprojective summands.

For arbitrary tame hereditary algebras, a "branch module" $Y$ captures the finite-dimensional summands (arising only finitely many ways and from non-homogeneous tubes). The infinite-dimensional part is always built from universal localizations, Prüfer modules, and possibly Lukas modules, leading to the canonical form

$$T(Y, \mathcal{A}) = Y \oplus R_\mathcal{V} \oplus (R_\mathcal{V} / R),$$

parametrized by branch module $Y$ and a subset $\mathcal{A}$ of the tubes, resulting in a bijection between $(Y, \mathcal{A})$ and the set of equivalence classes of large tilting modules (Hügel et al., 2010).

3. Complete Tilted Systems in Derived and Stability-Theoretic Settings

In the formalism of derived categories and stability conditions, "complete tilted systems" arise as canonical objects within hearts of tilted t-structures. For coherent systems over an integral curve $C$, the abelian category of coherent systems $(E, V, \varphi)$ admits torsion pairs and corresponding tilted hearts $\mathcal{H}_\alpha^\beta$. Objects in these hearts are termed tilted systems, and complete tilted systems are those for which the underlying system is complete, i.e., $\varphi: V \cong H^0(E)$ is an isomorphism (Jardim et al., 16 Nov 2025).

For these, stability under three-parameter stability conditions (central charge $Z_{\alpha, \beta, \gamma}$) is determined by a precise slope inequality:

• $E[1]$ for complete $E$ is $Z$-stable for all $\gamma > 1$,
• For incomplete $E$, a unique critical value $\gamma_0$ controls wall-crossing: $E[1]$ is stable for $\gamma > \gamma_0$ and unstable for $\gamma < \gamma_0$.

This framework enables a wall–chamber decomposition for moduli problems of tilted systems and establishes the geometric nature of these stability conditions in the sense of Bridgeland–Macrì.

4. Cluster-Tilting Analogs and Higher Calabi–Yau Categories

The theory of complete (almost-complete) cluster tilting objects in higher Calabi–Yau triangulated categories generalizes these notions to broader categorical settings. In generalized $m$-cluster categories arising from (strongly) $(m+2)$-Calabi–Yau dg algebras, every almost-complete $m$-cluster tilting object (i.e., with all but one summand) admits exactly $m+1$ complements, and these objects obey periodicity properties under silting mutation. The classification of complements and the construction of completions (via silting mutation or truncations of minimal cofibrant resolutions) parallels the "completion" phenomena for classical tilted systems (Guo, 2012).

5. Tilting Completion for Gentle Algebras and Bounds on Complements

In the case of gentle algebras of rank $n$, every almost-tilting module (pre-tilting module with $n-1$ summands and self-orthogonality) is partial-tilting and can be completed to a full tilting module. Uniquely for gentle algebras, the number of nonisomorphic complements is bounded: any almost-tilting module admits at most $2n$ complements. The proof employs the surface model (zigzag arcs on a marked surface), where completion corresponds to cuts and glueings in the surface, revealing that no further indecomposable direct summands can be adjoined beyond $n$ (Chang, 18 Dec 2024).

6. Duality and Structural Corollaries

The classification of cotilting modules (Buan–Krause) is recovered bilaterally: any infinite-dimensional cotilting module with an infinite indecomposable summand splits as a product of finite-dimensional modules (from non-homogeneous tubes), some number of adic modules, some number of Prüfer modules, and copies of the generic module. Each tube of rank $r$ contains exactly $r$ non-isomorphic indecomposable cotilting summands.

Further, the lattice of large tilting classes admits extremal elements: the largest is $\mathfrak{p}^\perp = \operatorname{Gen} L$ and the smallest is $\tau^\perp$, corresponding to sets built from Prüfer modules and the generic module. Noetherian and pure-injective tilting modules are also characterized in this structural hierarchy (Hügel et al., 2010).

7. Broader Implications and Open Directions

Complete tilted systems are foundational in the construction and classification of module categories, derived t-structures, and families of stability conditions. They are central to explicit moduli constructions, wall–crossing phenomena, and the global structure of stability manifolds. Open directions include describing the finer moduli of semistable objects under these new tilted stability conditions, elucidating wall-crossing formulae explicitly, and extending the classification to decorated sheaf categories and other algebraic and geometric settings (Jardim et al., 16 Nov 2025).

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Summary Table: Structural Components of Complete Tilted Systems

Setting	Building Blocks	Completion/Classification
Tame hereditary algebra	Universal localization, Prüfer, Lukas modules, branch modules	Complete decomposition via $(Y, \mathcal{A})$ parameters; unique up to equivalence (Hügel et al., 2010)
Gentle algebra	Zigzag arcs (surface model)	Every almost-tilting is completable; at most $2n$ complements (Chang, 18 Dec 2024)
Derived categories (curves)	Torsion pairs, tilted hearts	Complete tilted systems stable in all Bridgeland regions as per completeness of underlying system (Jardim et al., 16 Nov 2025)
Cluster category	Silting mutation, AR angles	Each almost-complete cluster-tilting object has $m+1$ complements (Guo, 2012)

The theory of complete tilted systems unifies tilting classification in module categories, the behavior of stability conditions for coherent systems, and the combinatorics of cluster and gentle algebras, with deep interconnections to derived and higher categorical settings.