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Tilting Completion and Self-Orthogonality Modules

This paper constructs specific finite-dimensional quasi-hereditary algebras to provide negative answers to two tilting-completion questions and disprove two major conjectures regarding self-orthogonal modules, while also establishing the equivalence between the Self-orthogonal Wakamatsu-tilting Conjecture and the Self-orthogonal Faithful Conjecture.

Original authors: Wen Chang, Quanyu Tang

Published 2026-08-24

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CC BY 4.0

Original authors: Wen Chang, Quanyu Tang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern mathematics, there is a field dedicated to understanding how complex structures are built from simpler, fundamental pieces. Imagine a universe of shapes and connections where the goal is to see how different parts fit together to form a whole. For decades, mathematicians have developed a powerful set of tools to map these connections, treating them like a language that can describe everything from the symmetries of crystals to the behavior of data in networks. At the heart of this language are special building blocks called modules. These are not physical objects, but rather abstract collections of rules and relationships that can be combined, split, and rearranged. A central question in this field has long been whether every valid, well-constructed partial arrangement of these blocks can be extended to form a complete, perfect structure. It was a comforting thought that if you had a solid foundation, you could always build the rest of the house.

For a long time, this idea held true in many specific cases, leading researchers to believe it might be a universal law of this mathematical world. The question was simple: if you have a collection of these building blocks that fits together perfectly without internal conflict, can you always find the missing pieces to complete the set? This belief guided the construction of theories that linked different areas of mathematics, acting as a bridge between algebra and geometry. The confidence in this rule was so strong that it became a standard assumption, a quiet expectation that the mathematical universe was orderly and that every good start could be finished.

However, a new study by Wen Chang and Quanyu Tang has shattered this expectation. The researchers have constructed specific examples where the answer is a definitive no. They have found situations where a collection of blocks fits together perfectly, satisfying all the necessary conditions to be a valid start, yet it is impossible to find the missing pieces to complete the structure. This is not a case of the pieces being lost or the math being too difficult to solve; the impossibility is built into the very nature of the arrangement. The team created two distinct types of these impossible scenarios using finite-dimensional algebras, which are mathematical systems with a limited number of rules. In the first scenario, they built a system where a collection of blocks was as large as the system itself, yet it could not be part of a complete set. In the second, they created a system where the collection was just one piece short of being the full size, yet it still could not be completed.

The significance of this discovery goes beyond just finding a missing puzzle piece. The researchers showed that these impossible structures also disprove two major conjectures that had been widely accepted by the mathematical community. One conjecture suggested that any self-contained, conflict-free collection of blocks of a certain size must be "faithful," meaning it interacts with every part of the system. The other suggested that such collections could always be completed into a specific type of perfect structure. By proving that these collections exist without being faithful and without being completable, the authors have shown that the mathematical landscape is more rugged and less predictable than previously thought. They demonstrated that the two conjectures were actually two sides of the same coin; if one fails, the other must fail as well, and they have proven that both have failed.

To achieve this, the team started with a complex geometric shape known as a rational surface, which can be thought of as a smooth, curved sheet. On this surface, they identified a specific sequence of line bundles, which are like layers of fabric wrapped around the shape. A previous mathematician had shown that these layers formed a sequence that was almost complete but had a gap. The new researchers took this sequence and translated it into the language of their algebraic blocks. They used a clever construction involving a "one-point extension," which is a method of adding a new dimension or a new rule to the system without breaking the existing connections. This process allowed them to transfer the properties of the geometric shape into the algebraic world, creating the exact counterexamples they needed.

The first example they built involved a system where the number of building blocks matched the number of fundamental types available in that system. In a perfect world, this would guarantee that the blocks could form a complete, self-contained structure. But in their construction, the blocks, while fitting together without conflict, were trapped in a configuration that prevented them from ever being part of a larger, complete set. The second example was even more striking: a system where the blocks were just one short of the total number of types. Intuitively, one might think that being so close to the full set would make completion easy. Yet, the researchers proved that even in this "almost-full" state, the blocks could not be completed. The missing piece simply did not exist within the rules of the system.

These findings have immediate consequences for how mathematicians understand the limits of their theories. The study confirms that the rules governing these algebraic structures are more subtle than previously believed. It shows that having a large, conflict-free collection of blocks is not enough to ensure that the collection can be expanded. The researchers also showed that the failure of one major conjecture automatically implies the failure of the other, linking two previously separate ideas into a single, unified truth. This means that the entire framework built on these assumptions needs to be revised. The work does not just add a new fact to the list; it removes a foundational pillar that many had relied upon.

The construction of these examples required a deep understanding of how different mathematical worlds connect. The team used a method that treated a sequence of mathematical objects as a single, complex module. They then applied a transformation that preserved the essential properties of the original sequence while changing the environment in which it lived. This allowed them to take a known geometric impossibility and turn it into an algebraic one. The result is a rigorous proof that the "tilting completion" question, which asks if every partial structure can be finished, has a negative answer in the general case. The paper does not suggest that this happens often or that it is a common occurrence; rather, it proves that it is possible, and that the possibility is enough to change the theory.

In the end, the work of Chang and Tang serves as a reminder that in mathematics, even the most intuitive rules can have exceptions. The idea that a good start guarantees a good finish is a comforting one, but the universe of algebraic structures does not always follow it. By finding these specific, concrete examples of failure, the researchers have provided a clearer, more accurate map of the territory. They have shown that the path to a complete structure is not always open, even when the starting point looks perfect. This discovery will likely lead to a re-evaluation of many existing theories and the development of new tools to navigate the gaps that have been revealed. The mathematical community now knows that the landscape is not as smooth as it once seemed, and that the search for completeness must account for the possibility of dead ends.

Technical Summary of "Tilting Completion and Full-Rank Self-Orthogonal Modules"

Problem Statement
The paper addresses two fundamental "tilting-completion" questions for finite-dimensional algebras, specifically concerning whether pretilting modules can be completed to tilting modules.

The Full-Rank Question ( $C_n$ ): Let $A$ be a finite-dimensional algebra with $n$ simple modules. If $M$ is a basic pretilting module with $|M| = n$ (i.e., it has the maximal possible number of indecomposable summands), must $M$ be a direct summand of a tilting module? In classical tilting theory (projective dimension $\le 1$ ), Bongartz's theorem affirms this. The question remains whether this holds for generalized tilting modules (arbitrary finite projective dimension).
The Almost-Full-Rank Question ( $C_{n-1}$ ): If $M$ is a basic pretilting module with $|M| = n-1$ (an "almost-tilting" module), must it admit a tilting complement?

Additionally, the paper investigates the Self-orthogonal Faithful Conjecture (proposed by Chen, Li, Zhang, and Zhao), which asserts that every self-orthogonal module $T$ with $|T| = n(A)$ is faithful. It also examines Enomoto's Self-orthogonal Wakamatsu-tilting Conjecture, which posits that every self-orthogonal module $T$ with $|T| = n(A)$ is Wakamatsu tilting.

Methodology
The authors construct counterexamples using a multi-step strategy that bridges derived categories and module categories:

Derived-to-Module Transfer: The construction begins with a non-full exceptional collection of maximal length on a rational surface (specifically, the blow-up of $\mathbb{P}^2$ at ten points), as established by Krah. Kalck observed that this yields a full-rank presilting complex $P^\bullet$ over a finite-dimensional algebra $A$ such that $\text{thick}(P^\bullet) \neq K^b(\text{proj } A)$ .
Tensor Product Construction: To transfer this complex to a module category, the authors define a new algebra $B = A \otimes_k R$ , where $R$ is the radical-square-zero path algebra of a linearly oriented quiver. They utilize an equivalence between $B$ -modules and complexes of $A$ -modules supported in a specific interval.
Ext-Homotopy Comparison: A key technical tool is the comparison between extension groups over $B$ and morphisms in the homotopy category of complexes over $A$ . Specifically, they prove that for a complex $P^\bullet$ viewed as a $B$ -module, $\text{Ext}^r_B(P^\bullet, P^\bullet) \cong \text{Hom}_{K^b}(P^\bullet, P^\bullet[r])$ . This allows them to preserve the self-orthogonality and projective dimension properties of the complex in the module setting.
One-Point Extensions: To generate counterexamples for the Faithful Conjecture and the almost-full-rank case, the authors employ one-point extension constructions.
- For the Faithful Conjecture, they use a non-monomorphic approximation of the full-rank module to construct a one-point extension $\Gamma$ . This forces the resulting module to be self-orthogonal and full-rank but non-faithful.
- For the Almost-Full-Rank case, they use a regular one-point extension to embed the full-rank counterexample into a larger algebra, increasing the algebra's rank by one while keeping the module's rank constant, thereby creating an almost-full-rank module with no completion.

Key Contributions and Results

Negative Answer to the Full-Rank Question ( $C_n$ ): The authors construct a finite-dimensional basic connected quasi-hereditary algebra $B$ and a faithful basic pretilting module $M$ with $|M| = n(B)$ that is not a direct summand of any tilting module (Theorem 1.2). This provides the first counterexample to the rank-tilting-completion property in the general finite-dimensional setting, even within the class of quasi-hereditary algebras.
Refutation of Enomoto's Conjecture: The module $M$ constructed above is shown not to be Wakamatsu tilting. Consequently, Enomoto's conjecture (that every self-orthogonal module of full rank is Wakamatsu tilting) is false, even for faithful basic modules over connected quasi-hereditary algebras (Corollary 5.2).
Negative Answer to the Self-orthogonal Faithful Conjecture: By applying a one-point extension reduction to the full-rank counterexample, the authors construct a basic self-orthogonal module $T$ with $|T| = n(\Gamma)$ such that $\text{Ann}_\Gamma(T) \neq 0$ (Theorem 1.3). This disproves the conjecture that full-rank self-orthogonal modules must be faithful.
Equivalence of Conjectures: The paper establishes that, as universal assertions over all finite-dimensional algebras, Enomoto's Self-orthogonal Wakamatsu-tilting Conjecture and the Self-orthogonal Faithful Conjecture are equivalent (Corollary 6.5). The failure of one implies the failure of the other via the one-point extension reduction.
Negative Answer to the Almost-Full-Rank Question ( $C_{n-1}$ ): The authors construct a finite-dimensional basic connected quasi-hereditary algebra $\Lambda$ and a basic pretilting module $L$ with $|L| = n(\Lambda) - 1$ that admits no tilting completion (Theorem 1.4). This demonstrates that the failure of tilting completion persists even when the module is missing only one summand from the full rank.

Significance
The paper resolves long-standing open questions regarding the completion of pretilting modules in the generalized setting. While classical tilting theory (Bongartz's theorem) guarantees completion for projective dimension $\le 1$ , this work demonstrates that for generalized tilting modules, the numerical condition $|M| = n(A)$ (or even $n(A)-1$ ) is insufficient to guarantee the existence of a tilting complement.

Furthermore, the results clarify the landscape of self-orthogonal modules by showing that full-rank self-orthogonal modules need not be Wakamatsu tilting nor faithful. The equivalence established between the two major conjectures regarding self-orthogonal modules unifies these problems, suggesting that their resolution (or lack thereof) is intrinsic to the structure of finite-dimensional algebras rather than being an artifact of specific module properties. The construction relies on deep connections between algebraic geometry (rational surfaces), derived categories (presilting complexes), and representation theory (tilting theory), providing a robust framework for generating counterexamples in tilting theory.

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