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Homological Invariants of Left and Right Serial Quiver Algebras

This paper investigates the relationship between delooping levels and finitistic dimensions in left and right serial quiver algebras, establishing that these algebras possess finite delooping levels calculable via a finite algorithm while demonstrating that the equality between right finitistic dimension and left delooping level holds unconditionally for right serial algebras but only under specific conditions for left serial ones, alongside introducing improved invariants like sub-derived and derived delooping levels.

Original authors: Ruoyu Guo

Published 2026-08-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ruoyu Guo

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

Imagine a vast, invisible city built not of brick and mortar, but of pure logic and relationships. In this city, called Representation Theory, mathematicians study "algebras"—complex rulebooks that dictate how different shapes (called modules) can fit together, break apart, or transform. Think of an algebra like a set of instructions for a game: you have pieces, you have moves, and you want to know how long the game can last before it gets stuck or repeats forever.

Two of the most important questions in this city are: "How deep can the game go?" and "Does the game ever end?" Mathematicians measure the depth of a game using something called projective dimension. If a game goes on forever, its depth is infinite; if it ends quickly, the depth is small. The finitistic dimension is a special number that tries to capture the maximum depth of all the games that do eventually end. It's like asking, "What is the longest possible winning streak in this game?"

For decades, mathematicians have been hunting for a way to predict this maximum depth without having to play every single game. They've invented various "rulers" to measure it. One such ruler is the delooping level, a tool that checks how many times you can "unwrap" a shape to see if it fits inside a bigger one. Another, more refined ruler is the derived delooping level, which looks at the whole history of how shapes were built, not just the final step. The big mystery is: Do these rulers always agree with the actual maximum depth of the game? Sometimes they do, but sometimes they give different answers, leaving mathematicians scratching their heads.

In this paper, the author, Ruoyu Guo, steps into this mathematical city to investigate a specific neighborhood called Left and Right Serial Quiver Algebras. To understand this neighborhood, imagine a map made of dots (vertices) and arrows (paths). In a "serial" neighborhood, the paths are very orderly. If you stand at any dot, you can only go in one direction (right serial) or come from only one direction (left serial). It's like a city where every street is a one-way lane that never splits or merges with another one-way lane in a messy way. These orderly maps make the math much easier to handle, like a well-organized library compared to a chaotic attic.

Guo's main job is to test the rulers (the delooping levels) against the actual game depth (the finitistic dimension) in this orderly neighborhood. The paper proves a satisfying result for one side of the neighborhood: Right Serial Algebras. Here, the ruler works perfectly. The paper shows that if you look at the "Right Serial" map, the maximum depth of the game is exactly equal to the reading on the "Left Delooping Level" ruler of the opposite map. It's as if measuring the height of a building from the left side gives you the exact same number as measuring its depth from the right side. This is a big deal because it means we can calculate a very hard number (the finitistic dimension) using a much easier algorithm (the delooping level).

However, the story gets twisty when we look at the Left Serial side. Here, the perfect match doesn't always happen. Guo provides a specific example (a small, finite map with 7 dots) where the rulers disagree. In this case, the "Derived Delooping Level" (the smarter, more detailed ruler) still gets the right answer, but the original "Delooping Level" ruler is too high. It's like using a tape measure that is slightly stretched; it gives a number, but it's not the true length. The paper demonstrates that for Left Serial algebras, you need an extra condition (like making sure the paths stay in specific zones of the map) for the simple ruler to work. Without that condition, the simple ruler fails, but the smarter "Derived" ruler saves the day.

The paper also introduces and explains these "smarter" rulers—the sub-derived and derived delooping levels. Think of the original ruler as a quick glance at a single step in a process. The derived ruler, however, looks at the entire chain of events that led to that step. Guo shows that this deeper look is often more reliable. In the tricky Left Serial example, the derived ruler correctly predicted the game's depth, while the simple ruler was off by one.

Ultimately, the paper doesn't claim to have solved the entire mystery of the finitistic dimension for every possible algebra in the universe. Instead, it successfully maps out a specific, orderly region where the rules are clear. It proves that for Right Serial algebras, the relationship is perfect. For Left Serial algebras, it shows that the relationship is usually good but can break, and it offers a better tool (the derived delooping level) to fix the break. The author leaves us with a few open questions, wondering if we can quantify exactly how much the rulers might disagree based on how many arrows are in the map, but for now, we have a clearer picture of how these mathematical shapes behave in their orderly, one-way streets.

Technical Summary: Homological Invariants of Left and Right Serial Quiver Algebras

Problem Statement
The paper addresses the finitistic dimension conjecture, a central open problem in the representation theory of finite-dimensional algebras. The conjecture posits that the little finitistic dimension ( $\text{findim } \Lambda$ ) and the big finitistic dimension ( $\text{Findim } \Lambda$ ) are finite for any finite-dimensional algebra $\Lambda$ . While the equality $\text{findim } \Lambda = \text{Findim } \Lambda$ is known to fail for general monomial algebras, the conjecture remains open regarding the finiteness of these dimensions.

A primary tool in this investigation is the delooping level ( $\text{dell}$ ), an invariant introduced by Igusa and Todorov. It is known that $\text{Findim } \Lambda \leq \text{dell } \Lambda^{\text{op}}$ . However, the delooping level has limitations: it is not necessarily finite for all algebras, and the set of modules with finite delooping level is not closed under extensions, submodules, or quotients. Consequently, $\text{dell } \Lambda^{\text{op}}$ can be strictly larger than $\text{Findim } \Lambda$ , and in some cases, the difference can be arbitrarily large. The paper seeks to refine the understanding of these invariants specifically for left and right serial quiver algebras, a subclass of monomial algebras with tractable syzygy structures.

Methodology
The author employs a combinatorial approach based on the structure of paths and relations in quiver algebras. The methodology relies on:

Right Completions: Utilizing the concept of "right completions" of paths (minimal paths that turn a given path into a zero relation) to analyze syzygies of modules over monomial algebras.
Reversal of Sequences: Establishing a correspondence between sequences of right completions in the opposite algebra $\Lambda^{\text{op}}$ and the projective dimensions of modules in $\Lambda$ . Specifically, the paper demonstrates how reversing a maximal sequence of right completions in $\Lambda^{\text{op}}$ yields a sequence of relations in $\Lambda$ that bounds the projective dimension.
Refined Invariants: The paper utilizes and motivates two improvements over the standard delooping level:
- Sub-derived delooping level ( $\text{sub-ddell}$ ): Defined via submodules, offering a tighter upper bound for $\text{Findim } \Lambda^{\text{op}}$ .
- Derived delooping level ( $\text{ddell}$ ): Defined via exact sequences, which behaves better under extensions and submodules, forming a torsion-free class.

Key Contributions and Results

Right Serial Algebras: The paper proves that for any right serial quiver algebra $\Lambda$ , the right little and big finitistic dimensions are equal and are exactly determined by the left delooping level and derived delooping level of the opposite algebra. Specifically:
$\text{findim } \Lambda = \text{Findim } \Lambda = \text{dell } \Lambda^{\text{op}} = \text{ddell } \Lambda^{\text{op}} < \infty$
This result is established by showing that for right serial algebras, the structure of the quiver (where each vertex has at most one outgoing arrow) ensures that the projective dimension of modules generated by paths corresponds precisely to the delooping level of simple modules in the opposite algebra.
Left Serial Algebras: For left serial algebras, the equality $\text{Findim } \Lambda = \text{dell } \Lambda^{\text{op}}$ does not hold in general. The paper provides a sufficient condition for the equality to hold: if every simple $\Lambda^{\text{op}}$ -module achieving the maximal delooping level has a corresponding sequence of right completions entirely supported on either the cyclic part or the tree part of the quiver (but not mixing both), then the equality holds.
Counterexamples and Refinements:
- The paper constructs a representation-finite left serial algebra (Example 4.7) where the sufficient condition fails. In this case, $\text{Findim } \Lambda = 1$ while $\text{Findim } \Lambda^{\text{op}} = 2$ .
- Crucially, for this counterexample, the derived delooping level of the algebra recovers the correct finitistic dimension of the opposite algebra: $\text{ddell } \Lambda = \text{Findim } \Lambda^{\text{op}} = 2$ . This demonstrates that while $\text{dell } \Lambda^{\text{op}}$ overestimates the dimension (being 3 in this case), $\text{ddell } \Lambda$ provides the precise bound for $\text{Findim } \Lambda^{\text{op}}$ .
- The paper demonstrates that $\text{sub-ddell}$ and $\text{ddell}$ can be strictly smaller than $\text{dell}$ while still equaling the finitistic dimension, validating their utility as improved invariants.
Recovery of Known Results: The proposed method recovers known results for Nakayama algebras and monomial algebras of acyclic quivers, where all these homological dimensions coincide.

Significance
The paper claims significance in the following areas:

Algorithmic Calculation: It provides a finite, easy-to-calculate algorithm for determining the finitistic dimension of right serial quiver algebras via the delooping level of the opposite algebra.
Expansion of Known Classes: It adds right serial quiver algebras and a specific subclass of left serial algebras to the list of algebras where the big finitistic dimension is described by the delooping level of the opposite algebra.
Validation of Refined Invariants: The work motivates and demonstrates the practical superiority of the derived delooping level ( $\text{ddell}$ ) and sub-derived delooping level ( $\text{sub-ddell}$ ) over the standard delooping level. It shows that while $\text{dell}$ may fail to capture the finitistic dimension in certain left serial cases, $\text{ddell}$ often succeeds, suggesting a path toward resolving the finitistic dimension conjecture for broader classes of algebras.
Structural Insight: By linking the combinatorial structure of quiver paths (specifically the mixing of cyclic and tree parts in completion sequences) to homological invariants, the paper offers a deeper structural understanding of why the delooping level bound fails in specific left serial scenarios.

The paper concludes by posing open questions regarding whether the difference between delooping levels and finitistic dimensions can be quantified by the maximum number of incoming/outgoing arrows in a quiver, and whether arbitrarily large differences can be constructed for left serial algebras.

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