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Singular equivalences and homological conjectures

This paper provides a complete description of the singularity categories and singular equivalences of centralizer matrix algebras, thereby verifying the Auslander–Reiten, Cartan determinant, and all major historical homological conjectures for these algebras over fields.

Original authors: Zhenxian Chen, Changchang Xi

Published 2026-03-24

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CC BY 4.0

Original authors: Zhenxian Chen, Changchang Xi

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 you are a detective trying to solve a massive, complex puzzle. In the world of mathematics, this puzzle is made of finite-dimensional algebras. These are abstract structures used to describe symmetries and relationships in everything from quantum physics to cryptography.

For a long time, mathematicians tried to solve these puzzles by looking at "quivers" (diagrams of arrows and dots). But the authors of this paper, Zhenxian Chen and Changchang Xi, had a different idea. They realized that every single one of these complex algebra puzzles can be built by looking at the "centralizer" of just two matrices.

Think of a matrix as a giant spreadsheet of numbers. A centralizer is the set of all other spreadsheets that "get along" perfectly with a specific one (meaning if you multiply them in either order, you get the same result).

The authors decided to simplify the problem even further. Instead of looking at two matrices, they focused on the centralizer of just one matrix. They call this a "Centralizer Matrix Algebra."

Here is the breakdown of their discovery, translated into everyday language:

1. The "Fingerprint" of a Matrix (I-Equivalence and Sg-Equivalence)

Imagine you have two different-looking keys. They might look totally different on the outside, but if they open the exact same lock, they are functionally identical.

In this paper, the authors invented two new ways to tell if two matrices are "functionally identical" in the context of these algebras:

I-Equivalence (Isomorphism): This is like checking if two keys are identical twins. If two matrices are I-equivalent, their centralizer algebras are exactly the same down to the last detail.
Sg-Equivalence (Singularity): This is a looser relationship. It's like checking if two keys have the same "rough shape" or "vibe," even if the teeth are slightly different. If two matrices are Sg-equivalent, their algebras share the same deep, hidden structure (called the "singularity category"), even if they look different on the surface.

The Big Discovery: The authors proved that you can tell if two centralizer algebras are "singly equivalent" just by looking at the "elementary divisors" (the prime factors) of the matrices. It's like saying, "If the DNA of these two matrices matches in a specific way, their resulting algebras are cousins."

2. The "Ghost" Structure (Singularity Categories)

To understand these algebras, the authors looked at their "Singularity Categories."

The Analogy: Imagine a building. The "perfect" parts of the building are the solid, load-bearing walls (these are the "perfect" modules). The "singularities" are the cracks, the weird corners, and the structural flaws.
The Singularity Category is a map that ignores the solid walls and focuses entirely on the flaws and weirdness.
The authors completely mapped out these "flaw maps" for centralizer matrix algebras. They showed that these maps are actually just collections of smaller, simpler maps (products of stable module categories). It's like realizing that a complex, jagged coastline is actually just a repeating pattern of small, simple waves.

3. Solving the "Impossible" Riddles (Homological Conjectures)

For decades, mathematicians have been stuck on a list of famous riddles called Homological Conjectures. These are guesses about how these algebraic structures behave. Some have been open for 50+ years.

The Cartan Determinant Conjecture: A guess about the "volume" of the algebra's structure.
The Auslander-Reiten Conjecture: A guess about whether certain "self-orthogonal" objects (objects that don't interact with themselves in a specific way) must be "projective" (a special, simple type of object).

The Victory: The authors proved that ALL of these famous conjectures are TRUE for centralizer matrix algebras.

Why? Because they showed that these algebras are "CM-finite" (they have a finite, manageable number of building blocks) and "Gorenstein" (they have a very nice, symmetric internal structure).
The Metaphor: It's like trying to prove that every house in a specific neighborhood has a working smoke detector. Instead of checking every single house one by one, the authors proved that the blueprint for this entire neighborhood guarantees that every house has one.

4. The "Permutation" Special Case

The paper also looked at Permutation Matrices (matrices that just shuffle rows around, like a deck of cards).

They found that for these specific matrices, the "Singular Equivalence" (the deep structural similarity) is very powerful.
The Twist: Usually, if two things are "Singularly Equivalent," they might not be "Derived Equivalent" (a stronger form of similarity). However, for permutation matrices, the authors found specific conditions where if they are Singularly Equivalent, they are also Derived, Stable, and Morita equivalent. It's like finding a special key that unlocks every type of door in the building, not just the back door.

Summary

In simple terms, this paper says:

"We found a new way to look at complex algebraic structures by focusing on the 'centralizer' of a single matrix. We created a new 'fingerprint' system to tell when these structures are deeply related. Using this system, we completely mapped out their hidden 'flaw' structures and proved that they solve almost every major unsolved riddle in the field of algebra."

They turned a chaotic, abstract mess of equations into a clean, organized system where the rules are known, the patterns are clear, and the old mysteries are finally solved.

1. Problem Statement

The paper addresses two primary problems in the representation theory of finite-dimensional algebras:

Classification of Singular Equivalences: While it is a known fact that every finite-dimensional algebra over a field is isomorphic to the centralizer of two matrices, the authors focus on the simpler case of centralizer matrix algebras (the centralizer of a single matrix). The goal is to completely describe the singularity categories ( $D_{sg}$ ) of these algebras and characterize when two such algebras are singularly equivalent (i.e., have equivalent singularity categories).
Verification of Homological Conjectures: Despite decades of effort, many major homological conjectures (such as the Auslander–Reiten, Cartan determinant, and finitistic dimension conjectures) remain open for general finite-dimensional algebras. The authors aim to verify these conjectures specifically for centralizer matrix algebras.

2. Methodology

The authors employ a combination of linear algebra, module theory over polynomial rings, and triangulated category theory.

Algebraic Structure: They utilize the isomorphism $S_n(c, R) \cong \text{End}_{R[c]}(R^n)$ , where $S_n(c, R)$ is the centralizer algebra of a matrix $c$ . This allows them to decompose the algebra based on the primary decomposition of the minimal polynomial of $c$ .
New Equivalence Relations: To characterize isomorphisms and singular equivalences, the authors introduce two new equivalence relations on square matrices over a field $R$ $R$ :
- I-equivalence: Defined via a bijection between maximal elementary divisors preserving the algebra structure of the quotient rings and the multiset of power indices. This characterizes algebra isomorphism.
- Sg-equivalence: Defined via equivalence classes of irreducible factors of maximal elementary divisors and combinatorial data of multiplicities (specifically involving multisets derived from power indices). This characterizes singular equivalence.
Singularity Category Decomposition: They describe the singularity category $D_{sg}(S_n(c, R))$ as a product of stable module categories of quotients of polynomial rings, specifically $R[x]/(f(x)^k)$ .
Gorenstein Projective Modules: The authors analyze the category of Gorenstein projective modules over endomorphism algebras of generators over quotients of polynomial algebras ( $R[x]/(f(x)^n)$ ). They prove these algebras are CM-finite (finitely many indecomposable Gorenstein projective modules) and 1-minimal Auslander-Gorenstein.

3. Key Contributions and Results

A. Characterization of Equivalences

Theorem 1.1 (Main Result on Equivalences): Let $A = S_n(c, R)$ $A = S_{n} (c, R)$ and $B = S_m(d, R)$ $B = S_{m} (d, R)$ .
- $A \cong B$ (as algebras) if and only if $c$ and $d$ are I-equivalent.
- $D_{sg}(A) \simeq D_{sg}(B)$ (as triangulated categories) if and only if $c$ and $d$ are Sg-equivalent.
- The paper establishes that singular equivalence does not necessarily preserve the number of non-semisimple blocks, distinguishing it from Morita, derived, and stable equivalences in this context.

B. Structure of Singularity Categories

Theorem 4.11: Provides a complete description of the singularity category of a centralizer matrix algebra:
$D_{sg}(S_n(c, R)) \simeq \prod_{f(x) \in I_c} \prod_{j \in D_{P_c(f(x))}} R[x]/(f(x)^j)\text{-mod}$
where $I_c$ is a specific set of irreducible polynomials derived from the elementary divisors of $c$ , and $D_{P_c}$ represents specific multisets of exponents.

C. Verification of Homological Conjectures

The authors prove that all historical homological conjectures listed in [2, Conjectures, p. 409] hold true for centralizer matrix algebras over fields. Specifically:

Theorem 6.2:
1. Centralizer matrix algebras are CM-finite and 1-minimal Auslander-Gorenstein.
2. The Auslander–Reiten Conjecture (ARC) and the Gorenstein Projective Conjecture (GPC) hold.
3. The Cartan Determinant Conjecture (CDC) holds: If the global dimension is finite, the determinant of the Cartan matrix is 1.
Corollary 6.6: The Cartan determinant is an invariant of Sg-equivalence. Furthermore, a centralizer matrix algebra is quasi-hereditary if and only if its singularity category is trivial (i.e., the algebra has finite global dimension).

D. Permutation Matrices

The paper investigates singular equivalences for centralizer algebras of permutation matrices.
Corollary 5.8: Under certain conditions on the characteristic of the field and the cycle types of permutations, if two permutation matrices are singularly equivalent, their "singular parts" (associated with singular cycles) are also singularly equivalent.
Example 5.10: Demonstrates that for permutation matrices, singular equivalence is strictly weaker than Morita or derived equivalence, providing concrete counterexamples where algebras are singularly equivalent but not Morita equivalent.

4. Significance

Unification of Conjectures: This paper is significant because it resolves a vast array of open homological conjectures for a broad class of algebras (centralizer matrix algebras) that can realize any finite-dimensional algebra (via the centralizer of two matrices). While the results apply to single-matrix centralizers, they provide a foundational step toward understanding the general case.
New Invariants: The introduction of Sg-equivalence provides a precise combinatorial tool to classify singular equivalences, which are notoriously difficult to characterize compared to derived equivalences.
Structural Insight: The explicit decomposition of the singularity category into products of stable module categories of local Nakayama algebras offers a clear geometric and algebraic picture of the "singular" part of these algebras.
Counterexamples: The work clarifies the hierarchy of equivalences, showing that singular equivalence is distinct from and does not imply Morita, derived, or stable equivalence in the context of centralizer algebras.

In summary, Chen and Xi provide a complete structural theory for the singularity categories of centralizer matrix algebras, introduce new equivalence relations to classify them, and use these structural insights to prove the validity of major homological conjectures for this class of algebras.