Stable equivalences and homological dimensions

This paper characterizes stable equivalences between centralizer matrix algebras over arbitrary fields using a new matrix equivalence relation, demonstrating that such equivalences induce Morita-type stable equivalences that preserve key homological dimensions and validate the Alperin–Auslander/Auslander–Reiten conjecture in this context.

The Gist

Imagine you are a master architect who designs buildings using a specific set of blueprints. In the world of mathematics, these "buildings" are called algebras, and the "blueprints" are often just matrices (grids of numbers).

This paper is about a special kind of building called a Centralizer Matrix Algebra. Think of these as buildings constructed by taking a single, specific blueprint (a matrix) and finding all the other blueprints that "get along" with it perfectly (they commute, meaning they can be swapped without changing the result).

The authors, Xiaogang Li and Changchang Xi, are asking a big question: "How can we tell if two of these buildings are essentially the same, even if they look different on the outside?"

In math, "essentially the same" is called Stable Equivalence. It's like saying two houses might have different paint colors or window shapes, but if you remove the "furniture" (projective parts) and look at the core structure, they are identical.

Here is the breakdown of their discovery using simple analogies:

1. The Problem: The "Fingerprint" Mystery

For a long time, mathematicians knew how to tell if two buildings were the same if they were built using Morita Equivalence (like checking if they have the same floor plan) or Derived Equivalence (checking if they have the same history of construction).

But Stable Equivalence is trickier. It's like trying to match two houses by only looking at their shadows. There was no clear rulebook (no "fingerprint") to say, "If House A's shadow looks like House B's shadow, they are stably equivalent."

The authors wanted to solve this mystery specifically for Centralizer Matrix Algebras.

2. The Solution: The "S-Equivalence" Rule

The authors invented a new way to compare the blueprints (matrices) themselves. They call it S-Equivalence.

Imagine you have two complex puzzles (matrices). To see if they are S-equivalent, you don't look at the whole picture. Instead, you break them down into their smallest, indivisible pieces (called elementary divisors).

• The Rule: Two matrices are S-equivalent if you can match their pieces up in pairs such that:
  1. The pieces are mathematically "twins" (isomorphic).
  2. The "power" or "weight" of the pieces matches up in a very specific, rhythmic pattern (either exactly the same or in a mirrored, shifted pattern).

The Big Discovery (Theorem 1.1):
The paper proves a magical connection: Two Centralizer Matrix Algebras are Stably Equivalent if and only if their blueprints (matrices) are S-Equivalent.

This is huge because it turns a hard, abstract algebra problem into a solvable linear algebra puzzle. Instead of staring at complex buildings, you just look at the numbers in the grid and check if they follow the S-Equivalence rule.

3. What Does This Preserve? (The "Invariants")

When two buildings are Stably Equivalent, they share certain "invariants"—properties that never change, no matter how you rearrange the furniture.

The authors show that if two of these algebras are Stably Equivalent, they also share:

• Global Dimension: How "deep" the building's foundation goes.
• Finitistic Dimension: How far you can build before hitting a wall.
• Dominant Dimension: How "stable" the structure is.

Basically, if the blueprints are S-equivalent, the buildings have the exact same structural integrity and depth, even if they look different.

4. The "Permutation" Special Case

The paper also looks at Permutation Matrices. Imagine a matrix that just shuffles numbers around (like a deck of cards being shuffled).

• The authors found that for these specific matrices, the "Stable Equivalence" depends entirely on the singular parts of the shuffle (the parts that get "stuck" or repeat in a specific way).
• If you take two shuffles, strip away the "regular" parts, and the remaining "stuck" parts are S-equivalent, then the whole algebras are Stably Equivalent.

5. Why Should You Care? (The "Conjecture" Check)

There is a famous unsolved mystery in math called the Auslander-Reiten Conjecture. It basically asks: "If two buildings are Stably Equivalent, do they have the same number of unique, non-repeating rooms (simple modules)?"

For decades, this was a guess. This paper proves that for Centralizer Matrix Algebras, the answer is YES. If the blueprints are S-equivalent, the buildings have the exact same number of unique rooms. This confirms the conjecture for this entire class of algebras.

Summary

Think of this paper as a new translation dictionary.

• Before: To compare two complex algebraic structures, you needed a PhD in abstract algebra and a lot of guesswork.
• Now: You just look at the matrices (the blueprints). If they pass the S-Equivalence test (matching their pieces and powers), you instantly know:
  1. The structures are Stably Equivalent.
  2. They have the same depth and stability.
  3. They have the same number of unique components.

The authors have turned a mysterious, high-level algebra problem into a clear, checkable rule based on the numbers in a grid. It's a bridge between the abstract world of "shadows" and the concrete world of "blueprints."

Technical Summary

Here is a detailed technical summary of the paper "Stable equivalences and homological dimensions" by Xiaogang Li and Changchang Xi.

1. Problem Statement

The paper addresses the fundamental problem of characterizing stable equivalences between centralizer matrix algebras.

• Context: Every finite-dimensional algebra over a field is isomorphic to the centralizer algebra of two matrices. Consequently, understanding the centralizer algebra of a single matrix, denoted $S_n(c, R) = \{a \in M_n(R) \mid ac = ca\}$, is foundational for the representation theory of general finite-dimensional algebras.
• The Gap: While Morita equivalences and derived equivalences for these algebras have been characterized using matrix equivalences (M-equivalence and D-equivalence), stable equivalences lack a concrete bimodule description or a clear criterion in terms of linear algebra.
• The Core Question: Given a field $R$ and two matrices $c \in M_n(R)$ and $d \in M_m(R)$, how can one determine if their centralizer algebras $S_n(c, R)$ and $S_m(d, R)$ are stably equivalent?
• Related Conjecture: The paper also investigates the Auslander–Reiten Conjecture (ARC), which posits that stably equivalent algebras have the same number of non-isomorphic, non-projective simple modules. While known for specific classes, its validity for general stable equivalences involving centralizer matrix algebras required proof.

2. Methodology

The authors employ a multi-step strategy combining representation theory, homological algebra, and linear algebra:

1. Block Decomposition: They decompose the centralizer algebra $S_n(c, R)$ into a product of indecomposable blocks based on the elementary divisors of the matrix $c$. Each block corresponds to a local symmetric Nakayama algebra $U_i = R[x]/(f_i(x)^{n_i})$.
2. Elimination of Nodes: To handle the complexities of stable equivalences (which do not generally preserve the number of blocks), the authors utilize the "node-elimination" procedure. They construct algebras without nodes (and without semisimple summands) that are stably equivalent to the original blocks.
3. Reduction to Endomorphism Algebras: The problem is reduced to studying stable equivalences between endomorphism algebras of generators over quotient polynomial rings ($R[x]/(f(x)^n)$).
4. Introduction of S-Equivalence: A new equivalence relation on square matrices, called S-equivalence, is introduced. This relation is defined purely in terms of linear algebra concepts (elementary divisors and power indices) rather than homological bimodules.
5. Lifting Equivalences: The authors prove that stable equivalences between the "Frobenius parts" (self-injective cores) of the algebras can be lifted to stable equivalences of the algebras themselves, provided specific conditions on the generators are met.

3. Key Contributions

A. Definition of S-Equivalence

The paper introduces a novel equivalence relation $c \mathrel{\mathcal{S}} d$ for matrices $c$ and $d$. Two matrices are S-equivalent if there exists a bijection $\pi$ between their sets of maximal reducible elementary divisors ($R_c$ and $R_d$) such that:

1. The corresponding quotient algebras are isomorphic: $R[x]/(f) \cong R[x]/(f^\pi)$.
2. The sets of power indices (exponents in the elementary divisors) satisfy a specific condition: either $P_c(f) = P_d(f^\pi)$ or $P_c(f) = J(P_d(f^\pi))$, where $J$ is a transformation related to the syzygy operator (specifically, $J(T) = \{m_1, m_1-m_2, \dots\}$ for a sorted set $T$).

B. Complete Characterization of Stable Equivalences

Theorem 1.1 establishes the main result:

Two centralizer matrix algebras $S_n(c, R)$ and $S_m(d, R)$ are stably equivalent if and only if the matrices $c$ and $d$ are S-equivalent.

This provides a computable, linear-algebraic criterion for stable equivalence, analogous to how Morita equivalence is characterized by matrix similarity.

C. Verification of the Auslander–Reiten Conjecture

The authors prove that the Auslander–Reiten Conjecture holds for stable equivalences between an arbitrary algebra and a centralizer matrix algebra.

Proposition 4.8: If $A$ is a centralizer matrix algebra and $B$ is stably equivalent to $A$, then $A$ and $B$ have the same number of non-projective simple modules.
This generalizes previous results that were limited to cases where both algebras were centralizer matrix algebras.

D. Preservation of Homological Dimensions

The paper demonstrates that for centralizer matrix algebras, stable equivalences are stronger than usual; they imply stable equivalences of Morita type.

Corollary 1.3: Stably equivalent centralizer matrix algebras over a common field share the same:

• Global dimension
• Finitistic dimension
• Dominant dimension
  This is significant because, in general, stable equivalences do not preserve these dimensions.

4. Key Results and Corollaries

• Permutation Matrices: For permutation matrices $c_\sigma$ and $c_\tau$, stable equivalence depends only on their p-singular parts (where $p$ is the characteristic of the field). If $S_n(c_\sigma, R)$ and $S_m(c_\tau, R)$ are stably equivalent, then the algebras generated by the singular parts $s(\sigma)$ and $s(\tau)$ are also stably equivalent.
• Morita Type: For centralizer matrix algebras, stable equivalence is equivalent to having non-semisimple parts that are stably equivalent of Morita type.
• Counter-examples: The paper provides examples showing that the converse of Corollary 1.2 (regarding permutation matrices) does not hold in general, highlighting the nuance between stable equivalence and Morita equivalence in specific contexts.

5. Significance

1. Bridging Linear Algebra and Representation Theory: The work successfully translates a deep categorical problem (stable equivalence) into a concrete linear algebra problem (S-equivalence of matrices). This makes the classification of these algebras accessible via standard matrix invariants.
2. Resolution of Open Problems: It confirms the Auslander–Reiten conjecture for a broad class of algebras (centralizer matrix algebras) and their stable counterparts, a significant step in representation theory.
3. Homological Invariants: By proving that stable equivalences preserve global, finitistic, and dominant dimensions for this specific class, the paper identifies a "rigid" behavior of centralizer matrix algebras that distinguishes them from general Artin algebras.
4. Methodological Innovation: The introduction of the "S-equivalence" relation and the technique of lifting Frobenius part equivalences provide new tools for studying stable equivalences in other contexts.

In summary, Li and Xi provide a complete and computable solution to the classification of stable equivalences for centralizer matrix algebras, resolving long-standing questions regarding homological invariants and the Auslander–Reiten conjecture within this framework.