Singular equivalences and homological conjectures

Abstract: The fact that each finite-dimensional algebra over a field is isomorphic to the centralizer of two matrices, has suggested to investigate representation theoretical problems of finite-dimensional algebras through centralizer algebras of matrices. The first natural question is to study the problems for the centralizer algebra of one matrix, called a centralizer matrix algebra. In this paper we give complete descriptions of the singularity categories and singular equivalences of centralizer matrix algebras, and verify the Auslander--Reiten (or Gorenstein projective) and Cartan determinant conjectures for centralizer matrix algebras. Consequently, all historical homological conjectures (the finitistic dimension, Wakamatsu tilting, tilting (projective) complement, strong Nakayama, generalized Nakayama and Nakayama conjectures) are true for centralizer matrix algebras over fields. Moreover, we prove some homological invariants of singular equivalences for centralizer matrix algebras.

• The paper classifies singular equivalences between centralizer matrix algebras by introducing Sg-equivalence, a computable relation based on irreducible factors, multiplicity data, and residue algebras.
• The paper decomposes each singularity category into stable module categories of local Nakayama algebras, yielding a precise criterion for equivalence and showing that singular equivalence can change the number of non-semisimple blocks.
• The paper proves that centralizer matrix algebras satisfy the principal Auslander–Reiten–Smalø homological conjectures, while also identifying quasi-heredity and Cartan determinants as invariants of singular equivalence.

Overview

This paper, by Zhenxian Chen and Changchang Xi, studies the singularity categories of centralizer matrix algebras over a field $R$. For a matrix $c \in M_n(R)$, the centralizer matrix algebra is $S_n(c,R) := \{x \in M_n(R) \mid cx = xc\}$. Since every finite-dimensional algebra over a field embeds as a centralizer of two matrices (Brenner), and since the one-matrix case already captures substantial structure, the authors develop a complete linear-algebraic classification of singular equivalences between such algebras. The main tools are two new equivalence relations on square matrices — $I$-equivalence and $Sg$-equivalence — defined via elementary divisors and combinatorial data of their multiplicities.

The paper's central results are: (1) an explicit triangle equivalence describing $$\underline{\mathrm{D}}^b(\mathrm{mod}\text{-}S_n(c,R))$$ as a product of stable module categories of local Nakayama algebras; (2) a theorem that two centralizer matrix algebras are singularly equivalent if and only if their defining matrices are $Sg$-equivalent; and (3) verification that all classical homological conjectures listed in Auslander–Reiten–Smalø hold for these algebras.

New equivalence relations on matrices

The authors introduce two equivalence relations defined purely in terms of linear algebraic data of a matrix $c$:

• $I$-equivalence: there exists a bijection between maximal elementary divisors $\mathcal{M}_c$ and $c \in M_n(R)$0 preserving $c \in M_n(R)$1 up to algebra isomorphism and the full multiset $c \in M_n(R)$2 of power indices.
• $c \in M_n(R)$3-equivalence: the number of equivalence classes of irreducible factors in $c \in M_n(R)$4 coincides with that of $c \in M_n(R)$5, together with matching data of multisets $c \in M_n(R)$6 and residue algebras $c \in M_n(R)$7.

These refine earlier notions ($c \in M_n(R)$8-, $c \in M_n(R)$9-, $S_n(c,R) := \{x \in M_n(R) \mid cx = xc\}$0-, and $S_n(c,R) := \{x \in M_n(R) \mid cx = xc\}$1-equivalence from prior work by Li and Xi). The paper establishes a hierarchy of implications among them and provides examples showing each converse fails in general. Notably, Example 3.5(2) exhibits matrices that are $S_n(c,R) := \{x \in M_n(R) \mid cx = xc\}$2-equivalent but neither $S_n(c,R) := \{x \in M_n(R) \mid cx = xc\}$3-equivalent nor $S_n(c,R) := \{x \in M_n(R) \mid cx = xc\}$4-equivalent, demonstrating that $S_n(c,R) := \{x \in M_n(R) \mid cx = xc\}$5-equivalence is genuinely distinct from previously studied relations.

Singularity categories of centralizer matrix algebras

The key structural result (Proposition 4.10) shows that for a generator $S_n(c,R) := \{x \in M_n(R) \mid cx = xc\}$6 over the local Nakayama algebra $S_n(c,R) := \{x \in M_n(R) \mid cx = xc\}$7, the endomorphism algebra $S_n(c,R) := \{x \in M_n(R) \mid cx = xc\}$8 satisfies

$S_n(c,R) := \{x \in M_n(R) \mid cx = xc\}$9

where $I$0 is a multiset determined by differences of consecutive elements of $I$1. This is proved by constructing Frobenius exact structures on subcategories $I$2 and showing their stable quotients decompose the Gorenstein stable category of $I$3.

As a consequence, the global dimension of $I$4 is finite if and only if $I$5 is an additive generator for $I$6, in which case $I$7 is the Auslander algebra of that quotient. This yields a clean trichotomy: gldim$I$8, $I$9, or $Sg$0.

Combining this block decomposition $Sg$1 with the above gives the main description:

$Sg$2

Classification of singular equivalences

The main theorem (Theorem 5.2) establishes:

1. $Sg$3 as $Sg$4-algebras iff $Sg$5 and $Sg$6 are $Sg$7-equivalent.
2. $Sg$8 as triangulated $Sg$9-categories iff they are equivalent as $$\underline{\mathrm{D}}^b(\mathrm{mod}\text{-}S_n(c,R))$$0-categories iff $$\underline{\mathrm{D}}^b(\mathrm{mod}\text{-}S_n(c,R))$$1 and $$\underline{\mathrm{D}}^b(\mathrm{mod}\text{-}S_n(c,R))$$2 are $$\underline{\mathrm{D}}^b(\mathrm{mod}\text{-}S_n(c,R))$$3-equivalent.

The proof of (2)(ii)$$\underline{\mathrm{D}}^b(\mathrm{mod}\text{-}S_n(c,R))$$4(iii) uses the fact that stable equivalences preserve non-semisimple blocks when the relevant Auslander–Reiten quivers are connected, reducing to the known classification of stable equivalences between Nakayama-type algebras.

An important consequence is that singular equivalences do not preserve the number of non-semisimple blocks in general. The authors exhibit $$\underline{\mathrm{D}}^b(\mathrm{mod}\text{-}S_n(c,R))$$5 indecomposable but singularly equivalent to $$\underline{\mathrm{D}}^b(\mathrm{mod}\text{-}S_n(c,R))$$6 having two blocks of infinite global dimension. This contrasts sharply with Morita, derived, and stable equivalences, which do preserve this invariant for centralizer matrix algebras.

Permutation matrices

For permutation matrices $$\underline{\mathrm{D}}^b(\mathrm{mod}\text{-}S_n(c,R))$$7, the elementary divisors are governed by cycle types and the characteristic $$\underline{\mathrm{D}}^b(\mathrm{mod}\text{-}S_n(c,R))$$8 of $$\underline{\mathrm{D}}^b(\mathrm{mod}\text{-}S_n(c,R))$$9: irreducible factors of $Sg$0 raised to powers $Sg$1. The authors prove (Corollary 5.8) that under the hypothesis "$Sg$2, or $Sg$3 and no cycle length has $Sg$4", singular equivalence between $Sg$5 and $Sg$6 implies singular equivalence between $Sg$7 and $Sg$8, where $Sg$9 denotes the product of $c$0-singular cycles.

They also show (Corollary 5.12) that under certain arithmetic conditions on cycle types, Morita, derived, stable, and singular equivalences between $c$1 and $c$2 are all equivalent conditions, characterized by divisibility of cycle lengths after removing $c$3-power factors.

Homological conjectures

The paper verifies several major conjectures for centralizer matrix algebras:

• CM-finiteness and 1-minimal Auslander–Gorenstein property (Proposition 4.10, Theorem 6.2): every $c$4 has finitely many indecomposable Gorenstein projective modules and satisfies $c$5.
• Auslander–Reiten conjecture and Gorenstein projective conjecture (Theorem 6.2(2)): follows from CM-finiteness via Marczinzik's result that self-orthogonal modules over CM-finite Gorenstein algebras have finite projective dimension.
• Cartan determinant conjecture (Theorem 6.2(3)): the Cartan determinant equals $c$6 whenever global dimension is finite, with an explicit formula

$c$7

when $c$8, and $c$9 otherwise.

• Quasi-heredity characterization (Lemma 6.4): $I$0 is quasi-hereditary iff gldim $I$1 iff $I$2 iff $I$3.

Combined with prior work verifying the finitistic dimension, Nakayama, strong Nakayama, generalized Nakayama, Wakamatsu tilting, and tilting complement conjectures, the paper establishes that all homological conjectures catalogued in [Auslander–Reiten–Smalø, p. 409] hold for centralizer matrix algebras over fields.

Invariants of singular equivalences

Two homological invariants are established (Corollary 6.5): if $I$4 and $I$5 are $I$6-equivalent, then (1) $I$7 is quasi-hereditary iff $I$8 is quasi-hereditary, and (2) the Cartan determinants coincide. These follow directly from the explicit formula for $I$9 in terms of $\mathcal{M}_c$0, which is preserved by $\mathcal{M}_c$1-equivalence.

Limitations and open questions

Several restrictions qualify the main results. The permutation-matrix corollary requires excluding the case $\mathcal{M}_c$2 with some $\mathcal{M}_c$3; Example 5.9 demonstrates concretely that without this hypothesis, singular equivalence of $\mathcal{M}_c$4 and $\mathcal{M}_c$5 need not descend to the $\mathcal{M}_c$6/$\mathcal{M}_c$7 level. Similarly, Corollary 5.12 assumes $\mathcal{M}_c$8 and one of three characteristic-dependent conditions. The classification of singular equivalences is specific to centralizer matrix algebras and does not extend to arbitrary finite-dimensional algebras. The authors pose as an open problem the classification of all basic tilting (respectively, silting) modules over $\mathcal{M}_c$9.

Conclusion

This paper provides a complete, computable classification of singular equivalences between centralizer matrix algebras via the new $c \in M_n(R)$00-equivalence relation on matrices, together with an explicit product decomposition of their singularity categories into stable module categories of local Nakayama algebras. It further establishes that all standard homological conjectures hold in this class, and identifies Cartan determinant and quasi-heredity as invariants of singular equivalence. The results position centralizer matrix algebras as a tractable class where the interplay between matrix equivalence relations and triangulated-category equivalences can be made fully explicit.