Extension dimensions under singular equivalences and recollements

Jinbi Zhang; Junling Zheng

REVIEW 2 major objections 4 minor 47 references

Extension dimensions under singular equivalences and recollements

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Two finite-dimensional algebras related by a singular equivalence of Morita type with level $l$ have extension dimensions that differ by at most $l$, and identical asymptotic syzygy dimensions.

desk verdict Theorem 4.3 is a solid new result; Theorem 3.5 has an adjunction/width mismatch that needs fixing before the recollement bounds are accepted. read the letter →

arxiv 2502.09009 v1 pith:VMIVKSEV submitted 2025-02-13 math.RT math.RA

Jinbi Zhang , Junling Zheng This is my paper · ORCID

classification math.RTmath.RA MSC 16E0516G1016E1018G80

keywords extensiondimensionsingularequivalenceofMoritatypewithlevelrecollementArtinalgebraderivedcategorysyzygyhomologicalwidthOmega-extension

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves quantitative control on extension dimensions, the invariant that measures how far an Artin algebra is from being representation-finite and vanishes exactly for representation-finite algebras. Its central result is that if two finite-dimensional algebras are singularly equivalent of Morita type with level $l$, then their extension dimensions differ by at most $l$, while the asymptotic invariants $\Omega$-ext.dim and $\mathrm{ext.dim}(\Omega^\infty(\mathrm{mod}))$ are exactly equal. For recollements of derived module categories, it establishes a parallel family of inequalities: the extension dimensions of the outer algebras are bounded by that of the middle algebra plus a homological width term, and the middle algebra's extension dimension is bounded by a combination of the outer ones, provided the recollement extends one step downward (and, for the sharper bounds, one step upward as well). These bounds matter because they turn loose structural relationships between algebras into concrete constraints on a hard-to-compute invariant.

What carries the argument

The argument is carried by three objects. The extension dimension $\mathrm{ext.dim}(A) = \inf\{m \ge 0 \mid A\text{-mod} \subseteq [M]_{m+1}\}$ for some module $M$, where $[M]_n$ is the iterated extension closure, measures how many steps of extensions are needed to build all modules from one. The homological width $w(X^\bullet)$ of a bounded complex of projectives is the minimum length of the interval over which its nonzero terms sit, and for modules of finite projective dimension it equals the projective dimension. The singular equivalence of Morita type with level $l$ is a pair of bimodules $(M,N)$ with $M\otimes_B N \simeq \Omega^l(A)$ and $N\otimes_A M \simeq \Omega^l(B)$ in the stable categories of bimodules. The proof of Theorem 4.3 funnels every $B$-module $Y$ through the filtration of $M \otimes_B Y$, applies $N\otimes_A -$ to translate the level-$l$ bimodule isomorphism into $\Omega^l_B(Y) \oplus$ projective, and then uses an inverse-syzygy lemma (Lemma 2.4(3)) to pull membership in an extension class back from $\Omega^l_B(Y)$ to $Y$ itself, at the cost of adding $l$ to the extension dimension.

What would settle it

Take a finite-dimensional algebra $A$ and a homological ideal $I$ whose bimodule projective dimension $d := \mathrm{pd}(A^e A/I)$ is finite, compute $\mathrm{ext.dim}(A)$ and $\mathrm{ext.dim}(A/I)$, and check whether $|\mathrm{ext.dim}(A) - \mathrm{ext.dim}(A/I)| > 2d$, which would refute Theorem 4.3. Similarly, for any recollement of derived module categories that extends one step downwards, compare $\mathrm{ext.dim}(A)$ with $2\,\mathrm{ext.dim}(B) + \mathrm{ext.dim}(C) + \max\{w(i^*(B)), w(j^*(A))\} + 2$; a single recollement violating this inequality would refute Theorem 3.5(1)(iii).

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Extended reading notes

Core claim

The paper's main finding is Theorem 4.3: if $A$ and $B$ are finite-dimensional algebras and $(M,N)$ is a singular equivalence of Morita type with level $l$, meaning $M\otimes_B N \simeq \Omega^l(A)$ and $N\otimes_A M \simeq \Omega^l(B)$ in the stable categories of bimodules, then $|\mathrm{ext.dim}(A) - \mathrm{ext.dim}(B)| \le l$, $\Omega$-ext.dim$(A) = \Omega$-ext.dim$(B)$, and $\mathrm{ext.dim}(\Omega^\infty(A\text{-mod})) = \mathrm{ext.dim}(\Omega^\infty(B\text{-mod}))$. For recollements of derived module categories (Theorem 3.5), it establishes inequalities such as $\mathrm{ext.dim}(B) \le \mathrm{ext.dim}(A) + w(i^*(B))$ and $\mathrm{ext.dim}(A) \le \mathrm{ext.dim}(B) + \mathrm{ext.dim}(C) + \max\{w(i^*(B)), w(j^*(A))\} + 1$, where $w(-)$ is the homological width of a complex; these require the recollement to extend one step downwards, with the finer ones requiring an upward extension as well.

Load-bearing premise

The recollement half of the paper rests on the condition that the recollement extend one step downwards, equivalently that $i^*(B)$ be isomorphic in the derived category to a bounded complex of projective $A$-modules, a hypothesis that fails for many recollements; the singular-equivalence half rests on the existence of the two bimodules $M$ and $N$ satisfying the level-$l$ syzygy identities.

Editorial extensions

If this is right

A homological ideal $I \subseteq A$ with finite bimodule projective dimension yields $|\mathrm{ext.dim}(A) - \mathrm{ext.dim}(A/I)| \le 2\,\mathrm{pd}(A^e A/I)$ and equal $\Omega$-extension dimensions (Corollary 4.4).
A bounded extension $B \subseteq A$ with $(A/B)^{\otimes p} = 0$ yields $|\mathrm{ext.dim}(A) - \mathrm{ext.dim}(B)| \le 2\,\mathrm{pd}(B^e A/B) + p - 1$ and equal $\Omega$-extension dimensions (Corollary 4.5).
In a recollement extending one step downwards, both outer extension dimensions are bounded by the middle one plus the width of the corresponding complex, and the middle one is bounded by $2\,\mathrm{ext.dim}(B) + \mathrm{ext.dim}(C) + \max\{w(i^*(B)), w(j^*(A))\} + 2$ (Theorem 3.5(1)).
When the recollement also extends upwards, the bound improves to $\mathrm{ext.dim}(A) \le \mathrm{ext.dim}(B) + \mathrm{ext.dim}(C) + \max\{w(i^*(B)), w(j^*(A))\} + 1$, and when one outer algebra has finite global dimension the $\Omega$-extension dimension of the middle algebra equals that of the other outer algebra (Theorem 3.5(2)).
Derived equivalences are a special case: for an equivalence $F : D(A) \to D(B)$, one gets $|\mathrm{ext.dim}(A) - \mathrm{ext.dim}(B)| \le w(F(A))$ with equal $\Omega$-extension dimensions (Corollary 3.4).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Read as a metric statement, the theorem makes extension dimension a $1$-Lipschitz invariant on the graph whose vertices are algebras and whose edges are level-$l$ singular equivalences; the paper does not pursue this reading, but it suggests defining the distance between algebras as the minimal level of such an equivalence and asking whether the bound is ever attained.
The mechanism behind the bound, pushing a filtration through a tensor product and pulling it back through a syzygy, suggests that the inequality should be tight in families where the level grows, for instance along bounded extensions with growing $p$; computing those extension dimensions would test whether the $l$ in Theorem 4.3 is genuinely necessary.
An unproved converse is implicit: if two algebras have extension dimensions differing by at most one, one might ask whether they are necessarily linked by a level-one singular equivalence or by a recollement of width one; the paper establishes only the forward direction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. The singular-equivalence result, Theorem 4.3, is the real payoff: a singular equivalence of Morita type with level l gives |ext.dim(A)-ext.dim(B)| ≤ l and invariance of Ω-ext.dim and ext.dim(Ω^∞). The proof is mostly self-contained, and the use of Heller's theorem to control projective summands is sound. The recollement half, Theorem 3.5, is more fragile. The inequalities are plausible and would be useful if proven, but the proof as written has an adjunction/width mismatch. In (1)(i), Lemma 3.3(2) is applied with G = i_*, so the lemma returns the width w(i^*(A)), not the stated w(i_*(B)). For a triangular matrix algebra these widths are not equal: i_*(B) = Ae1 is projective, width 0, while i^*(A) = e1A can have infinite projective dimension as a B-module. So the cited lemma does not prove the stated inequality. The same kind of mismatch may affect the 'analogous argument' steps in (1)(ii) and (2). Also, Lemma 3.3 has a small overstrong vanishing claim; the later argument seems to use a stronger bound, so that one is likely harmless.

What is genuinely new: the level-l bound for singular equivalences extends the authors' earlier l = 0 case, and the recollement inequalities with homological widths are new statements. The paper is careful about not claiming more than it proves, except for the gap above. Citation pattern is fine: self-citations are for auxiliary lemmas and prior work, not for the main theorems.

Who it is for: people working on extension dimensions, Igusa-Todorov distances, and recollement techniques for Artin algebras. Theorem 4.3 alone is worth a cite; the recollement part needs a check.

Recommendation: send it to peer review, but make the referee look hard at the adjunction issue in Theorem 3.5 and at the 'analogous' steps. If the recollement bounds cannot be repaired, the paper still has a solid Section 4. My own verdict: conditional, leaning positive.

Referee Report

2 major / 4 minor

Summary. This paper studies the behavior of extension dimensions of Artin algebras under two types of links between derived module categories: recollements and singular equivalences of Morita type with level. The authors introduce homological width for triangle functors and use it to state inequalities comparing extension dimensions of algebras in a recollement that extends one step downwards (and upwards), including bounds involving the widths of the induced functors. They also prove that a singular equivalence of Morita type with level l implies |ext.dim(A) − ext.dim(B)| ≤ l, along with invariance of the Ω-extension dimension and of the extension dimension of the category of infinite syzygies. Several corollaries are derived for triangular matrix algebras, homological epimorphisms, exact contexts, and bounded extensions of algebras.

Significance. The results are potentially valuable: extension dimension measures how far an algebra is from being representation-finite, and an invariance statement under singular equivalences with level, as in Theorem 4.3, would be a clean and useful addition. The proof of Theorem 4.3 is detailed and largely self-contained, using standard syzygy and stable-category techniques. The recollement inequalities in Theorem 3.5, if correct, would provide a flexible tool for bounding extension dimensions. However, the central recollement proof contains a width-mismatch in an application of Lemma 3.3; this is a load-bearing gap that must be repaired before the recollement claims are supported. The paper also makes extensive use of 'analogous arguments,' which complicates verification of the remaining parts of Theorem 3.5. The Section 4 contribution appears sound and is the strongest part of the manuscript.

major comments (2)

[§3, Theorem 3.5(1)(i)] The proof of (1)(i) says that it follows directly from Lemma 3.3(2). Lemma 3.3(2) requires a fully faithful functor G : D(B) → D(A) with left adjoint F : D(A) → D(B), and yields ext.dim(B) ≤ ext.dim(A) + w(F(A)). In the recollement (3.29), the fully faithful functor from D(B) into D(A) is i_*, whose left adjoint is i^*. Thus the lemma produces the term w(i^*(A)), not the stated w(i_*(B)). These two widths are not generally equal: for the canonical recollement of a triangular matrix algebra A = (B M \ 0 C), i_*(B) is the projective module Ae1, so w(i_*(B)) = 0, while w(i^*(A)) = pd_B(e1A) can be positive or infinite. The same misapplication appears to affect part (1)(ii) and the 'analogous arguments' in Theorem 3.5(2)(i)–(v). The authors should either correct the statement to use the width that the lemma actually yields, or provide a different proof that genuinely gives the stated bound w(i_*(B)).
[§3, Lemma 3.3(2), near Eq. (3.11)-(3.12)] The proof claims that Ω^p_D(F(P^•)) = 0 for p ≥ −inf(F) − inf(G). However, F(P^•) is shown to lie in K^{[−sup(G)+inf(G)+1, −inf(G)+sup(G)]}(A-proj), whose width is controlled by w(G), not by w(F). The claimed vanishing bound therefore appears stronger than what the displayed interval justifies; the required bound should plausibly involve w(G). This step is load-bearing because the isomorphism (3.12) is used to transfer the syzygy filtration from F(X) to Y, and the same pattern recurs in later arguments. Please clarify or correct this estimate.

minor comments (4)

[§2, Preliminaries] There are several typographical errors: 'nonegative' should be 'nonnegative', 'provied' should be 'provided', 'trianle' should be 'triangle', and in the definition of A-inj it says 'all finitely generated projective A-modules' where 'injective' is clearly intended.
[§2, Lemma 3.1(2)] The proof contains an apparent misprint: 'inf(F(D(A)) = −t' is missing a closing parenthesis, and the final displayed inequality reads 'cw(G(D(B))) ≥ cw(G(D(B)))', which is tautological; presumably one occurrence should be cw(F(D(A))).
[§1 and §3, Theorem statements] The displayed bounds contain a typographical mismatch in the max expression: 'max{w(i∗(B), w(j∗(A))}' has an extra closing brace and missing a parenthesis; the intended expression is max{w(i_*(B)), w(j_*(A))}.
[§3, Theorem 3.5] Several steps are delegated to 'an analogous argument' (e.g., (3.46), (3.49), (3.50)), and at least one of those analogues inherits the same adjunction/width issue flagged in the major comments. The authors should spell out these arguments or clearly state which adjoint is being used in each case.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main theorems are derived from their stated hypotheses, and the few self-citations are auxiliary lemmas that are not load-bearing for the central claims.

full rationale

The derivation chain is not circular. Theorem 4.3 is proved directly: the defining bimodule isomorphisms M⊗_B N ≃ Ω^l_{A^e}(A) and N⊗_A M ≃ Ω^l_{B^e}(B) of a singular equivalence of Morita type with level are used exactly as hypotheses, and the bound |ext.dim(A)−ext.dim(B)| ≤ l is obtained by applying the exact functor N⊗_A− to a resolution chain and then applying the syzygy lemma 2.4(3). The level l appears as an additive term in the constructed filtration, not as a renamed conclusion. Theorem 3.5 is likewise derived from Lemma 3.3, whose proof is self-contained via Lemma 3.1 and Lemma 3.2, both proved in the paper; the recollement hypotheses are explicit assumptions, and the inequalities are established rather than assumed. The only self-citations are to earlier papers of the authors for elementary syzygy-containment facts (Lemma 2.1 from [45], Lemma 2.2(1) from [42], Lemma 2.3 from [44]) and, in the applications, for prior existence theorems of singular equivalences with level (Corollary 4.5 via [33]). These citations import auxiliary technical facts or supply hypotheses for corollaries; they do not smuggle in the main conclusions, and the central theorems do not reduce to them. The possible width-index mismatch in the application of Lemma 3.3 to Theorem 3.5(1) is a proof-correctness concern, not a circularity concern: it questions whether a stated inequality follows from the cited lemma, not whether the inequality is assumed or defined into existence. Overall, the paper shows no significant circularity; the score of 1 reflects only the presence of minor auxiliary self-citations.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. The central claims rest on the standard definitions of extension dimension, recollement, and singular equivalence with level, together with standard results on syzygies (from [39]) and Heller's theorem (from [21]). The recollement corollaries additionally invoke existence theorems from [16] and [11], and the singular-equivalence corollaries invoke construction theorems from [32] and [33].

assumptions (5)

domain assumption A is an Artin algebra (or finite dimensional k-algebra over a field) and all modules are finitely generated
Needed to define extension dimension and to have finite length modules; stated in Section 2.
domain assumption Recollement axioms (R1)-(R4) hold for the derived module categories
Definition 2.14; the inequalities in Theorem 3.5 are stated only for such recollements.
standard math Standard properties of syzygy and cosyzygy complexes in derived categories (Lemmas 2.9 and 2.12)
Cited from [39] and used throughout the proofs.
standard math Heller's theorem that objects isomorphic in the stable category are direct summands of each other up to projectives ([21, Theorem 2.2])
Used in the proof of Theorem 4.3 to transfer stable isomorphisms to module-level isomorphisms with projective summands.
standard math Existence of recollements for idempotent ideals and for exact contexts ([16] and [11])
Used in Corollaries 3.6 through 3.12 to instantiate the abstract recollement result.

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Pith. "Pith review of Extension dimensions under singular equivalences and recollements." pith.science (2026). https://pith.science/paper/VMIVKSEV

@misc{pith2026250209009,
  author       = {Pith},
  title        = {Pith review of: Extension dimensions under singular equivalences and recollements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VMIVKSEV}},
  note         = {Machine review of arXiv:2502.09009}
}

read the original abstract

The extension dimensions of an Artin algebra give a reasonable way of measuring how far an algebra is from being representation-finite. In this paper we mainly study extension dimensions linked by recollements of derived module categories and singular equivalences of Morita type with level, and establish a series of new inequalities and relationships among their extension dimensions.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Angeleri H ¨ugel, S

L. Angeleri H ¨ugel, S. Koenig , Q. H. Liu , and D. Yang , Ladders and simplicity of derived module categories, J. Algebra 472 (2017) 15-66

work page 2017

[2] [2]

Auslander , M.I

M. Auslander , M.I. Platzeck and G. Todorov , Homological theory of idempotent ideals, Trans. Amer. Math. Soc. 332 (1992) 667–692

work page 1992

[3] [3]

A vramovand S

L. A vramovand S. Iyengar , Constructing modules with prescribed cohomological support, Illinois. J. Math. 51 (2007) 1-20

work page 2007

[4] [4]

A. A. Beilinson , J. Bernstein and P. Deligne , Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), 5-171, Ast´ erisque, 100, Soc. Math. France, Paris, 1982

work page 1981

[5] [5]

the representation dimension of artin alge- bras

A. Beligiannis , Some ghost lemmas, survey for “the representation dimension of artin alge- bras”, Bielefeld, http://www.mathematik.uni-bielefeld.de/ ∼sek/2008/ghosts.pdf, 2008

work page 2008

[6] [6]

Beligiannis and I

A. Beligiannis and I. Reiten , Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (883) (2007) 1-207

work page 2007

[7] [7]

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work page 1994

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work page Pith review arXiv