On preservation of relative resolutions for poset representations

Shunsuke Tada; Toshitaka Aoki

REVIEW 2 major objections 4 minor 1 cited by

On preservation of relative resolutions for poset representations

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For aligned interior systems, induction preserves interval covers and interval resolutions, so interval resolution dimensions are unchanged when passing from a subposet to its ambient poset.

desk verdict Solid preservation theorem for interval resolutions over aligned subposets; the poset-dimension computations are less polished, with a broken reference in the s=2 tilde-A case. read the letter →

arxiv 2506.21227 v1 pith:LCK5LB2V submitted 2025-06-26 math.RT math.ATmath.CT

Toshitaka Aoki , Shunsuke Tada This is my paper · ORCID

classification math.RTmath.ATmath.CT MSC 16G2055N3118G2516E10

keywords persistencemodulesintervalresolutionsGaloisconnectionsinteriorsystemscontractionfunctorsresolutionglobaldimensionmultiparameterpersistenthomology

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 establishes that for a class of full subposets called aligned interior systems, the induction functor sends interval covers to interval covers and interval resolutions to interval resolutions, so the interval resolution dimension of an induced module equals that of the original. This matters for multiparameter persistent homology because interval resolutions are a relative homological invariant of persistence modules, and the result gives a way to compute them over a small aligned subposet and then extend the answer to the large poset. The argument rests on a contraction functor, the left adjoint to induction, which over aligned interior systems sends interval modules to interval-decomposable modules, yielding an adjunction between interval-decomposable categories. The paper then uses this to prove stabilization results for interval resolution global dimensions of finite posets and to compute those dimensions for tree-type posets, A-tilde-type posets, and low-dimensional minimal examples.

What carries the argument

The load-bearing object is the aligned interior system: a full subposet $Q$ whose inclusion has a right adjoint (the floor function $\lfloor \cdot \rfloor_Q$) and whose fibers $\lceil y \rceil_Q = \{a \in P \mid \lfloor a \rfloor_Q = y\}$ are filtered and satisfy $\lceil y{\downarrow} \rceil_Q = (\lceil y \rceil_Q){\downarrow}$. The contraction functor $\mathrm{Cont}_Q$ is the left Kan extension along the floor function; alignment gives $\mathrm{Cont}_Q M(y) = \mathrm{colim}(M|_{\lceil y \rceil_Q})$, which makes it exact and lets it send interval modules to interval-decomposable modules. The induction functor $\mathrm{Ind}_Q$, being pullback along the floor function, sends interval modules to interval modules. Together they restrict to an adjunction between the categories of interval-decomposable modules, and that adjunction is what transports interval covers and resolutions.

What would settle it

Compute interval resolution global dimensions directly for a finite poset $P$ with an equioriented $A_4$-type segment and compare with $P'$ obtained by deleting $\ell_4$; a single pair with different dimensions would falsify Theorem 5.2(a). Alternatively, rerun the Section 6.3 minimality classification over a field of characteristic different from $2$; a table entry whose interval resolution global dimension changes would show the low-dimensional classification is field-sensitive.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that an aligned interior system $Q \subseteq P$ produces an adjoint pair $\mathrm{Cont}_Q \dashv \mathrm{Ind}_Q$ that respects the interval-decomposable part of both representation categories: for every interval $S$ of $Q$, $\mathrm{Ind}_Q I_S$ is the interval module $I_{\lceil S \rceil_Q}$, and for every interval $T$ of $P$, $\mathrm{Cont}_Q I_T$ is the interval-decomposable module $I_{T_Q}$. Because both functors preserve interval decomposability, an interval cover or interval resolution of a module $M$ over $Q$ is carried by the exact, fully faithful induction functor to an interval cover or interval resolution of $\mathrm{Ind}_Q M$, and the interval resolution dimension is unchanged. This is Theorem 4.10, and it holds without assuming the base poset is locally finite, an upper semilattice, or finite.

Load-bearing premise

The finite-poset computations assume that representations of a finite poset are modules over its incidence algebra, that $\mathrm{int\text{-}res\text{-}gl.dim}\, P = \sup_S \mathrm{int\text{-}res\text{-}dim}\, \Gamma_S$ holds for every poset in the tables, and (in Section 6.3) that the ground field is the two-element field, so a failure of any of these would break Sections 5 and 6 even though Section 4 stands independently.

Editorial extensions

If this is right

For any finitely presentable module $M$ over $Q$, an interval cover $V \to M$ induces an interval cover $\mathrm{Ind}_Q V \to \mathrm{Ind}_Q M$, and the same holds for full interval resolutions.
Interval resolution dimensions are invariant under induction: $\mathrm{int\text{-}res\text{-}dim}\, M = \mathrm{int\text{-}res\text{-}dim}\, \mathrm{Ind}_Q M$ for every finitely presentable module $M$ over $Q$.
Finite aligned subgrids of products of totally ordered sets are aligned interior systems, so the reduction to a subgrid applies to the usual finitely presentable multiparameter persistence modules.
For finite tree-type posets, the interval resolution global dimension is $\#\mathrm{leaf}(P)-2$, so it depends only on the underlying graph of the Hasse diagram.
For $\widetilde{A}$-type posets, the interval resolution global dimension is $1$ when there are two sinks and $2$ when there are at least three sinks; Section 6.3 gives a partial classification of minimal posets of global dimension $2$ over the two-element field.

Reading between the lines

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

A practical algorithm suggested by the proof: given a finitely presentable module, choose a small aligned interior system containing the generators and relations of its presentation, compute the interval resolution of its contraction there, then induce back; the cost savings can be large when the small system is much smaller than the ambient poset.
The adjunction mechanism is not specific to intervals: any class of modules closed under $\mathrm{Cont}_Q$ and $\mathrm{Ind}_Q$ in the same way would be carried by the same adjunction, so analogous preservation statements may hold for other relative resolutions.
The stabilizing operation of Theorem 5.2 could plausibly be iterated to a normal form for finite posets, and the Section 6 computations suggest that interval resolution global dimension depends only on coarse graph data in many cases; this is an extension, not a claim of the paper.

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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

Colleague,

This is a real advance on the contraction-functor program. The central result, Theorem 4.10, says that for an aligned interior system Q ⊆ P the induction functor preserves interval covers and interval resolutions, so you can compute interval resolutions over the smaller poset and push them back up. The proof is detailed and the setup is clean: aligned interior systems genuinely generalize the finite aligned subgrids in BBH25, and Proposition 4.4, showing that Cont_Q sends interval modules to interval-decomposable modules, is solid. The paper is also honest: it gives a counterexample for non-aligned systems and does not overclaim.

I agree with the reader's conditional verdict. The soft spots are all in Sections 5 and 6. The stress-test note is correct: Proposition 6.5 opens its s=2 case with a citation to 'Proposition 5.2', and no such proposition exists. If the authors meant Theorem 5.2, the hypotheses do not line up—an tilde-A poset with two sinks is a cycle, so the A_n-type segment there need not have leaf endpoints and the removals in Theorem 5.2 are not available. As written, the '1 if s=2' leg of the formula is unproved. That is a real gap in a stated classification result, though it does not touch Theorem 4.10. Also, Proposition 5.8 defers the m=1 case with 'similarly', and Section 6.3 is a large hand-checked case analysis over F_2 that any referee would find hard to verify line by line. These are fixable, or at least properly flaggable, but they should not be waved through.

The tree-type formula (Proposition 6.4) is nice and the proof is short. The stabilization theorem 5.2(a) has a careful case analysis. No circularity in the reliance on AENY23 and BBH25; those are legitimate cited tools.

For whom: anyone in multiparameter persistence who wants computational reductions, and poset representation theorists. I would bring it to a reading group and would cite Theorem 4.10. The paper deserves a serious referee; the right outcome is likely major revision with the tilde-A gap fixed, not a desk reject.

Referee Report

2 major / 4 minor

Summary. The paper studies Galois connections between posets whose left adjoint is the inclusion of a full subposet, called an interior system. It introduces the associated induction and contraction functors, proves that for aligned interior systems both functors preserve interval-decomposability, and uses this to show that induction preserves interval covers and interval resolutions. The paper then applies these ideas to finite posets, proving a stabilization theorem for interval resolution global dimensions, computing formulas for tree-type and tilde-A-type posets, and offering a partial classification of posets of interval resolution global dimension 2 under an assumption that the ground field has two elements.

Significance. If the main results hold, the paper gives a clean and broadly applicable reduction principle: interval resolutions over a large poset can be computed over a smaller aligned subposet and induced back. The authors are explicit about the new notion of aligned interior systems, they prove the central preservation statements in detail, and they provide parameter-free formulas for interval resolution global dimensions of tree-type and tilde-A-type posets. The proof of Theorem 4.10 is careful and does not depend on the finite-poset machinery used later, which is a genuine strength. The finite-poset sections are more computational in character and contain at least one missing case that needs to be repaired before the paper can be accepted.

major comments (2)

[§6.2, proof of Proposition 6.5] The proof of the case s = 2 states that the claim 'can be shown by using Proposition 5.2', but there is no Proposition 5.2 in the manuscript; Section 5 contains Theorem 5.2, Proposition 5.5, and Proposition 5.8, but not Proposition 5.2. If the intended reference is Theorem 5.2, it does not apply to this situation: an ~A-type poset with two sinks is a cycle, so a length-four segment running through its vertices has internal vertices of degree 2 but its endpoints are not leaves in the sense of Definition 5.1. Thus the stated formula for s = 2 is not established as written. Please supply a direct proof for the two-sink case or a correct reference that covers it.
[§5.2, proof of Proposition 5.8] The proof of Proposition 5.8 is written only for the case m = 2 and dismisses m = 1 with the sentence 'the case m = 1 can be shown similarly'. Since Theorem 5.2(b) relies on the m = 1 case when reflecting a leaf, this omitted case is load-bearing. The analogous argument is plausible, but the manuscript should either spell out the m = 1 verification or state explicitly that the m = 2 proof applies verbatim with the obvious notational changes.

minor comments (4)

[§6.3, proof of Proposition 6.11(2)] The proof refers to 'Table 6.7', but the relevant statement is Example 6.7, not a numbered table. Please correct the cross-reference.
[§6.3, Proposition 6.11] The completeness statement in Proposition 6.11 is proved under the standing assumption from the beginning of Section 6.3 that the ground field is F_2. This assumption should be recorded in the statement of the proposition itself, since the earlier results in the paper, such as Proposition 6.5, are not restricted to F_2.
[§4.2, Definition 4.9] The first sentence of Definition 4.9 reads 'For 0 ≠ M ∈ fprep M'; this should be 'M ∈ fprep P'.
[§4.2, proof of Theorem 4.10(a)] In the right-minimality part of the proof, the isomorphism Hom_P(G(V), G(V)) ≅ Hom_Q(FG(V), FG(V)) is asserted without explanation. This is valid because F is an equivalence when restricted to the essential image of the fully faithful functor G, but that fact should be stated explicitly for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central preservation theorems are self-contained, and the only flagged issue is a non-circular missing reference in Proposition 6.5.

full rationale

The paper's central claims, especially Theorem 4.10 and Proposition 4.4, are derived directly from adjointness, Kan extensions, filtered colimits, and universal properties, not from any fitted parameter or self-citational premise. The induction functor is described by Equation (3.3), the contraction functor by (3.4), and Proposition 4.4 computes Cont_Q(I_T) as I_{T_Q} using Lemma 2.4 and Proposition 3.5; there is no step where the conclusion is assumed in the construction. Theorem 4.10 then follows from exactness and full faithfulness of Ind_Q together with the restricted adjunction on interval modules, again with independent proof. Section 5 invokes the incidence algebra equivalence and the external formula int-res-gl.dim P = sup_S int-res-dim Γ_S from [AENY23, Prop. 3.15]; this is an imported external result, not a renaming or a fitted input. Section 6 uses standard Auslander-Reiten theory, tilting theory, and string-algebra combinatorics; self-citations such as [AET25, Theorem 4.1] and [AET25, Theorem 5.1] are used as background classification results, not as the load-bearing derivation of the new preservation theorem. The proof of Proposition 6.5 contains the sentence 'The claim for s = 2 can be shown by using Proposition 5.2,' but the manuscript contains no Proposition 5.2, and the intended Theorem 5.2 does not obviously apply to the two-sink tilde-A cycle. This is a missing or erroneous reference and a possible proof gap in a classification formula, but it is not circularity: a missing proof is not an equivalence between a prediction and its input. Because no derivation step reduces to its own assumptions by construction, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 1 invented entities

The central claims rest on standard category theory (Kan extensions, filtered colimits, decomposition of pfd modules), and for Sections 5 and 6 on the module theory of incidence algebras, AR theory, and tilting. No free parameters are fitted; the only new objects are definitions (aligned interior systems), whose properties are proven in the paper, not assumed. The Section 6.3 classification additionally assumes the ground field is F_2, as the paper explicitly states.

assumptions (7)

standard math Kan extensions along order-preserving maps f: Q -> P exist for persistence modules and give adjoint triples Lan_f ⊣ f^* ⊣ Ran_f.
Invoked in Section 2.2 to define induction, restriction, coinduction, and contraction functors; standard in category theory.
standard math Every pointwise finite-dimensional persistence module over a poset has a unique direct-sum decomposition into indecomposables (Botnan-Crawley-Boevey).
Used in Section 2.1 to justify decomposition and in the definition of interval-decomposability.
standard math Filtered colimits commute with finite limits in the category of vector spaces.
Used in Proposition 3.6 to prove exactness of contraction functors over aligned interior systems.
domain assumption For a finite poset P, the category rep P is equivalent to mod k[P], modules over the incidence algebra of P.
Used in Section 5 to apply Auslander-Reiten theory and tilting; stated but not proved in the paper.
domain assumption APR-tilting and the Brenner-Butler theorem for finite-dimensional algebras, and the Auslander-Reiten formula, hold as in [ASS06].
Used in Lemma 5.9 and Proposition 5.8 to prove reflection invariance of interval resolution global dimensions.
domain assumption The cited formula int-res-gl.dim P = sup_{S ∈ Int(P)} int-res-dim Γ_S (from [AENY23, Prop. 3.15]) is correct.
Used as Proposition 5.5 to reduce global dimensions to the modules Γ_S; not re-derived in the paper.
domain assumption In Section 6.3, the base field k is the field with two elements.
Explicitly stated at the beginning of Section 6.3; the classification of minimal posets of interval resolution global dimension 2 is proven only over F_2.

invented entities (1)

Aligned interior systems
purpose: A class of full subposets Q of P satisfying conditions (AL1) and (AL2), over which contraction functors have a simple filtered-colimit description and preserve interval-decomposability.
Introduced in Definition-Proposition 3.4. Existence is shown by examples, but there is no falsifiable prediction outside the theory; its role is internal to the paper's framework.

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Cite this review

Pith. "Pith review of On preservation of relative resolutions for poset representations." pith.science (2026). https://pith.science/paper/LCK5LB2V

@misc{pith2026250621227,
  author       = {Pith},
  title        = {Pith review of: On preservation of relative resolutions for poset representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LCK5LB2V}},
  note         = {Machine review of arXiv:2506.21227}
}

read the original abstract

The concept of Galois connections (i.e., adjoint pairs between posets) is ubiquitous in mathematics. In representation theory, it is interesting because it naturally induces the adjoint quadruple between the categories of persistence modules (representations) of the posets via Kan extensions. One of central subjects in multiparameter persistent homology analysis is to understand structures of persistence modules. In this paper, we mainly study a class of Galois connections whose left adjoint is the canonical inclusion of a full subposet. We refer to such a subposet as an interior system, with its corresponding right adjoint given by the floor function. In the induced adjoint quadruple, we call the left Kan extension along its floor function the contraction functor. From its construction, it is left adjoint to the induction functor. Under this setting, we firstly prove that this adjoint pair gives an adjoint pair between finitely presentable persistence modules. Moreover, we introduce a special class of interior systems called aligned interior systems, and prove that both induction and contraction functors over them preserve interval-decomposability of modules. Then, we use them to analyze interval covers and resolutions. We also compute interval resolution global dimensions for certain classes of finite posets.

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Stabilization of the Spread-Global Dimension
math.RT 2025-06 accept novelty 7.0 of 10

For any fixed finite poset Q, the spread-global dimension of T×Q is bounded independently of the total order T, and for grids the bound is attained at k=1+4|Q|.

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Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Escolar, Ken Nakashima, and Michio Yoshiwaki

Hideto Asashiba, Emerson G. Escolar, Ken Nakashima, and Michio Yoshiwaki. Approximation by interval-decomposables and interval resolutions of persistence modules. J. Pure Appl. Algebra , 227(10):Paper No. 107397, 20, 2023

work page 2023

[2] [2]

Escolar, and Shunsuke Tada

Toshitaka Aoki, Emerson G. Escolar, and Shunsuke Tada. Summand-injectivity of interval covers and monotonicity of interval resolution global dimensions. J. Appl. Comput. Topol. , 9(2):Paper No. 13, 2025

work page 2025

[3] [3]

Interval replacements of persistence modules

Hideto Asashiba, Etienne Gauthier, and Enhao Liu. Interval replacements of persistence modules. arXiv preprint arXiv:2403.08308 , 2024

work page arXiv 2024

[4] [4]

Bipath Persistence as Zigzag Persistence

\'A ngel Javier Alonso and Enhao Liu. Bipath persistence as zigzag persistence. arXiv preprint arXiv:2501.00322 , 2024

work page Pith review arXiv 2024

[5] [5]

Relative K oszul coresolutions and relative B etti numbers

Hideto Asashiba. Relative K oszul coresolutions and relative B etti numbers. J. Pure Appl. Algebra , 229(3):Paper No. 107905, 33, 2025

work page 2025

[6] [6]

Elements of the representation theory of associative algebras: V ol

Ibrahim Assem, Daniel Simson, and Andrzej Skowro\' n ski. Elements of the representation theory of associative algebras: V ol. 1: Techniques of representation theory , volume 65 of London Mathematical Society Student Texts . Cambridge University Press, Cambridge, 2006

work page 2006

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Faisceaux pervers

Alexander A Beilinson, Joseph Bernstein, and Pierre Deligne. Faisceaux pervers. Ast \'e risque , 100:5--171, 1982

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Benjamin Blanchette, Thomas Br\"ustle, and Eric J. Hanson. Homological approximations in persistence theory. Canad. J. Math. , 76(1):66--103, 2024

work page 2024