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arxiv: 2605.06617 · v1 · submitted 2026-05-07 · 🧮 math.AC

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Connectedness in Codimension One and the Non-S₂ Locus

Likun Xie
Authors on Pith no claims yet

Pith reviewed 2026-05-08 03:34 UTC · model grok-4.3

classification 🧮 math.AC
keywords S2-sheafcodimension one connectednessnon-S2 locuscanonical moduleS2-ificationdeficiency modulelattice idealtoric ring
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The pith

Coherent S2-sheaves and graded S2-modules decompose canonically according to the connected components in codimension one of their support.

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

The paper shows that coherent S2-sheaves and finitely generated graded S2-modules break down into direct summands corresponding to the connected components in codimension one of their supports. This decomposition supplies criteria linking indecomposability of an S2-object to connectedness in codimension one of its support. It extends the Hochster-Huneke correspondences for complete local rings among connectedness in codimension one, indecomposability of canonical modules, and localness of S2-ifications. For a local ring A admitting a canonical module, both the canonical module and the S2-ification End_A(ω_A) therefore decompose into summands that are the canonical modules and S2-ifications of the quotient rings associated to those components. The same viewpoint realizes the non-S2 locus of an equidimensional unmixed ring as the support of the cokernel C in the exact sequence 0 to A to End_A(ω_A) to C to 0, with the natural map of deficiency modules identifying the canonical module of C with the S2-hull of the deficiency module of A.

Core claim

Finite S2-objects—coherent S2-sheaves and finitely generated graded S2-modules—admit a canonical decomposition according to the connected components in codimension 1 of their support. This structural principle extends the Hochster-Huneke correspondences and yields criteria for indecomposability. Consequently, if A is local and admits a canonical module ω_A, there are canonical decompositions of both ω_A and End_A(ω_A) whose indecomposable summands are the canonical modules and S2-ifications of the quotient rings by the codimension-1 components. For A equidimensional and unmixed, the non-S2 locus equals Supp C via the S2-ification sequence 0 to A to End_A(ω_A) to C to 0, and the map K^{dim C+

What carries the argument

The canonical decomposition of S2-objects indexed by the connected components in codimension 1 of their support; this decomposition carries the structural principle, supplies the indecomposability criteria, and realizes the non-S2 locus as Supp C through the S2-ification sequence.

If this is right

  • An S2-object is indecomposable precisely when its support is connected in codimension one.
  • The canonical module ω_A of a local ring decomposes into the canonical modules of the quotient rings by the codimension-1 components.
  • The S2-ification End_A(ω_A) decomposes into the S2-ifications of those same quotient rings.
  • For equidimensional unmixed rings the non-S2 locus equals the support of the cokernel C in the S2-ification sequence.
  • Under suitable conditions, codimension-1 connectedness of the non-S2 locus is detected by the deficiency module K^{dim C+1}(A).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The componentwise decomposition suggests that S2-hulls and S2-ifications can be computed separately on each codimension-1 piece of the support.
  • The identification of the non-S2 locus with Supp C supplies a module-theoretic test for connectedness properties in graded and toric settings.
  • The same viewpoint may be used to relate connectedness of other loci, such as the non-Cohen-Macaulay locus, to suitable deficiency modules.

Load-bearing premise

The ring A is local and admits a canonical module, and for the non-S2 locus statements it is equidimensional and unmixed.

What would settle it

A concrete counterexample would be a coherent S2-sheaf whose support has two distinct connected components in codimension one yet remains indecomposable as a sheaf, or a local equidimensional unmixed ring where the support of C in the sequence 0 to A to End_A(ω_A) to C to 0 fails to coincide with the non-S2 locus.

read the original abstract

We formulate a structural principle for finite $S_2$-objects: coherent $S_2$-sheaves and finitely generated graded $S_2$-modules decompose canonically according to the connected components in codimension $1$ of their support. This gives criteria relating indecomposability of $S_2$-objects to connectedness in codimension $1$ of their supports, and extends the Hochster--Huneke correspondences for complete local rings between connectedness in codimension $1$, indecomposability of canonical modules, and localness of the $S_2$-ifications. As a consequence, if $A$ is a local ring admitting a canonical module $\omega_A$, there are canonical decompositions of both $\omega_A$ and the $S_2$-ification $\operatorname{End}_A(\omega_A)$ whose indecomposable summands are the canonical modules and $S_2$-ifications of the quotient rings associated to the connected components in codimension $1$. We then apply this viewpoint to the non-$S_2$ locus. For $A$ equidimensional and unmixed, this locus is naturally realized as $\operatorname{Supp}_A C$ via the $S_2$-ification sequence $0 \to A \to \operatorname{End}_A(\omega_A) \to C \to 0$. The natural map between deficiency modules $K^{\dim C+1}(A)\to K^{\dim C}(C)$ identifies the canonical module $K^{\dim C}(C)$ with the $S_2$-hull of $K^{\dim C+1}(A)$. Under suitable conditions, this allows codimension-$1$ connectedness of the non-$S_2$ locus to be detected by the deficiency module $K^{\dim C+1}(A)$. We illustrate the theory with examples and apply it to codimension $2$ lattice ideals, obtaining connectedness-in-codimension-$1$ results for the non-$S_2$ loci of certain toric and lattice rings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper formulates a structural principle asserting that coherent S_2-sheaves and finitely generated graded S_2-modules admit canonical decompositions indexed by the connected components in codimension one of their supports. This extends the Hochster--Huneke correspondences between codimension-one connectedness, indecomposability of canonical modules, and localness of S_2-ifications for complete local rings. For a local ring A admitting a canonical module ω_A, the authors obtain corresponding decompositions of both ω_A and its S_2-ification End_A(ω_A). For equidimensional unmixed A they realize the non-S_2 locus as Supp C via the exact sequence 0 → A → End_A(ω_A) → C → 0, identify the image of the natural map K^{dim C+1}(A) → K^{dim C}(C) with the S_2-hull of K^{dim C+1}(A), and derive criteria for codimension-one connectedness of the non-S_2 locus from properties of the deficiency module. The results are illustrated by examples and applied to codimension-two lattice ideals, yielding connectedness statements for the non-S_2 loci of certain toric and lattice rings.

Significance. If the central claims are verified, the work supplies a clean structural decomposition for S_2-objects that directly generalizes the Hochster--Huneke dictionary and furnishes a concrete realization of the non-S_2 locus via the cokernel C together with a link to deficiency modules. These tools are likely to be useful in the study of singularities, connectedness questions, and explicit computations for toric and lattice rings. The presence of concrete applications to lattice ideals and the explicit hypotheses (local, canonical module, equidimensional and unmixed) strengthen the practical utility of the framework.

major comments (2)
  1. [§4] §4 (or the section containing the proof of the decomposition of End_A(ω_A)): the statement that the indecomposable summands are precisely the S_2-ifications of the quotient rings associated to the codimension-one connected components requires an explicit verification that the summands are indeed S_2 and that no further splitting occurs; without this check the extension of the Hochster--Huneke correspondence remains formal.
  2. [The paragraph following the S_2-ification sequence] The paragraph following the S_2-ification sequence (near the discussion of deficiency modules): the claim that the natural map identifies K^{dim C}(C) with the S_2-hull of K^{dim C+1}(A) is load-bearing for the connectedness criterion; the manuscript should supply the precise definition of “S_2-hull” used here and confirm that the map is surjective onto that hull under the equidimensional unmixed hypothesis.
minor comments (3)
  1. [Abstract] The abstract is information-dense; splitting the sentence that begins “We then apply this viewpoint to the non-S_2 locus” into two shorter sentences would improve readability.
  2. [Introduction] Notation for deficiency modules is introduced as K^i without a parenthetical reminder of the standard definition; adding one sentence in the introduction would assist readers outside the immediate subfield.
  3. [Applications section] In the applications to lattice ideals, the precise codimension-two hypothesis on the ideals should be restated when the connectedness conclusion is stated, to make the result self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. The comments help clarify the presentation of the structural results and their applications. We address each major comment below and have incorporated revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (or the section containing the proof of the decomposition of End_A(ω_A)): the statement that the indecomposable summands are precisely the S_2-ifications of the quotient rings associated to the codimension-one connected components requires an explicit verification that the summands are indeed S_2 and that no further splitting occurs; without this check the extension of the Hochster--Huneke correspondence remains formal.

    Authors: We appreciate this observation. The proof in §4 constructs the summands via the canonical decomposition of the support into codimension-one connected components and shows that each summand is the S_2-ification of the corresponding quotient ring. In the revised manuscript we have added an explicit lemma verifying that these summands satisfy the S_2 property (by direct appeal to the definition of S_2-ification and the fact that the components are equidimensional) and that the decomposition is maximal: any further splitting would contradict the codimension-one connectedness of the supports. This makes the extension of the Hochster--Huneke correspondence fully rigorous. revision: yes

  2. Referee: [The paragraph following the S_2-ification sequence] The paragraph following the S_2-ification sequence (near the discussion of deficiency modules): the claim that the natural map identifies K^{dim C}(C) with the S_2-hull of K^{dim C+1}(A) is load-bearing for the connectedness criterion; the manuscript should supply the precise definition of “S_2-hull” used here and confirm that the map is surjective onto that hull under the equidimensional unmixed hypothesis.

    Authors: We agree that both the definition and the surjectivity statement require explicit treatment. In the revised version we have inserted a precise definition of the S_2-hull of a module M as the smallest S_2-module containing the image of M under the natural map, and we prove that, under the standing equidimensional and unmixed hypotheses on A, the natural map K^{dim C+1}(A) → K^{dim C}(C) is surjective onto this hull. The argument uses the exact sequence 0 → A → End_A(ω_A) → C → 0 together with the fact that C is supported in codimension at least 2. This clarification directly supports the subsequent connectedness criterion for the non-S_2 locus. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rest on standard module properties

full rationale

The paper formulates a structural principle for S2-objects decomposing by codimension-1 connected components of support, extending the external Hochster-Huneke correspondences via the explicitly stated S2-ification sequence 0 → A → End(ω_A) → C → 0. All claims (decompositions of ω_A and End(ω_A), identification of non-S2 locus with Supp C, and the map on deficiency modules) are derived under upfront hypotheses (local ring admitting canonical module, equidimensional unmixed) using standard facts about canonical modules and S2-modules; no step reduces by definition to its inputs, no fitted parameters are renamed as predictions, and no load-bearing self-citation chain appears. The argument is self-contained against external commutative-algebra benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is pure commutative algebra and introduces no free parameters, new entities, or ad-hoc axioms beyond standard background results on S2 modules, canonical modules, and local rings.

axioms (2)
  • domain assumption Existence of canonical module ω_A for the local ring A
    Invoked when stating the decompositions of ω_A and End_A(ω_A)
  • domain assumption A is equidimensional and unmixed
    Required for realizing the non-S2 locus as Supp C and the identification of deficiency modules

pith-pipeline@v0.9.0 · 5679 in / 1447 out tokens · 36146 ms · 2026-05-08T03:34:52.717179+00:00 · methodology

discussion (0)

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Reference graph

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