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arxiv: 2605.17987 · v1 · pith:F7JMOZLHnew · submitted 2026-05-18 · 🧮 math.RA · math.RT

Separability for relative extensions of object unital strongly groupoid graded rings

Zaqueu Cristiano , Patrik Lundstr\"om This is my paper

Pith reviewed 2026-05-20 00:12 UTC · model grok-4.3

classification 🧮 math.RA math.RT
keywords separabilitygroupoid-graded ringsrelative trace mapcentralizerwide subgroupoidobject unitalcrossed products
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The pith

R/R_Δ is separable iff a relative trace map applied to centralizer elements yields the identity for each connected component.

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

The paper proves an if-and-only-if criterion for the separability of the relative extension R/R_Δ. Here R is an object unital ring that is strongly graded by a groupoid Γ, and Δ is a wide subgroupoid of Γ. The condition requires that for every object e in Γ, there is a connected f and an element r in the centralizer of the isotropy part R_Λ inside R_0 such that the relative trace at f of r is the local identity. This unifies and extends previous separability results from group-graded rings and matrix rings to the more general groupoid-graded setting, with applications to crossed products and twisted groupoid rings.

Core claim

If R is object unital and strongly graded by a groupoid Γ with Δ a wide subgroupoid, then the quotient R/R_Δ is separable exactly when for each object e there exist f in the same connected component and r in the centralizer C_{R_0}(R_Λ) satisfying tr^f_{Γ/Δ}(r) = 1_{R_f}.

What carries the argument

The relative trace map tr_{Γ/Δ}^f from the centralizer C_{R_0}(R_Λ) to the component R_f, which aggregates the graded pieces across the cosets of Δ in Γ.

If this is right

  • Separability criteria for object crossed products follow directly from the trace condition.
  • Results on classical groupoid rings and matrix rings are recovered as special cases.
  • The criterion applies to crossed product algebras arising from infinite separable field extensions.
  • The criterion generalizes previous results on separability for group-graded rings and matrix rings.

Where Pith is reading between the lines

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

  • The condition could be used to study separability in infinite groupoid gradings where direct computation is hard.
  • This might connect to questions of projectivity or freeness in noncommutative Galois theory for groupoids.
  • Testing the criterion on specific examples like infinite cyclic groupoids could reveal new separable extensions.

Load-bearing premise

R must be object unital and strongly graded by Γ with Δ wide so that the relative trace map and centralizer are defined and the grading respects the quotient.

What would settle it

Finding a concrete example of an object unital strongly groupoid-graded ring where the trace condition fails for some component but the extension R/R_Δ is still separable would disprove the claim.

read the original abstract

We prove that if $R$ is a ring that is object unital and strongly graded by a groupoid $\Gamma$, and if $\Delta$ is a wide subgroupoid of $\Gamma$, then $R/R_\Delta$ is separable if and only if, for each $e \in \Gamma_0$, there exist $f \in [e]$ and $r \in C_{R_0}(R_\Lambda) := \{ x \in R_0 \mid xy = yx \text{ for all } y \in R_\Lambda \}$ with ${\rm tr}_{\Gamma/\Delta}^f(r) = 1_{R_f}$. Here, $\Gamma_0$ denotes the set of objects of $\Gamma$, $[e]$ the connected component of $\Gamma_0$ containing $e$, $\Lambda$ the isotropy groupoid of $\Delta$, and ${\rm tr}_{\Gamma/\Delta}^f$ the relative trace map at $f$. This result simultaneously generalizes earlier theorems on separability for matrix rings and group-graded rings due to DeMeyer-Ingraham, N{\v a}st{\v a}sescu, Van den Bergh, Van Oystaeyen, Miyashita, Theohari-Apostolidi, and Vavatsoulas, as well as results on groupoid-graded rings due to Cala, Lundstr\"{o}m, and Pinedo. As an application, we consider separability for object crossed products, including object twisted groupoid rings, classical groupoid rings and matrix rings, as well as crossed product algebras defined by infinite separable field extensions.

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

1 major / 0 minor

Summary. The paper proves that if R is object unital and strongly graded by a groupoid Γ with Δ a wide subgroupoid, then the relative extension R/R_Δ is separable if and only if for each e ∈ Γ₀ there exist f ∈ [e] and r ∈ C_{R₀}(R_Λ) such that tr_{Γ/Δ}^f(r) = 1_{R_f}. The result generalizes separability theorems for matrix rings, group-graded rings, and groupoid-graded rings, and applies to object crossed products including infinite separable extensions.

Significance. If the central equivalence is fully established, the explicit criterion in terms of the relative trace and centralizer unifies and extends prior results of DeMeyer-Ingraham, Năstăsescu-Van den Bergh-Van Oystaeyen, Miyashita, and others to the groupoid setting while accommodating infinite cases; this supplies a concrete, checkable condition that should prove useful for applications in graded ring theory and separable algebras.

major comments (1)
  1. Abstract / Main Theorem: the abstract asserts a complete if-and-only-if proof, yet the visible text supplies only the statement of the characterization without derivation steps, explicit verification of the equivalence, or handling of infinite cases; this is load-bearing for the soundness of the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for recognizing the unifying value of the central criterion for separability in the groupoid-graded setting. We address the single major comment below.

read point-by-point responses

  1. Referee: Abstract / Main Theorem: the abstract asserts a complete if-and-only-if proof, yet the visible text supplies only the statement of the characterization without derivation steps, explicit verification of the equivalence, or handling of infinite cases; this is load-bearing for the soundness of the central claim.

    Authors: The abstract states the main result in the conventional summary form. The full manuscript contains the detailed proof of the if-and-only-if statement, including explicit verification of both directions of the equivalence and the treatment of infinite cases via the applications to object crossed products and infinite separable extensions. To make the logical structure more immediately visible, we will insert a concise proof outline immediately after the statement of the main theorem in the revised version. revision: yes

Circularity Check

0 steps flagged

Minor self-citation in cited prior generalizations; central iff characterization remains independently derived

full rationale

The paper proves a new if-and-only-if characterization of separability for the relative extension R_Δ ⊂ R under object-unital strong groupoid grading. The statement invokes the relative trace tr_{Γ/Δ}^f and centralizer C_{R_0}(R_Λ) drawn from earlier literature (including works co-authored by Lundström). These are standard tools required for the setup and enter the theorem statement directly as hypotheses. The abstract explicitly frames the result as a proof that simultaneously generalizes prior theorems by DeMeyer-Ingraham, Năstăsescu et al., Cala-Lundström-Pinedo and others. No load-bearing step in the given claim reduces by the paper's own equations to a fitted input, self-definition, or unverified self-citation chain; the central equivalence retains independent content beyond the cited background.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the standard definitions of rings, groupoids, strong grading, object-unitality, wide subgroupoids, the centralizer, and the relative trace map; these are invoked without new postulates or fitted constants.

axioms (3)
  • domain assumption R is object unital
    Invoked in the opening sentence of the main theorem to ensure local identities behave correctly under the grading.
  • domain assumption R is strongly graded by the groupoid Γ
    Required for the decomposition R = ⊕ R_γ and for the trace map to be surjective onto the relevant components.
  • domain assumption Δ is a wide subgroupoid of Γ
    Ensures the quotient grading and the relative trace are defined for every object.

pith-pipeline@v0.9.0 · 5837 in / 1522 out tokens · 61520 ms · 2026-05-20T00:12:38.672614+00:00 · methodology

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Reference graph

Works this paper leans on

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