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arxiv: 2604.08069 · v2 · submitted 2026-04-09 · 🧮 math.RA

Differential graded Brauer groups over dg-rings

Xiaoxiao Xu , Alexander Zimmermann This is my paper

Pith reviewed 2026-05-10 17:43 UTC · model grok-4.3

classification 🧮 math.RA
keywords Brauer groupdifferential graded algebradg-ringdg-fieldgraded commutative ringhomological algebraAzumaya algebra
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The pith

A Brauer group is defined for differential graded algebras over dg-rings and computed explicitly for all dg-fields after their classification.

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

The paper introduces a Brauer group for differential graded algebras over differential graded base rings that are graded-commutative or commutative. It draws on an explicit classification of dg-fields to calculate the group in every case. A sympathetic reader would care because this supplies a concrete invariant for dg-algebras analogous to the classical Brauer group that classifies central simple algebras. If the definition holds, it turns an abstract equivalence relation into explicit group values that distinguish dg-algebras up to Morita-type equivalence in the homological setting. The result gives a working tool for comparing dg-algebras where none existed before in this form.

Core claim

We define a Brauer group for differential graded algebras over differential graded graded-commutative or commutative base rings. Based on previous work we give an explicit classification of dg-fields, and compute the so-defined Brauer group in each case explicitly.

What carries the argument

The dg-Brauer group defined for differential graded algebras, whose explicit values are obtained directly from the classification of dg-fields.

If this is right

  • The Brauer group takes explicit, computable values for every dg-field over the allowed bases.
  • The group functions as an invariant that distinguishes dg-algebras up to the relevant equivalence.
  • The same definition and computation apply uniformly whether the base ring is graded-commutative or commutative.
  • The construction extends the classical Brauer group construction to the differential graded setting in a direct way.

Where Pith is reading between the lines

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

  • The explicit values could be used to decide when two dg-algebras are Morita equivalent in the dg-sense.
  • The same method might extend to compute Brauer groups for dg-algebras that are not fields.
  • This supplies a concrete starting point for studying Brauer groups in derived algebraic geometry.

Load-bearing premise

The classification of dg-fields obtained in previous work is accurate and the newly defined Brauer group is well-defined and computable from that classification.

What would settle it

A specific dg-field whose Brauer group value contradicts the one computed from the classification, or a dg-algebra for which the group fails to be well-defined.

read the original abstract

We define a Brauer group for differential graded algebras over differential graded graded-commutative or commutative base rings. Based on previous work we give an explicit classification of dg-fields, and compute the so-defined Brauer group in each case explicitly.

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 / 2 minor

Summary. The manuscript defines a Brauer group for differential graded algebras over differential graded graded-commutative or commutative base rings. Drawing on prior work, it supplies an explicit classification of dg-fields and computes the associated Brauer group explicitly in each case.

Significance. If the definition is well-posed and the computations are correct, the work supplies concrete, case-by-case results for Brauer groups in the dg-setting. This is a useful extension of classical theory, particularly for applications in derived algebraic geometry or homotopical algebra, where explicit descriptions are often scarce. The reliance on a prior classification enables the explicitness, which is a strength provided the foundational step is independently grounded.

major comments (1)
  1. [Computation section (following the classification)] The explicit computations of the Brauer group rest directly on the classification of dg-fields imported from previous work. A brief self-contained summary of the classification (or at least the properties used to derive the group elements and relations) should be included in the relevant computation section, as this step is load-bearing for the central claim of explicit computation.
minor comments (2)
  1. [Definition of the Brauer group] The distinction between graded-commutative and commutative dg-base rings is stated in the abstract but should be recalled with a short sentence when the Brauer group is defined, to avoid any ambiguity for readers.
  2. [Throughout] Notation for dg-algebras, dg-fields, and the Brauer group operation should be checked for consistency across the introduction, definitions, and computations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the helpful suggestion to improve the self-contained nature of the computations. We will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Computation section (following the classification)] The explicit computations of the Brauer group rest directly on the classification of dg-fields imported from previous work. A brief self-contained summary of the classification (or at least the properties used to derive the group elements and relations) should be included in the relevant computation section, as this step is load-bearing for the central claim of explicit computation.

    Authors: We agree that a brief, self-contained summary of the dg-field classification (including the key cases and properties relevant to the Brauer group elements and relations) will make the computation section more accessible. We will insert this overview at the start of the computation section in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces a definition of the Brauer group for differential graded algebras over dg base rings, references prior work for an explicit classification of dg-fields, and then computes the group explicitly in each case. No equations, definitions, or computational steps within the provided abstract and structure reduce to each other by construction, nor does the derivation rely on self-citation chains that substitute for independent justification. The classification is treated as input from previous results rather than derived internally via fitting or renaming, leaving the central construction self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the correctness of a dg-field classification taken from previous work and on standard properties of differential graded algebras and Brauer groups; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Standard properties of differential graded algebras and the classical Brauer group construction extend appropriately to the dg-setting.
    Invoked when defining the dg-Brauer group and when computing it from the dg-field classification.
  • domain assumption The classification of dg-fields from previous work is complete and accurate.
    Directly stated as the basis for explicit computation of the Brauer group in each case.

pith-pipeline@v0.9.0 · 5312 in / 1266 out tokens · 62152 ms · 2026-05-10T17:43:42.298348+00:00 · methodology

discussion (0)

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Reference graph

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