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Partial generalized crossed products and a seven-term exact sequence

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Pith's one-line read For every partial Galois extension of commutative rings by a finite group, the paper proves that the six homomorphisms built in earlier work form a seven-term exact sequence relating partial Galois cohomology, Picard-type groups, and the…

desk verdict Solid partial analogue of the CHR sequence, but exactness at H^1(G,α*,PicS(R)) has a missing step that needs a referee's attention. read the letter →

arxiv 1908.05820 v1 pith:I7NTMOQU submitted 2019-08-16 math.RA math.OA

M. Dokuchaev , A. Paques , H. Pinedo , I. Rocha This is my paper · ORCID
classification math.RAmath.OA MSC 13B0513A5016H0516K5016S3516W2220M18
keywords partialactionGaloisextensioncohomologyPicardgroupBrauersemigroupgeneralizedcrossedproductseven-termexactsequence
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

This paper proves that the six homomorphisms constructed in its predecessor form a seven-term exact sequence for every partial Galois extension $R \supseteq R^\alpha$ of commutative rings with $G$ finite. The sequence runs from $H^1(G,\alpha,R)$ through the Picard group of the invariant subring, a semigroup of rank-at-most-one projectives, $H^2(G,\alpha,R)$, the relative Brauer group $B(R/R^\alpha)$, $H^1(G,\alpha^*,\mathrm{PicS}(R))$, and finally $H^3(G,\alpha,R)$. Exactness means each image is exactly the kernel of the next map, giving the partial-action analogue of the Chase–Harrison–Rosenberg theorem for global Galois extensions. If correct, the result supplies the same style of descent control over invertible modules and Azumaya algebras that the global sequence gives, with the relative Brauer group described through twisted partial crossed products.

What carries the argument

The load-bearing object is the partial generalized crossed product $\Delta(F,\alpha,R,\Phi,G)=\bigoplus_{g\in G} J_g$, assembled from a unital partial representation $\Phi=f\Phi_0$ of $G$ into the Picard semigroup $\mathrm{PicS}_{R^\alpha}(R)$ and a commuting factor set $F=\{f_{g,h}:J_g\otimes J_h\to D_g\otimes J_{gh}\}$ of $R$-$R$-bimodule isomorphisms satisfying diagram (26). This object extends the classical generalized crossed product construction to partial actions: it packages a 1-cocycle $f$ with values in $\mathrm{PicS}(R)$ into an Azumaya $R^\alpha$-algebra that contains $R$ as a maximal commutative subalgebra, and the exactness proofs use it to identify the image of the Brauer-group map $\phi_5$ and the kernel of $\phi_6$. The factor-set diagram is what makes the multiplication associative, and Lemma 7.6, which computes $\phi_5$ of such a product as the inverse cocycle, is the hinge connecting Brauer classes to partial cohomology.

What would settle it

Build a partial Galois extension $R/R^\alpha$ and a $1$-cocycle $f\in Z^1(G,\alpha^*,\mathrm{PicS}(R))$ whose associated partial generalized crossed product $\Delta(F,\alpha,R,\Phi_f,G)$ is Brauer-equivalent to $R^\alpha$ while $f$ is not a coboundary. Lemma 7.6 forces $\phi_5([\Delta])=\mathrm{cls}(f^{-1})$, so exactness at $H^1(G,\alpha^*,\mathrm{PicS}(R))$ would demand that $f$ be a coboundary; any such example would refute Theorem 7.7.

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Extended reading notes

Core claim

On its own terms, the paper's result is that the six maps constructed in [17]—between partial Galois cohomology $H^i(G,\alpha,R)$, the Picard group $\mathrm{Pic}(R^\alpha)$, the inverse semigroup $\mathrm{PicS}(R)$ of finitely generated projective modules of rank at most one, and the relative Brauer group $B(R/R^\alpha)$—fit together into an exact seven-term sequence. Exactness is proved one term at a time: Theorem 3.2 at $H^1(G,\alpha,R)$, Theorem 3.10 at $\mathrm{Pic}(R^\alpha)$, Theorem 3.13 at $\mathrm{PicS}(R)^{\alpha^*}\cap\mathrm{Pic}(R)$, Theorem 4.2 at $H^2(G,\alpha,R)$, Theorem 5.7 at $B(R/R^\alpha)$, and Theorem 7.7 at $H^1(G,\alpha^*,\mathrm{PicS}(R))$. The proof is constructive through the middle: every class in the kernel at the Brauer group is represented by a twisted partial crossed product, and every 1-cocycle with values in $\mathrm{PicS}(R)$ contributes a partial generalized crossed product. Read sympathetically, the paper establishes the complete partial Galois analogue of the Chase–Harrison–Rosenberg sequence: the obstruction patterns for descent of invertible modules and for Azumaya algebras are the same in the partial setting as in the global one.

Load-bearing premise

The theorem assumes that $R$ is a commutative ring with a finite group $G$ acting partially so that $R$ over its invariant subring $R^\alpha$ is a partial Galois extension in the strong sense that there is a finite set of pairs $x_i,y_i\in R$ with $\sum_i x_i\alpha_g(y_i1_{g^{-1}})=\delta_{1,g}$ for every $g\in G$; if no such coordinate system exists, the identifications behind Lemma 3.1, Theorems 3.10 and 3.13, and Proposition 7.3 no longer hold.

Editorial extensions

If this is right

  • A partial Galois analogue of Hilbert's theorem 90 holds: if $\mathrm{Pic}(R^\alpha)=0$, then $H^1(G,\alpha,R)=0$ (Corollary 7.8).
  • When $\mathrm{Pic}(R)=0$, the map $\phi_4$ becomes an isomorphism $H^2(G,\alpha,R)\cong B(R/R^\alpha)$, so every relative Brauer class is represented by a twisted partial crossed product (Corollary 7.9).
  • The new sequence specializes to the original Chase–Harrison–Rosenberg sequence when the action is global, making the classical theorem a special case (Corollary 7.10).
  • Exactness at $H^1(G,\alpha^*,\mathrm{PicS}(R))$ says that a class of partially invertible modules comes from an Azumaya algebra split by $R$ precisely when its partial $3$-cocycle obstruction vanishes.
  • Every class in the kernel of $\phi_5$ is realized as a twisted partial crossed product $R\star_{\alpha,\omega}G$, and Proposition 7.3 shows such products are Azumaya over $R^\alpha$ with $R$ as a maximal commutative subalgebra.

Reading between the lines

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

  • The exactness at $\mathrm{Pic}(R^\alpha)$ and $H^1(G,\alpha,R)$ can be read as a partial Galois descent theorem for invertible modules: after tensoring up to $R$, the fixed-point condition in $\mathrm{PicS}(R)$ is the descent condition, and $H^1(G,\alpha,R)$ parametrizes the ways a descended module is glued together; this torsor reading is not spelled out in the paper but follows directly from The
  • Remark 7.11 raises whether the image of $\phi_6$ is exactly the subgroup of locally trivial $3$-classes $H^3_{\mathrm{lt}}(G,\alpha,R)$; if that equality holds, the last arrow of the sequence would be a precise obstruction map, and the seven-term sequence would give a cohomological classification of locally trivial partial $3$-cocycles.
  • Because the proofs localize at primes of $R^\alpha$ and use that $R_\mathfrak{p}$ is semilocal, the sequence should be effectively computable in examples: when $\mathrm{Pic}(R^\alpha)$ vanishes locally, the middle terms collapse and $B(R/R^\alpha)$ is governed by twisted partial crossed products with trivial $f(g)=[D_g]$, reducing the Brauer group to a finite-group cohomology calculation.
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Editorial analysis

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

Referee Report

1 major / 4 minor

Summary. The paper proves a seven-term exact sequence generalizing the Chase–Harrison–Rosenberg sequence to partial Galois extensions of commutative rings. For a finite group G and a partial Galois extension R ⊇ R^α, the authors define homomorphisms φ1,...,φ6 involving partial Galois cohomology, the Picard group of R^α, the inverse semigroup PicS(R), and the relative Brauer group B(R/R^α), and prove exactness at the first five positions in Theorems 3.2, 3.10, 3.13, 4.2, and 5.7. The final exactness at H^1(G,α*,PicS(R)) is handled in Theorem 7.7 via partial generalized crossed products. Corollaries include a partial Hilbert 90 theorem, a crossed product theorem when Pic(R)=0, and recovery of the classical sequence for global actions.

Significance. If the main theorem is correct, this is a substantial contribution: it gives a partial-action analogue of the Chase–Harrison–Rosenberg exact sequence, with explicit constructions and with the classical statement recovered as a special case. The detailed proofs of exactness at the earlier positions and the concrete corollaries (Hilbert 90 and crossed product theorem) are valuable. The central obstruction to accepting the paper as it stands is a missing argument in the proof of Theorem 7.7, described below; once that is supplied, the result would merit publication.

major comments (1)
  1. [Section 7, Theorem 7.7] The sentence "if f ∈ Z^1(G,α*,PicS(R)) satisfies cls(f) ∈ ker(φ6), then ∆ = ⊕_g J_g ... is a partial generalized crossed product" is not justified. By the definition at the beginning of Section 7, a partial generalized crossed product requires a factor set F = {f_{g,h}: J_g⊗J_h → D_g⊗J_{gh}} for which diagram (26) commutes. The isomorphisms provided in Section 6 have an associativity defect ω_f, and the hypothesis only says [ω_f] = 1 in H^3(G,α,R). The proof does not show that a coboundary ρ with δρ = ω_f can be absorbed by replacing the f_{g,h} with modified isomorphisms, nor does it cite a theorem to that effect. This absorption step is exactly what is needed for exactness at H^1(G,α*,PicS(R)); without it, the reverse inclusion in Theorem 7.7 is not established.
minor comments (4)
  1. [Abstract and Section 7 heading] The abstract contains the typo "wich" for "which", and the Section 7 heading has "exactnesss" for "exactness".
  2. [Lemma 7.6] The displayed identity α*_g(f(h)[D_{g^{-1}}]) f(gh)^{-1}f(g) = [D_g][D_{gh}] is valid, but it deserves a derivation: from the 1-cocycle condition f(gh)[D_g] = f(g)α*_g(f(h)[D_{g^{-1}}]), commutativity of PicS(R), and f(gh)f(gh)^{-1} = [D_{gh}], one obtains the identity by multiplying by f(gh)^{-1}. Adding this one-line justification would avoid confusion.
  3. [Remark 7.11] The phrase "The fact that ϕ6(cls(f)) = cls(ω) ∈ H^3(G,α,R) ..." is tautological as written; the intended point appears to be that the class is locally trivial. Please rephrase to state that the construction takes values in the locally trivial classes.
  4. [Sections 6 and 7] The paper would be easier to check if it stated explicitly which of the isomorphisms f_{g,h} constructed in Section 6 are used in the definition of φ6, and how they relate to the factor set F constructed in Proposition 7.5. This is a readability issue rather than a mathematical error.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the seven-term exactness is not assumed and does not reduce to its inputs.

full rationale

The paper's central claim is the exactness of the seven-term sequence. The exactness proofs are carried out inside the paper: Theorems 3.2, 3.10, 3.13, 4.2, 5.7, and 7.7 prove the successive image/kernel inclusions using the partial Galois coordinate system and the partial cohomology formalism. No parameter is fitted and no 'prediction' is read back from the target conclusion. The heavy reliance on [13], [14], and [17] is real but non-circular: those works supply the definitions of partial Galois extensions, partial cohomology, PicS(R), and the six homomorphisms, with stated assumptions that do not include exactness of the seven-term sequence. The cited results are parameter-free published theorems, used as building blocks rather than as a restatement of the conclusion. The one mathematically delicate point, Theorem 7.7, asserts without proof that cls(f) in ker phi_6 makes Delta = direct sum J_g a partial generalized crossed product; this is a possible proof gap (the commuting of diagram (26) does not follow immediately from triviality of the 3-cocycle omega_f), but it is a correctness issue, not a circular reduction, since no equation in the paper identifies the kernel condition with the factor-set condition by construction. Accordingly, no circular step can be exhibited under the required standard.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The theorem has no empirical free parameters. It rests on standard algebra, the partial Galois extension hypothesis, and a body of cited results, several from the same authors' prior work. The partial generalized crossed product is a new mathematical object introduced to carry the proof.

assumptions (5)
  • standard math ZFC; standard module and ring theory (Hom-tensor relation, faithful projective cancellation, localization, semilocal ring Pic=0).
    Invoked as background in Section 2 and throughout, for example in Lemma 2.1, Theorem 3.13, and Proposition 7.3.
  • domain assumption R is a commutative ring and G is a finite group.
    Stated in Section 2: 'From now on G will denote a finite group, R a commutative ring and R ⊇ R^α a partial Galois extension.'
  • domain assumption R ⊇ R^α is an alpha-partial Galois extension with a partial Galois coordinate system satisfying sum_i x_i alpha_g(y_i 1_{g^{-1}}) = delta_{1,g}.
    This is the main hypothesis, used in Lemma 3.1, Theorems 3.10, 3.13, 5.7, 7.7, and Proposition 7.3.
  • standard math Definitions and basic results of partial actions, twisted partial actions, and partial cohomology from [11], [13], and [14] are correct.
    The paper builds directly on these; they are cited as established, not proved here.
  • standard math Cited external theorems: Auslander-Goldman [1], DeMeyer-Ingraham [7], Ford [22], Lam [25], [26], and Paques-Sant'Ana [27] are correct.
    Used for Azumaya algebras, separability, local Picard groups, and centralizer facts.
invented entities (1)
  • Partial generalized crossed product Delta(F,alpha,R,Phi,G)
    purpose: A ring built from a partial cocycle f and a factor set F, used to prove exactness at H^1(G,alpha*,PicS(R)).
    New mathematical construction introduced in Section 7; it has no empirical content, but it is well-defined and used to prove the theorem.

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Pith. "Pith review of Partial generalized crossed products and a seven-term exact sequence." pith.science (2026). https://pith.science/paper/I7NTMOQU

@misc{pith2026190805820,
  author       = {Pith},
  title        = {Pith review of: Partial generalized crossed products and a seven-term exact sequence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I7NTMOQU}},
  note         = {Machine review of arXiv:1908.05820}
}
read the original abstract

For a partial Galois extension of commutative rings we give a seven term exact sequence which generalize the Chase-Harrison-Rosenberg sequence.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The third partial cohomology group and existence of extensions of semilattices of groups by groups

    math.GR 2019-09 accept novelty 6.0 of 10

    A partial abstract kernel of a semilattice of groups by a group admits an admissible extension exactly when its obstruction in H^3 vanishes, and if it does, the extensions are classified by H^2(G,C(A)).

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [17]

    Partial Galois cohomology and related homomorphisms (expanded version)

    M. Dokuchaev, A. Paques, H. Pinedo, Partial Galois coho mology, extensions of the Picard group and related homomorphisms, Quart. J. Math. 70, (2019), (2), 737-766, expanded version in arXiv:1804.03762

    work page Pith review arXiv 2019

  2. [1]

    Auslander, O

    M. Auslander, O. Goldman, The Brauer group of a commutati ve ring Trans. Amer. Math. Soc. (1960) 52: 367-409

    work page 1960

  3. [2]

    Azumaya, Completely faithful modules and self inject ive rings Nagoya Math

    G. Azumaya, Completely faithful modules and self inject ive rings Nagoya Math. (1966) 27: 697-708

    work page 1966

  4. [3]

    Batista, Partial actions: What they are and why we care , Bull

    E. Batista, Partial actions: What they are and why we care , Bull. Belg. Math. Soc. - Simon Stevin (2017) 24: (1), 35-71

    work page 2017

  5. [4]

    Batista, A

    E. Batista, A. D. M. Mortari, M. M. Teixeira, Cohomology f or partial actions of Hopf algebras, J. Algebra, 528 (2019), 339–380

    work page 2019

  6. [5]

    Chase, D.K

    S.U. Chase, D.K. Harrison, A. Rosenberg, Galois theory a nd Galois cohomology of commutative rings, Mem. Amer. Math. Soc. 52 (1965) 15–33. A SEVEN-TERM EXACT SEQUENCE 33

    work page 1965

  7. [6]

    S. U. Chase, A. Rosenberg, Amitsur cohomology and the Bra uer group, Mem. Am. Math. Soc. 52 (1965) 34–79

    work page 1965

  8. [7]

    DeMeyer, E

    F. DeMeyer, E. Ingraham, Separable algebras over commut ative rings, Lect. notes in math. , 181, Springer (1971)

    work page 1971

Show all 28 references
  1. [8]

    Dokuchaev, Partial actions: a survey, Contemp

    M. Dokuchaev, Partial actions: a survey, Contemp. Math. , 537 (2011), 173–184

    2011

  2. [9]

    Dokuchaev, Recent developments around partial actio ns, S˜ ao Paulo J

    M. Dokuchaev, Recent developments around partial actio ns, S˜ ao Paulo J. Math. Sci. (2019) 13: (1), 195-247

    2019

  3. [10]

    Dokuchaev, R

    M. Dokuchaev, R. Exel, Associativity of crossed produc ts by partial actions, enveloping actions and partial representations. Trans. Amer. Math. Soc. (2005) 357: 1931-1952

    2005

  4. [11]

    Dokuchaev, R

    M. Dokuchaev, R. Exel, J. J. Sim´ on, Crossed products by twisted partial actions and graded alge- bras, J. Algebra (2008) 320: 3278-3310

    2008

  5. [12]

    Dokuchaev, R

    M. Dokuchaev, R. Exel, P. Piccione, Partial representa tions and partial group algebras, J. Algebra, 226 (1) (2000), 505–532

    2000

  6. [13]

    Dokuchaev, M

    M. Dokuchaev, M. Ferrero, A. Paques, Partial actions an d Galois theory, J. Pure Appl. Algebra (2007) 208: 77-87

    2007

  7. [14]

    Dokuchaev, M

    M. Dokuchaev, M. Khrypchenko, Partial cohomology of gr oups, J. Algebra (2015) 427: 142-182

    2015

  8. [15]

    Dokuchaev, M

    M. Dokuchaev, M. Khrypchenko, Twisted partial actions and extensions of semilattices of groups by groups, Int. J. Algebra Comput. , 27, (7) (2017), 887–933

    2017

  9. [16]

    Dokuchaev, M

    M. Dokuchaev, M. Khrypchenko, Partial cohomology of gr oups and extensions of semilattices of abelian groups, J. Pure Appl. Algebra , 222, (2018), 2897–2930

    2018

  10. [18]

    Dokuchaev, N

    M. Dokuchaev, N. Sambonet, Schur’s theory for partial p rojective representations, Israel J. Math. (to appear), arXiv:1711.06739 (2017)

    2017 arXiv

  11. [19]

    Exel, Twisted partial actions: a classification of re gular C ∗ -algebraic bundles, Proc

    R. Exel, Twisted partial actions: a classification of re gular C ∗ -algebraic bundles, Proc. London Math. Soc. , 74 (1997), (3), 417–443

    1997

  12. [20]

    Exel, Partial Dynamical Systems, Fell Bundles and Applications , Mathematical Surveys and Monographs, vol

    R. Exel, Partial Dynamical Systems, Fell Bundles and Applications , Mathematical Surveys and Monographs, vol. 224, American Mathematical Society, 2017

    2017

  13. [21]

    Ferrero, Partial actions of groups on algebras, a sur vey, S˜ ao Paulo J

    M. Ferrero, Partial actions of groups on algebras, a sur vey, S˜ ao Paulo J. Math. Sci. 3 (2009), (1), 95 –107

    2009

  14. [22]

    Ford, Separable Algebras, Am

    T. Ford, Separable Algebras, Am. Math. Soc. (2017) 187

    2017

  15. [23]

    Kanzaki, On generalized crossed product and Brauer g roup, Osaka J

    T. Kanzaki, On generalized crossed product and Brauer g roup, Osaka J. Math (1968) 5: 175-188

    1968

  16. [24]

    Kennedy, C

    M. Kennedy, C. Schafhauser, Noncommutative boundarie s and the ideal structure of reduced crossed products, Duke Math. J. (to appear), arXiv:1710.02200 (2017)

    2017 arXiv

  17. [25]

    T. Y. Lam, Lectures on Modules and Rings, 2 nd Edition Springer (1998)

    1998

  18. [26]

    T. Y. Lam, A first Course in Noncommutative Rings, 2 nd Edition Springer (1990)

    1990

  19. [27]

    Paques, A

    A. Paques, A. Sant’Ana, When is a crossed product by a twi sted partial action Azumaya?, Comm. Algebra (2010) 38: 1093-1103

    2010

  20. [28]

    Pinedo, Partial projective representations and the partial Schur multiplier: a survey, Bolet ´ ın de Matem´ aticas, 22 (2015), (2), 167–175

    H. Pinedo, Partial projective representations and the partial Schur multiplier: a survey, Bolet ´ ın de Matem´ aticas, 22 (2015), (2), 167–175. 34 M. DOKUCHAEV, A. PAQUES, H. PINEDO, AND I. ROCHA Instituto de Matem ´atica e Estat´ıstica, Universidade de S ˜ao P aulo, Rua do Ma...

    2015

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