Pith. sign in

REVIEW 3 major objections 5 minor 25 references

Skew Group Algebras, (Fg) and Self-injective Rad-Cube-Zero Algebras

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that self-injective algebras with radical cubed zero satisfy the (Fg) finite-generation condition exactly when their type is Dynkin or extended Dynkin, with a single outer-automorphism condition in the remaining tilde-A…

desk verdict A useful reduction and a real step toward completing the (Fg) classification, but the final ~An case in Corollary 2.15 has a proof gap that needs a lifting lemma. read the letter →

arxiv 2411.16179 v1 pith:NFPV2XOJ submitted 2024-11-25 math.RT

Mads Hustad Sand{o}y This is my paper · ORCID
classification math.RT MSC 16D5016E4016S35
keywords self-injectivealgebraradicalcubezero(Fg)propertyHochschildcohomologysupportvarietiesskewgroupseparableequivalenceNakayamaautomorphism
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 completes the classification of self-injective algebras whose radical cubed is zero and which satisfy (Fg), a finite-generation condition on Hochschild cohomology needed to define support varieties. For fields of characteristic not 2 in which the order of the Nakayama automorphism is invertible, the answer is: an algebra of this kind is (Fg) if and only if its type is Dynkin, or extended Dynkin other than $\tilde A_n$, or a specific $\tilde A_n$ case governed by the quiver shape and the outer order of the Nakayama automorphism. The main step is a reduction: each such algebra is (Fg) if and only if a certain symmetric radical-cube-zero algebra built from it by a skew group algebra construction is (Fg), and the symmetric case was already classified. This reduction, combined with prior work on the $\tilde A_n$ family, gives the complete answer promised in the title.

What carries the argument

The argument relies on three constructions. First, separable equivalence, defined in [16], transfers the (Fg) property between an algebra and a skew group algebra $\Lambda G$ when the order of $G$ is invertible in $k$ (Proposition 2.1). Second, the 2-quasi-Veronese $\Lambda^{[2]}$ is a smash product $\Lambda\#\mathbb Z_2^*$, so for $\operatorname{char} k \neq 2$, $\Lambda$ is (Fg) if and only if $\Lambda^{[2]}$ is (Proposition 2.8). Third, $\Lambda^{[2]}$ is a twisted trivial extension $\Delta_\sigma A$ of a bipartite hereditary algebra $A$, whose Nakayama automorphism is $\sigma^{-1}$. For $A$ tame hereditary of type other than $\tilde A_n$, $\sigma$ can be chosen of finite order because the vertex-fixing part of the outer automorphism group is trivial; the cyclic group generated by the Nakayama automorphism then makes the skew group algebra symmetric (Proposition 2.4), putting one in the already-classified weakly symmetric case [9]. This chain of equivalences produces the theorem.

What would settle it

Take a type $\tilde A_n$ radical-cube-zero self-injective algebra whose quiver matches [8, Proposition 5.4], choose a commutativity relation with a coefficient that is not a root of unity, and compute directly the Noetherianity of its Hochschild cohomology; the paper predicts (Fg) holds, so a failure would refute Corollary 2.15(3).

Watch

Extended reading notes

Core claim

The central claim is that (Fg) is invariant under separable equivalence and under passage to the 2-quasi-Veronese, a graded normal form, when the characteristic differs from 2. For a connected Frobenius algebra $\Lambda$ with $\operatorname{rad}^3\Lambda = 0 \neq \operatorname{rad}^2\Lambda$, the 2-quasi-Veronese $\Lambda^{[2]}$ is a twisted trivial extension of a bipartite hereditary algebra $A$, and its Nakayama automorphism is essentially the inverse of the twisting automorphism. Since the vertex-fixing part of the outer automorphism group of a tame hereditary algebra of type other than $\tilde A_n$ is trivial, the twisting automorphism can be chosen of finite order; adjoining the cyclic group it generates makes the skew group algebra symmetric radical-cube-zero, hence (Fg) by the weakly symmetric classification in [9]. For type $\tilde A_n$, the earlier thesis [21] provides the precise boundary. Corollary 2.15 states the resulting classification: (Fg) holds exactly for Dynkin type, for extended Dynkin type other than $\tilde A_n$, and for $\tilde A_n$ either when the quiver matches one of the two shapes in [8, Proposition 5.4 or 6.4], or when the quiver is the remaining shape and the Nakayama automorphism has finite order as an outer automorphism.

Load-bearing premise

The load-bearing premise is that 2 and the order of the Nakayama automorphism are invertible in the field, since the proof passes (Fg) through a $\mathbb Z_2$ smash product and a cyclic skew group algebra whose group order must be invertible.

Editorial extensions

If this is right

  • Every connected Frobenius algebra with $\operatorname{rad}^3\Lambda = 0 \neq \operatorname{rad}^2\Lambda$ of extended Dynkin type other than $\tilde A_n$ satisfies (Fg), provided the field is not characteristic 2 and the Nakayama order is invertible.
  • For type $\tilde A_n$, (Fg) is equivalent to having one of the two quiver shapes from [8, Proposition 5.4 or 6.4], or the remaining shape with the Nakayama automorphism of finite order as an outer automorphism.
  • Because (Fg) implies finite complexity, the listed algebras are exactly the ones in this family with a well-behaved support variety theory via Hochschild cohomology.
  • The classification is independent of the choice of Nakayama automorphism within its inner class; only the outer order matters.

Reading between the lines

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

  • Beyond the paper: if one can show that the smash-product equivalence in Proposition 2.7 preserves (Fg) even in characteristic 2, the whole classification would extend verbatim; a counterexample would show the restriction is intrinsic.
  • Beyond the paper: the reduction suggests a general recipe — for any self-injective algebra whose Nakayama automorphism generates a finite group up to inner automorphisms, (Fg) might be equivalent to (Fg) of a symmetric skew group algebra, potentially beyond radical-cube-zero algebras.
  • Beyond the paper: the outer-order condition in the $\tilde A_n$ case may be a repackaging of the root-of-unity condition on the single non-$\pm1$ commutativity coefficient known from [21]; a direct comparison of the two invariants would make the classification more transparent.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops a reduction strategy for deciding whether a self-injective radical-cube-zero algebra satisfies the finite-generation condition (Fg). The main idea is to pass from a Frobenius algebra Λ to its 2-quasi-Veronese Λ[2], identify Λ[2] as a Z2-smash product, and then use skew group algebras and separable equivalence to reduce the problem to the weakly symmetric case, where the classification is known. The paper states a reduction theorem (Theorem 2.13) and a full classification (Corollary 2.15) in terms of the Dynkin type of the separated quiver, the quiver shape in the ~A_n case, and the Nakayama automorphism. The exposition is concise and depends on several external results, with some proofs sketched or omitted.

Significance. If the stated classification is correct, the paper gives a short, conceptual proof of a result that previously required long case-by-case computations in Said's thesis, and it covers the exceptional types that were not accessible there. The use of separable equivalence and the 2-quasi-Veronese to transfer (Fg) is a genuine methodological contribution. The paper is also careful about characteristic assumptions and is explicit that the complete answer is conditional on them. However, the final ~A_n case contains a load-bearing gap in the proof of Corollary 2.15(4), so the classification is not fully established as written.

major comments (3)
  1. [Corollary 2.15(4), Theorem 2.13(2), Proposition 2.4] The 'if' direction of Corollary 2.15(4) is not proved. Theorem 2.13(2) requires a Nakayama automorphism of finite order as an automorphism, so that the cyclic group G generated by it can be formed and Proposition 2.4 can be applied. Finite order of the outer class [ν] in Out(Λ) only yields ν^m = Inn(u) for some unit u; it does not by itself produce a finite-order representative of that outer class. The proof of the 'only if' direction shows that (Fg) implies finite outer order, but the converse is exactly what the 'if' direction needs. The text also does not connect the finite outer order condition to the root-of-unity coefficient q of [21], which is the criterion that actually gives (Fg) in the Prop. 5.1 case. Please either prove that finite outer order implies the existence of a finite-order representative in this setting (equivalently, that q is a root of unity), or reformulate condition (4) in terms of q or of a finite-order representative.
  2. [Proposition 2.12, Proposition 2.10] The reduction for types other than ~A_n depends on Proposition 2.12, whose proof is only sketched. The claim that any twisted trivial extension of a tame hereditary algebra not of type ~A_n can be endowed with a finite-order Nakayama automorphism rests on Proposition 2.10, whose proof is explicitly omitted, and on an unproved extension of [19, Proposition 1.7] to disconnected algebras. Since this is a load-bearing step in the proof of Theorem 2.13(1), the omitted proof and the disconnected-case justification should be supplied, or the relevant statements should be quoted with precise references verifying all hypotheses.
  3. [Proposition 2.1] The proof of Proposition 2.1 is a sketch: it cites [16, Theorem 4.1] and asserts that the symmetric assumption can be circumvented by substituting N for D(M), but it does not carry out the functorial correspondence between the Ext modules or verify the Noetherianity transfer in the non-symmetric setting. Since all subsequent transfer steps (Proposition 2.8 and Theorem 2.13) rely on this proposition, the argument should be written out in full or the proposition should be stated as a known theorem with all hypotheses explicitly checked.
minor comments (5)
  1. [Theorem 2.13, Corollary 2.15] The phrase 'order of the Nakayama automorphism' is ambiguous: it could mean the order of the automorphism itself or the order of its outer class. Proposition 2.4 requires the former, while Corollary 2.15(4) is stated in terms of the latter. This ambiguity should be resolved explicitly.
  2. [Proposition 2.10] The proof of Proposition 2.10 is omitted. If the statement is as elementary as it appears, a short proof should be included; if it is meant to be quoted, a reference should be given.
  3. [Section 2, paragraph on [19, Proposition 1.7]] The assertion that the split exact sequence of [19, Proposition 1.7] remains valid without the connectedness assumption whenever Out0(A) is trivial is made without proof. Please add a justification or a precise reference.
  4. [Proof of Corollary 2.15] The sentence declaring the 'if' direction for items (2)-(4) to follow from 'the above theorem in combination with the main result of [21]' is imprecise: item (3) is a direct consequence of [21], while item (4) is not a consequence of Theorem 2.13 as written.
  5. [Throughout] There are several minor typos: 'for all all λ′' in the nondegeneracy check, 'autmorphism' in the proof of Corollary 2.15, 'any twisted trivial extensions' in Proposition 2.12, and a duplicated 'Acknowledgments' heading. These should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main (Fg) reduction is a genuine transfer argument ending at the external weakly-symmetric classification; the sole self-citation is an auxiliary graded-Frobenius lemma, and the ~An outer-order issue is a correctness gap, not a circular step.

full rationale

The derivation of Theorem 2.13 is a genuine transfer argument: (Fg) is shown to be invariant under the Z2-smash product / 2-quasi-Veronese (Prop. 2.8), under separable equivalence (Prop. 2.1), and under passing to basic versions (Prop. 2.5); the endpoint is a weakly symmetric radical-cube-zero algebra whose (Fg) status is taken from the external classification [9]. The type dichotomy and the ~An cases are taken from [8] and [21], respectively, not from this paper's conclusions. No parameter is fit to the target data and no equation is defined in terms of the property being proved. The only self-citation, [12, Lemma 2.2], identifies the graded bimodule structure DΛ⟨−2⟩ ≃ 1Λν of a Frobenius algebra; it is an auxiliary lemma with assumptions unrelated to the (Fg) classification, so it does not carry the target result. I therefore find no circular step. A separate correctness gap (not circularity) appears in Corollary 2.15(4) and Theorem 2.13(2): the corollary states the final ~An case using finite outer order of the Nakayama automorphism, while the cited sufficiency theorem requires an actual finite-order Nakayama automorphism to form the cyclic skew group algebra; no lifting lemma is supplied. This affects soundness of that branch but is not a self-referential reduction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper is a pure-math note and contains no fitted constants. The central claim rests on a chain of external results: the (Fg) classification for weakly symmetric radical-cube-zero algebras [9]; Said's thesis [21] for the ~An family; Linckelmann's separable equivalence theorem [16]; Reiten-Riedtmann's skew group algebra theorem [20]; the automorphism-group exact sequence of Miyachi-Yekutieli [19]; and a graded Frobenius structure lemma from the author's own preprint [12]. The self-citation [12] is used but is not load-bearing for the main reduction.

assumptions (6)
  • domain assumption A self-injective radical-cube-zero algebra of infinite representation type is Koszul, so it is isomorphic as a graded algebra to its associated graded with respect to the radical filtration.
    Invoked in Section 2 to justify replacing Λ by its 2-quasi-Veronese Λ[2]; cited to [17] and [3, Proposition 2.5.1].
  • domain assumption For a graded Frobenius algebra with socle in degree 2, DΛ⟨−2⟩ ≃ 1Λν as graded bimodules, and hence Λ[2] is a twisted trivial extension of A = Λ[2]_0.
    Used before Proposition 2.6; cited to [12, Lemma 2.2], a preprint coauthored by the author. This is the main self-citation.
  • domain assumption Every radical-cube-zero weakly symmetric algebra satisfies (Fg).
    The reduction target in Theorem 2.13; this is the main theorem of [9].
  • domain assumption For type ~An radical-cube-zero self-injective algebras, (Fg) holds exactly in the cases described by [8, Proposition 5.4 or 6.4] or [8, Proposition 5.1] with the commutativity coefficient a root of unity.
    Needed for the 'only if' direction and cases (3)-(4) of Corollary 2.15; this is the main result of the thesis [21], which is not available in the text.
  • domain assumption The split exact sequence 1 → Out0(A) → Out(A) → Aut(Q0;d) → 1 for hereditary A, extended to the disconnected case when Out0(A) is trivial.
    Used to derive Proposition 2.12, which supplies the finite-order Nakayama automorphism for types not ~An; the disconnected extension is asserted from the proof of [19, Proposition 1.7] but not proved.
  • domain assumption The (Fg) property is invariant under Morita equivalence.
    Used in the paragraph before Proposition 2.8 and in Theorem 2.13 to pass between an algebra and its basic version; cited to [14].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Skew Group Algebras, (Fg) and Self-injective Rad-Cube-Zero Algebras." pith.science (2026). https://pith.science/paper/NFPV2XOJ

@misc{pith2026241116179,
  author       = {Pith},
  title        = {Pith review of: Skew Group Algebras, (Fg) and Self-injective Rad-Cube-Zero Algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NFPV2XOJ}},
  note         = {Machine review of arXiv:2411.16179}
}
read the original abstract

We classify self-injective radical cube zero algebras with respect to whether they satisfy certain finite generation conditions sufficient to have a fruitful theory of support varieties defined via Hochschild cohomology in the vein of (Erdmann et al, 2004) and (Snashall and Solberg, 2004). Using skew group algebras and Linckelmann's notion of separable equivalence, we obtain results that complement the existing partial classification of (Said, 2015) and complete the classification begun in (Erdmann and Solberg, 2011) and (Said, 2015) up to assumptions on the characteristic of the field.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [2]

    Smalø, Representation theory of Artin al- gebras, Cambridge Studies in Advanced Mathematics, vol

    Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin al- gebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambr idge University Press, Cambridge, 1997. Corrected reprint of the 1995 original. MR 1476671

    work page 1997

  2. [21]

    Thesis, 2015

    Shumbana Said, Radical cube zero self injective algebras and support varieti es., Ph.D. Thesis, 2015

    work page 2015

  3. [1]

    Ibrahim Assem, Andrzej Skowronski, and Daniel Simson, Elements of the representation theory of associative algebras: Volume 1: Techniques of repr esentation theory, London Math- ematical Society Student Texts, Cambridge University Pres s, 2006

    work page 2006

  4. [3]

    Alexander Beilinson, Victor Ginzburg, and Wolfgang Soe rgel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. MR1322847 (96k:17010)

    work page 1996

  5. [4]

    Petter Andreas Bergh, Separable equivalences, finitely generated cohomology and finite tensor categories, Math. Z. 304 (2023), no. 3, Paper No. 49, 21. MR4610380

    work page 2023

  6. [5]

    Miriam Cohen and Susan Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282 (1984), no. 1, 237–258. MR728711

    work page 1984

  7. [6]

    Dugas, Periodic resolutions and self-injective algebras of finite t ype, J

    Alex S. Dugas, Periodic resolutions and self-injective algebras of finite t ype, J. Pure Appl. Algebra 214 (2010), no. 6, 990–1000. MR2580674

    work page 2010

  8. [7]

    1, 67–87

    Karin Erdmann, Miles Holloway, Rachel Taillefer, Nicol e Snashall, and Øyvind Solberg, Support varieties for selfinjective algebras , K-Theory 33 (2004), no. 1, 67–87. MR2199789

    work page 2004

Show all 25 references
  1. [8]

    Pure Appl

    Karin Erdmann and Øyvind Solberg, Radical cube zero selfinjective algebras of finite com- plexity, J. Pure Appl. Algebra 215 (2011), no. 7, 1747–1768. MR2771643

    2011

  2. [9]

    Pure Appl

    , Radical cube zero weakly symmetric algebras and support variet ies, J. Pure Appl. Algebra 215 (2011), no. 2, 185–200. MR2720683

    2011

  3. [10]

    Al- gebra 526 (2019), 112–165

    Simone Giovannini and Andrea Pasquali, Skew group algebras of Jacobian algebras , J. Al- gebra 526 (2019), 112–165. MR3915328

    2019

  4. [11]

    Green, Nicole Snashall, and Øyvind Solberg, The Hochschild cohomology ring of a selfinjective algebra of finite representation type , Proc

    Edward L. Green, Nicole Snashall, and Øyvind Solberg, The Hochschild cohomology ring of a selfinjective algebra of finite representation type , Proc. Amer. Math. Soc. 131 (2003), no. 11, 3387–3393. MR1990627

    2003

  5. [12]

    Johanne Haugland and Mads Hustad Sandøy, Higher Koszul duality and connections with n-hereditary algebras, 2021

    2021

  6. [13]

    Algebra 346 (2011), 116–146

    Thorsten Holm and Alexander Zimmermann, Deformed preprojective algebras of type L: Külshammer spaces and derived equivalences , J. Algebra 346 (2011), 116–146. MR2842073

    2011

  7. [14]

    Julian Külshammer, Chrysostomos Psaroudakis, and Øys tein Skartsæterhagen, Derived in- variance of support varieties , Proc. Amer. Math. Soc. 147 (2019), no. 1, 1–14. MR3876726

    2019

  8. [15]

    Algebra 446 (2016), 103–131

    Thierry Lambre, Guodong Zhou, and Alexander Zimmerman n, The Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorp hism is a Batalin-Vilkovisky algebra, J. Algebra 446 (2016), 103–131. MR3421088 SKEW GROUP ALGEBRAS, ( FG) AND SELF-INJECTIVE RAD-CU...

    2016

  9. [16]

    Markus Linckelmann, Finite generation of Hochschild cohomology of Hecke algebr as of fi- nite classical type in characteristic zero , Bull. Lond. Math. Soc. 43 (2011), no. 5, 871–885. MR2854558

    2011

  10. [17]

    Roberto Martínez-Villa, Introduction to Koszul algebras, Rev. Un. Mat. Argentina 48 (2007), no. 2, 67–95 (2008). MR2388890

    2007

  11. [18]

    Hiroyuki Minamoto and Izuru Mori, The structure of AS-Gorenstein algebras , Adv. Math. 226 (2011), no. 5, 4061–4095. MR2770441

    2011

  12. [19]

    129 (2001), no

    Jun-ichi Miyachi and Amnon Yekutieli, Derived Picard groups of finite-dimensional heredi- tary algebras, Compositio Math. 129 (2001), no. 3, 341–368. MR1868359

    2001

  13. [20]

    Algebra 92 (1985), no

    Idun Reiten and Christine Riedtmann, Skew group algebras in the representation theory of Artin algebras, J. Algebra 92 (1985), no. 1, 224–282. MR772481

    1985

  14. [22]

    London Math

    Nicole Snashall and Øyvind Solberg, Support varieties and Hochschild cohomology rings , Proc. London Math. Soc. (3) 88 (2004), no. 3, 705–732. MR2044054

    2004

  15. [23]

    Øyvind Solberg, Support varieties for modules and complexes , Trends in representation the- ory of algebras and related topics, 2006, pp. 239–270. MR225 8047

    2006

  16. [24]

    Kota Yamaura, Realizing stable categories as derived categories , Adv. Math. 248 (2013), 784–819. MR3107527

    2013

  17. [25]

    Zimmermann, On the use of Külshammer type invariants in representation th eory, Bull

    A. Zimmermann, On the use of Külshammer type invariants in representation th eory, Bull. Iranian Math. Soc. 37 (2011), no. 2, 291–341. MR2890588 Department of mathematical sciences, NTNU, Trondheim, Nor w ay Email address : mads.sandoy@ntnu.no

    2011

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.