Total acyclicity of complexes over group algebras

Ioannis Emmanouil; Olympia Talelli

Total acyclicity of complexes over group algebras

T0 review · 0 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper defines a class of groups, Z, over whose group algebras Gorenstein projective modules are Gorenstein flat, the Gorenstein projective cotorsion pair is complete, and every acyclic complex of projective, flat, or injective modules…

desk verdict A genuinely unifying closure framework for Gorenstein properties over group algebras, with solid proofs and only minor external dependencies; worth serious review. read the letter →

arxiv 2505.10920 v1 pith:KEM3COBU submitted 2025-05-16 math.RT math.GRmath.KT

Ioannis Emmanouil , Olympia Talelli This is my paper · ORCID

classification math.RTmath.GRmath.KT MSC 16E6516E1020C0718G25

keywords GorensteinhomologicalalgebragroupalgebrastotallyacycliccomplexesflatmodulesprojectivecotorsionpairsLHoperationPhi

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 aims to establish that one well-chosen class of groups, denoted Z, makes Gorenstein homological algebra over the group ring behave like ordinary homological algebra. For a group G in Z, every Gorenstein projective kG-module is Gorenstein flat, the Gorenstein projective cotorsion pair is complete, and every acyclic complex of projective, flat, or injective kG-modules is totally acyclic. The paper proves that Z is closed under two hierarchical group-class operations, LH and Phi-inj, and that every group whose group algebra is weakly Gorenstein regular belongs to Z, including all finite groups and, over the integers, all groups of type Phi. A sympathetic reader would care because this one framework subsumes previously scattered results for hierarchically decomposable groups and type-Phi groups and produces many new groups with all these properties.

What carries the argument

The engine of the paper is a universally valid structural fact (Proposition 2.1): if M is a cokernel of an acyclic complex of flat modules over any ring, then there is a short exact sequence $0 \to M \to K \to N \to 0$ with K flat and N a cokernel of an acyclic complex of projective modules. This reduces questions about flat complexes to questions about projective complexes, and it yields the equivalences $\mathcal{P}(R) = \mathrm{PGF}(R)$ iff $\mathcal{F}(R) = \mathrm{GFlat}(R)$ used to show that total acyclicity for injective complexes implies total acyclicity for projective and flat complexes. Around this engine the paper places the criterion that a module is Gorenstein flat exactly when it has finite Gorenstein flat dimension and its Pontryagin dual is Gorenstein injective, together with the completeness of cotorsion pairs for projectively coresolved Gorenstein flat and Gorenstein flat modules, and the continuous group-class operations LH and Phi-inj with their transfinite hierarchical closures.

What would settle it

A concrete way to test the central claim is to take a group G in the LH-closure of Z and exhibit a ZG-module M whose Pontryagin dual is Gorenstein injective but whose Gorenstein flat dimension is infinite; even a module built over an uncountable ascending union of Z-subgroups, with each restriction Gorenstein flat, would refute Theorem 5.2(i).

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Extended reading notes

Core claim

Let k be a commutative ring, let D denote Pontryagin duality, and let X, Y, Z be the group classes defined by: X consists of groups for which the cokernels of every acyclic complex of projective, respectively flat, kG-modules are projectively coresolved Gorenstein flat, respectively Gorenstein flat; Y consists of groups for which every acyclic complex of injective kG-modules is totally acyclic; and Z = Y ∩ {G : $D^{-1}\mathrm{GInj}(kG) \subseteq \mathrm{GFlat}(kG)$}. The paper's central claim is that Z is closed under LH and Phi-inj, that Z contains every G for which kG is weakly Gorenstein regular (in particular all finite groups, and all type-Phi groups when k = Z), and that membership in Z implies all four desired conclusions: $\mathrm{GProj}(kG) \subseteq \mathrm{GFlat}(kG)$, the GProj cotorsion pair is complete, and every acyclic complex of projective, flat, or injective kG-modules is totally acyclic. The proof shows, more generally, that X is LH-, Phi-proj-, and Phi-flat-closed and that Y is LH- and Phi-inj-closed, with Z the common refinement where the Pontryagin-dual condition forces Gorenstein flatness.

Load-bearing premise

The load-bearing premise is that the two quoted results on Gorenstein flatness—the finite-dimension-plus-dual criterion and the filtered-colimit theorem—apply to every coefficient ring k treated in the paper; the paper cites them without restating their hypotheses, so a failure of either for some k would leave the LH- and Phi-closure proofs incomplete.

Editorial extensions

If this is right

Every group in the hierarchical closure of the weakly Gorenstein regular groups (the class called ZGor) satisfies all four properties, so the framework delivers new examples beyond the previously treated LHF and type-Phi groups.
For any Z-group G, every kG-module has Gorenstein projective approximations on both sides, since the Gorenstein projective cotorsion pair is complete.
For any Z-group G, acyclic complexes of projective, flat, and injective kG-modules are all totally acyclic, so complete resolutions and the constructions that depend on them are available.
If G is in Z and H is a subgroup, then restrictions of Gorenstein projective (respectively flat, injective) kG-modules to kH are again Gorenstein projective (respectively flat, injective), with the flat and injective statements requiring only weaker assumptions on H.
The tensor product of a Gorenstein projective kG-module with a k-free module, or of a Gorenstein flat kG-module with a k-flat module, is again Gorenstein projective, respectively flat, under the diagonal action; the Hom module into a Gorenstein injective module is Gorenstein injective when the first argument is k-flat.

Reading between the lines

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

If the two cited Gorenstein-flat results hold without extra hypotheses on k, then the class Z is a natural maximal framework for these four properties; a converse question the paper does not settle is whether every group with all four properties must lie in Z.
The paper's own open request for a group in X_fin or Y_fin that is neither LHF nor type Phi suggests a concrete test: computable examples among solvable or locally finite groups could show the new hierarchy is strictly larger than the old ones.
The reduction of flat to projective behavior in Proposition 2.1 holds for arbitrary rings, so the group-theoretic operations are needed only to propagate the injective-side and duality conditions; analogous hierarchies could be built for other dualities or for the sfp-injective dimension variants mentioned in the paper.
Since Z is closed under LH and Phi-inj but closure under extensions or direct products is not addressed, testing those operations on Z would indicate whether the class is stable under the remaining standard group constructions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take on Emmanouil–Talelli, arXiv:2505.10920. The paper defines three group classes X, Y, Z, over a fixed commutative ring k, where the group algebra kG behaves well in Gorenstein homological algebra: acyclic complexes of projective, flat, or injective modules are totally acyclic; Gorenstein projective modules are Gorenstein flat; the Gorenstein projective cotorsion pair is complete; and modules whose Pontryagin dual is Gorenstein injective are Gorenstein flat. The main results are closure theorems: X is closed under LH, Φ_proj, Φ_flat; Y and Z are closed under LH and Φ_inj; and Z contains all groups G for which kG is weakly Gorenstein regular, in particular all type Φ groups and all finite groups. That genuinely unifies and generalizes earlier scattered results.

What is actually new is the class framework and the closure proofs. The proofs are careful and use standard tools: transfinite induction à la Kropholler, cotorsion pairs, and induction/restriction arguments. I did not find internal gaps or circularity. The class definitions are natural, and the corollaries showing that Z-groups have all four desirable properties are clearly organized.

Soft spots: the two weight-bearing external citations in Theorem 5.2 — Bouchiba's criterion (Prop. 1.1) and [38, Cor. 4.12] for direct-limit closure of the Gorenstein flat class — are quoted without discussion of their hypotheses. The reader flagged these. I checked the assumptions in the paper; both are standard in the stated generality. Bouchiba's theorem holds over arbitrary rings, and Šaroch–Šťovíček's result is exactly the filtered-colimit closure of GFlat(R). So the dependence is bibliographic, not mathematical. If either citation were misquoted, the LH-closure proof for Z would need repair, but I see no reason to doubt them.

Minor: the paper gives no new example groups beyond those already in the literature; the authors explicitly ask for such groups in the remarks. That is honest and does not undermine the theorems. The notation is dense, especially the Φ-family of operations, but the organization helps. The citation pattern is fine — self-citations are to background tools and earlier related results, not to hide assumptions.

Who it is for: specialists in Gorenstein homological algebra and cohomological group theory. It is a serious paper and the main theorems look solid. I would send it to a referee, not desk-reject; I expect it to pass with small revisions. I would bring it to the reading group.

Referee Report

0 major / 5 minor

Summary. The paper develops a unified framework for four related conjectures in Gorenstein homological algebra over group algebras kG: (a) Gorenstein projective modules are Gorenstein flat, (b) modules whose Pontryagin dual is Gorenstein injective are Gorenstein flat, (c) the Gorenstein projective cotorsion pair is complete, and (d) acyclic complexes of projective, flat, or injective modules are totally acyclic. The authors introduce group classes X, Y, and Z whose defining conditions are these desirable properties, and prove that X is closed under LH, Φ_proj and Φ_flat (Theorem 3.3), Y is closed under LH and Φ_inj (Theorem 4.2), and Z is closed under LH and Φ_inj (Theorem 5.2). They also show that any group with weakly Gorenstein regular group algebra lies in Z (Proposition 5.5), that finite groups lie in Z under mild hypotheses on the coefficient ring (Proposition 5.3), and that all four properties hold for groups in these classes (Propositions 3.2, 4.1, 5.1). For k=Z this recovers and generalizes earlier results of Dembegioti–Talelli, Biswas, Mazza–Symonds, and Ren–Yang.

Significance. If the results stand, this is a valuable unifying contribution to Gorenstein homological algebra over group algebras. The paper places previously scattered results into a common framework and substantially extends the classes of groups for which the four properties are known. The proofs are carefully structured, with transparent use of cotorsion-pair machinery, transfinite induction à la Kropholler, and the Saroch–Stovíček theory of PGF and Gorenstein flat modules. The manuscript also provides explicit hierarchical descriptions of the resulting group classes. I found the central derivations sound and the external citations appropriate; in particular, Bouchiba's criterion (Proposition 1.1) and the filtered-colimit closure of Gorenstein flat modules used in Theorem 5.2 are quoted in their correct general form. The paper is a worthy successor to the earlier literature it generalizes.

minor comments (5)

[Abstract] The abstract contains several typographical errors: 'Gorentein' should be 'Gorenstein' (appears twice), and 'co ntrolled' and 'a cyclic' should be 'controlled' and 'acyclic'.
[Introduction, Theorem] The statement 'In particular, Z contains all groups of type Φ' is not justified in the text; it would be helpful to add a reference or a sentence explaining that for k=Z, groups of type Φ have weakly Gorenstein regular (indeed Gorenstein regular) group algebra because Z has finite global dimension.
[§1, Proposition 1.2(iii)] In the proof, the functor Tor_1^{kG}(I, -) is applied with I an injective left kG-module, which is formally a right-module argument; the authors should briefly state that left and right kG-modules are identified via the standard anti-automorphism of kG, to avoid a left-right confusion for the reader.
[§3, Theorem 3.3(i)] The uncountable cardinality step is somewhat terse: the assertion 'we may conclude as in [5, Lemma 5.6] that res_H^G M has PGF-dimension ≤ 1' is correct but would benefit from a short explanation of how the continuous ascending union of subgroups interacts with the PGF-dimension bound.
[§5, Theorem 5.2(i)] In the uncountable step, the isomorphism res_H^G M ≅ colim_α ind_{H_α}^H res_{H_α}^G M is used without proof; adding a sentence justifying this standard fact would make the argument easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the paper's closure theorems for the class Z are genuine implications built on independent external results.

full rationale

The paper's central claim is that the class Z = Y ∩ {G : D^-1 GInj(kG) ⊆ GFlat(kG)} is closed under LH and Φ_inj, and that it contains groups with weakly Gorenstein regular group algebras. The class Z is defined by the very properties under study, but the theorems are substantive statements about that class, not tautologies. Theorem 5.2(i) proves LHZ ⊆ Z by first using Z ⊆ Y and the independently established LH-closure of Y (Theorem 4.2), then proving D^-1 GInj(kG) ⊆ GFlat(kG) via transfinite induction on HZ-subgroups. The base case uses the defining property of Z for subgroups in Z, which is legitimate. The uncountable ascending-union step invokes [38, Corollary 4.12], an external result on filtered colimits of Gorenstein flat modules, and the finiteness-to-membership step uses Bouchiba's criterion (Proposition 1.1), also external. Theorem 5.2(ii) similarly reduces flatness of a module M with DM Gorenstein injective to finiteness of Gorenstein flat dimension via Bouchiba's criterion, using Φ_inj ⊆ Φ_flat (Lemma 1.5) to get finite flat dimension of the corresponding kernel. Proposition 5.5, showing weakly Gorenstein regular group algebras give Z-groups, relies on [39, Corollary 5.9] for I(kG) = GInj(kG) and on Bouchiba's criterion for the duality condition; neither input is derived from the target result. Where self-citations occur, they concern background or prior definitions: [40] introduces groups of type Φ, [16] concerns LHF-group conjectures for ZG, and [21]/[22] are cited for technical criteria in Sections 2 and the introduction. These citations are not load-bearing for the closure theorems themselves, and no fitted parameter is renamed as a prediction. The paper is self-contained in the sense that every stated implication is proved from the cited external theorems and the definitions, with no equation identified where the conclusion is equivalent to its input by construction. No circular step can be exhibited, so the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new postulated entities. Its results rest on a body of established theorems in Gorenstein homological algebra and group theory; the most delicate are Bouchiba's duality criterion and the closure results for Gorenstein flat modules under filtered colimits.

assumptions (7)

domain assumption Bouchiba's criterion: a module M is Gorenstein flat iff M has finite Gorenstein flat dimension and DM is Gorenstein injective.
Stated as Proposition 1.1 and cited to [9, Theorem 4(5)]; used in Theorem 5.2 to deduce Gorenstein flatness from finiteness plus duality. No coherence hypothesis is stated.
domain assumption Completeness and heredity of the cotorsion pairs (PGF, PGF^perp), (GFlat, GFlat^perp), and (perp-GInj, GInj).
From Saroch-St'ovicek [38] and Cortes-Izurdiaga-Saroch [15]; used throughout to produce the short exact sequences (3), (4), (5), and (7).
domain assumption Neeman's theorem on the homotopy category of flat modules and pure-acyclicity.
Used in Section 2, especially Proposition 2.1, to relate cokernels of acyclic complexes of flat modules to those of projective modules.
domain assumption GFlat is closed under the filtered colimits that appear in the cardinal induction ([38, Corollary 4.12]).
Used in Theorem 5.2(i) to pass from Gorenstein flatness over smaller subgroups to Gorenstein flatness over their union.
domain assumption If sfli k < infinity, then I(k) = GInj(k) ([39, Corollary 5.9]).
Used in Propositions 4.4, 5.3, and 5.5 to show finite and weakly Gorenstein regular ring coefficients make the trivial group lie in Y and Z.
standard math Standard induction/restriction/coinduction facts for group algebras, including restriction preserving injectives and coinduction preserving Gorenstein injectives.
Used in subgroup-closure arguments, e.g., Lemma 3.1, Proposition 4.1, and Proposition 5.1.
standard math Eklof's lemma and Benson's continuous-union lemma.
Used in Lemma 4.3 to bound the Gorenstein injective dimension of a module over an ascending union of subgroups.

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Pith. "Pith review of Total acyclicity of complexes over group algebras." pith.science (2026). https://pith.science/paper/KEM3COBU

@misc{pith2026250510920,
  author       = {Pith},
  title        = {Pith review of: Total acyclicity of complexes over group algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KEM3COBU}},
  note         = {Machine review of arXiv:2505.10920}
}

read the original abstract

In this paper, we study group algebras over which modules have a controlled behaviour with respect to the notions of Gorenstein homological algebra, namely: (a) Gorenstein projective modules are Gorenstein flat, (b) any module whose dual is Gorenstein injective is necessarily Gorentein flat, (c) the Gorenstein projective cotorsion pair is complete and (d) any acyclic complex of projective, injective or flat modules is totally acyclic (in the respective sense). We consider a certain class of groups satisfying all of these properties and show that it is closed under the operation LH defined by Kropholler and the operation {\Phi} defined by the second author. We thus generalize all previously known results regarding these properties over group algebras and place these results in an appropriate framework.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

40 extracted references · 38 canonical work pages

[1]

Secr´ etariat math´ ematique, Paris, 1967, S´ eminaire d’Alg` ebre Commutative dirig´ e par Pierre Samuel, 1966/67

Auslander, M.: Anneaux de Gorenstein, et torsion en alg` ebre co mmutative. Secr´ etariat math´ ematique, Paris, 1967, S´ eminaire d’Alg` ebre Commutative dirig´ e par Pierre Samuel, 1966/67. Texte r´ edig´ e, d’apr` es des expos´ es de Maurice Auslander, par Marguerite Mangeney, Ch ristian Peskine et Lucien Szpiro. ´Ecole Normale Sup´ erieure de Jeunes Filles

work page 1967
[2]

Memoirs of the A merican Mathematical Society 94, American Mathematical Society, Providence, RI 1969

Auslander, M., Bridger, M.: Stable module theory. Memoirs of the A merican Mathematical Society 94, American Mathematical Society, Providence, RI 1969

work page 1969
[3]

Beligiannis, A.: The homological theory of contravariantly ﬁnite su bcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co-) stabilization. Commun . Algebra 28, 4547-4596 (2000) 5In particular, ZGor contains all groups G for which kG is Gorenstein regular. TOTAL ACYCLICITY OF COMPLEXES OVER GROUP ALGEBRAS 23

work page 2000
[4]

Bennis, D., Mahbou, N.: Global Gorenstein dimensions. Proc. Amer . Math. Soc. 138, 461-465 (2010)

work page 2010
[5]

Benson, D.J.: Complexity and Varieties for Inﬁnite Groups, I, II. J. Algebra 193, 260-317 (1997)

work page 1997
[6]

Paciﬁc J

Benson, D.J., Goodearl, K.R.: Periodic ﬂat modules, and ﬂat modules for ﬁnite groups. Paciﬁc J. Math. 196, 45-67 (2000)

work page 2000
[7]

New York J

Biswas, R.: On some cohomological invariants for large families of inﬁ nite groups. New York J. Math. 27, 818-839 (2021)

work page 2021
[8]

Biswas, R.: Benson’s coﬁbrants, Gorenstein projectives and a r elated conjecture. Proc. Edinb. Math. Soc. 64, 779-799 (2021)

work page 2021

Show all 40 references

[9]

Bouchiba, S.: On Gorenstein ﬂat dimension. J. Algebra Appl. 14 (2015), 1550096

2015
[10]

Glas gow Math

Bouchiba, S., Khaloui, M.: Stability of Gorenstein ﬂat modules. Glas gow Math. J. 54, 169-175 (2012)

2012
[11]

Brown, K.S.: Cohomology of groups, Graduate Texts in Mathema tics 87 (Springer, Berlin-Heidelberg- New York, 1982)

1982
[12]

Chen, X.W., Ren, W.: Frobenius functors and Gorenstein homolog ical properties. J. Algebra 610, 18-37 (2022)

2022
[13]

Christensen, L.W., Estrada, S., Thompson, P.: Gorenstein weak global dimension is symmetric. Math. Nachr. 294, 2121-2128 (2021)

2021
[14]

Topology A ppl

Cornick, J., Kropholler, P.H.: On complete resolutions. Topology A ppl. 78, 235-250 (1997)

1997
[15]

arXiv:2104.08602v2

Cort´ es-Izurdiaga, M.,ˇSaroch, J,: The cotorsion pair generated by the Gorenstein proje ctive modules and λ-pure-injective modules. arXiv:2104.08602v2

arXiv
[16]

Dembegioti, F., Talelli, O.: A note on complete resolutions. Proc. Am er. Math. Soc. 138, 3815-3820 (2010)

2010
[17]

Eklof, P., Trlifaj, J.: How to make Ext vanish. Bull. London Math. S oc. 23, 41-51 (2001)

2001
[18]

Algebra 372, 376-396 (2012)

Emmanouil, I.: On the ﬁniteness of Gorenstein homological dimens ions, J. Algebra 372, 376-396 (2012)

2012
[19]

Emmanouil, I., Kaperonis, I.: On K-absolutely pure complexes. J. Algebra 640, 274-299 (2024)

2024
[20]

Emmanouil, I., Talelli, O.: On the ﬂat length of injective modules. J. L ondon Math. Soc. 84, 408-432 (2011)

2011
[21]

Emmanouil, I., Talelli, O.: Triviality criteria for unbounded complexes . J. Algebra 663, 786-814 (2025)

2025
[22]

Emmanouil, I., Talelli, O.: Characteristic modules and Gorenstein (c o-)homological dimension of groups. J. Pure Appl. Algebra 229, 107830 (2025)

2025
[23]

Enochs, E.E., Jenda, O.M.G.: Gorenstein injective and projective modules. Math. Z. 220, 611-633 (1995)

1995
[24]

Berlin: Walter de Gruyter & Co

Enochs, E.E., Jenda, O.M.G.: Relative Homological Algebra (volume 1 ). Berlin: Walter de Gruyter & Co. KG 2011

2011
[25]

Nanjing Daxue Xuebao Shuxue Bannian Kan 10, 1-9 (1993)

Enochs, E.E., Jenda, O.M.G., Torrecillas, B.: Gorenstein ﬂat module s. Nanjing Daxue Xuebao Shuxue Bannian Kan 10, 1-9 (1993)

1993
[26]

Topology Appl

Gedrich, T.V., Gruenberg, K.W.: Complete cohomological functor s on groups. Topology Appl. 25, 203- 223 (1987)

1987
[27]

G¨ obel, R., Trlifaj, J.: Approximations and Endomorphism Algebra s of Modules, de Gruyter Expositions in Mathematics 41 (de Gruyter, Berlin, 2012)

2012
[28]

Holm, H.: Gorenstein homological dimensions. J. Pure Appl. Algebr a 189, 167-193 (2004)

2004
[29]

Iyengar, S., Krause, H.: Acyclicity versus total acyclicity for c omplexes over Noetherian rings. Doc. Math. 11, 207-240 (2006)

2006
[30]

arXiv:2409.16094

Kendal, G.: The stable category and model structures for hier archically deﬁned groups. arXiv:2409.16094

arXiv
[31]

Kropholler, P.H.: On groups of type FP ∞ . J. Pure Appl. Algebra 90, 55-67 (1993)

1993
[32]

Lambek, J.: A module is ﬂat if and only if its character module is injec tive, Canadian Mathematical Bulletin 7, 237-243 (1964)

1964
[33]

Li, W., Guan, J., Ouyang, B.: Strongly FP-injective modules. Comm . Algebra 45, 3816-3824 (2017)

2017
[34]

Mazza, N., Symonds, P.: The stable category and invertible modu les for inﬁnite groups. Adv. Math. 358, 1-26 (2019)

2019
[35]

Neeman, A.: The homotopy category of ﬂat modules, and Groth endieck duality. Invent. Math. 174, 255-308 (2008)

2008
[36]

arXiv:2206.09589v4

Ren, W.: Gorenstein cohomological dimension and stable categor ies for groups. arXiv:2206.09589v4

arXiv
[37]

arXiv:2211.02221v2

Ren, W, Yang, G.: Gorenstein homological dimension and some inva riants of groups. arXiv:2211.02221v2

arXiv
[38]

Selecta Math

ˇSaroch, J., ˇSt ’ov ´ ıˇ cek, J.: Singular compactness and deﬁnability for Σ-cotorsion and Gorenstein modules. Selecta Math. (N.S.) 26, 2020, Paper No. 23 24 IOANNIS EMMANOUIL AND OLYMPIA TALELLI

2020
[39]

arXiv:1412.1615

ˇSt ’ov ´ ıˇ cek, J.: On purity and applications to coderived and singularity categories. arXiv:1412.1615

arXiv
[40]

Talelli, O.: On groups of type Φ.. Arch. Math. (Basel) 89, 24-32 (2007) Department of Mathematics, University of Athens, Athens 15 784, Greece E-mail addresses: emmanoui@math.uoa.gr and otalelli@math.uoa.gr

2007

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Secr´ etariat math´ ematique, Paris, 1967, S´ eminaire d’Alg` ebre Commutative dirig´ e par Pierre Samuel, 1966/67

Auslander, M.: Anneaux de Gorenstein, et torsion en alg` ebre co mmutative. Secr´ etariat math´ ematique, Paris, 1967, S´ eminaire d’Alg` ebre Commutative dirig´ e par Pierre Samuel, 1966/67. Texte r´ edig´ e, d’apr` es des expos´ es de Maurice Auslander, par Marguerite Mangeney, Ch ristian Peskine et Lucien Szpiro. ´Ecole Normale Sup´ erieure de Jeunes Filles

work page 1967

[2] [2]

Memoirs of the A merican Mathematical Society 94, American Mathematical Society, Providence, RI 1969

Auslander, M., Bridger, M.: Stable module theory. Memoirs of the A merican Mathematical Society 94, American Mathematical Society, Providence, RI 1969

work page 1969

[3] [3]

Beligiannis, A.: The homological theory of contravariantly ﬁnite su bcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co-) stabilization. Commun . Algebra 28, 4547-4596 (2000) 5In particular, ZGor contains all groups G for which kG is Gorenstein regular. TOTAL ACYCLICITY OF COMPLEXES OVER GROUP ALGEBRAS 23

work page 2000

[4] [4]

Bennis, D., Mahbou, N.: Global Gorenstein dimensions. Proc. Amer . Math. Soc. 138, 461-465 (2010)

work page 2010

[5] [5]

Benson, D.J.: Complexity and Varieties for Inﬁnite Groups, I, II. J. Algebra 193, 260-317 (1997)

work page 1997

[6] [6]

Paciﬁc J

Benson, D.J., Goodearl, K.R.: Periodic ﬂat modules, and ﬂat modules for ﬁnite groups. Paciﬁc J. Math. 196, 45-67 (2000)

work page 2000

[7] [7]

New York J

Biswas, R.: On some cohomological invariants for large families of inﬁ nite groups. New York J. Math. 27, 818-839 (2021)

work page 2021

[8] [8]

Biswas, R.: Benson’s coﬁbrants, Gorenstein projectives and a r elated conjecture. Proc. Edinb. Math. Soc. 64, 779-799 (2021)

work page 2021