Pith. sign in

REVIEW 2 major objections 5 minor 41 references

Group class operations and homological conditions

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · 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 closing weakly Gorenstein regular groups under the operations LH and Φ_inj preserves virtual Gorensteinness of the group algebra, and closing finite groups under LH, Φ, and Φ_flat preserves Moore's conjecture.

desk verdict Genuinely new closure theorems for virtually Gorenstein algebras and Moore's conjecture, but Theorem A rests on an unpublished preprint [23] that must be made available before the main proof is independently checkable. read the letter →

arxiv 2505.00513 v2 pith:SZDIP2Z3 submitted 2025-05-01 math.GR math.KT

Ioannis Emmanouil , Wei Ren This is my paper · ORCID
classification math.GRmath.KT MSC 20J0516E6518G25
keywords virtuallyGorensteinalgebrasMoore'sconjectureLH-hierarchyPhi-operationprojectivemodulesinjectiverelativeprojectivitygroupcohomology
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 two closure theorems for hierarchical group-class operations. Theorem A says that if you start with any group whose group algebra is weakly Gorenstein regular and repeatedly apply the operation LH and the operation $\Phi_{\mathrm{inj}}$, every group you reach still has a virtually Gorenstein group algebra. Theorem B says that if you start with finite groups and repeatedly apply LH, $\Phi$, and $\Phi_{\mathrm{flat}}$, every group you reach satisfies Moore's conjecture, meaning a module that becomes projective after restriction to a suitable finite-index subgroup was already projective. These results matter because the two operations construct very large classes of groups, so the theorems move two homological properties from finite and LHF groups to a much wider universe.

What carries the argument

The machinery is the pair of group-class operations. LH starts from a class $C$ and closes under groups that admit a cellular action on a finite-dimensional contractible complex whose isotropy subgroups come from earlier stages of a transfinite induction, then adds all groups whose finitely generated subgroups lie in that hierarchy; $\Phi$ puts $G$ into $\Phi C$ when a $kG$-module has finite projective dimension exactly when its restrictions to $C$-subgroups have bounded projective dimension, with $\Phi_{\mathrm{flat}}$ and $\Phi_{\mathrm{inj}}$ obtained by replacing projective dimension by flat or injective dimension. The paper couples these operations with Ext$^1$-orthogonality: the technical core shows that membership in the right orthogonal of Gorenstein projective modules, or the left orthogonal of Gorenstein injective modules, is inherited under restriction from base-class subgroups to subgroups produced by any of the operations (Theorems 3.1 and 3.4). The orthogonal classes are then intersected over base subgroups to prove the virtual Gorenstein equality (Theorem 4.2). For Moore's conjecture, the operative object is the finite-index subgroup $H$ satisfying Moore's condition; Proposition 5.2 and Theorem 5.3 show projectivity, flatness, or injectivity of a restricted module climbs the LH, $\Phi$, and $\Phi_{\mathrm{flat}}$ hierarchy.

What would settle it

Exhibit a group $G$ that is a single $\Phi_{\mathrm{inj}}$-step or LH-step away from a base class for which some $kG$-module violates the relevant Ext$^1$-orthogonality equality, or an $\overline{F}$-group with a module projective on every Moore-condition finite-index restriction but not globally projective; either would refute Theorem 3.1, Theorem 4.2, or Theorem B. A cheaper check is to verify Theorem 2.1's closure claim in the companion preprint on the smallest nontrivial classes.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that virtual Gorensteinness of group algebras and Moore's conjecture both propagate through the closure process. If $G$ lies in $\overline{W}$, the smallest class containing all groups with $kG$ weakly Gorenstein regular and closed under LH and $\Phi_{\mathrm{inj}}$, then $kG$ is virtually Gorenstein: $GProj(kG)^\perp = {}^\perp GInj(kG)$, i.e. the classes Ext$^1$-orthogonal to Gorenstein projective and to Gorenstein injective modules coincide. If $G$ lies in $\overline{F}$, the smallest class containing all finite groups and closed under LH, $\Phi$ and $\Phi_{\mathrm{flat}}$, then $G$ satisfies Moore's conjecture over $k$. The mechanism is an Ext$^1$-orthogonality transfer: a module whose restrictions to base-class subgroups lie in the orthogonal complement of Gorenstein modules has the same property after restriction to any subgroup built by LH, $\Phi$, $\Phi_{\mathrm{flat}}$ or $\Phi_{\mathrm{inj}}$ (Theorems 3.1 and 3.4), and intersecting these conditions over base subgroups yields the desired equality for the whole group (Theorem 4.2). For Moore's conjecture, the proof propagates projectivity of restricted modules up the same hierarchy using dimension shifting and the adjunction between induction and restriction (Theorem 5.3).

Load-bearing premise

The argument's load-bearing premise is Theorem 2.1, imported from the companion preprint [23]: the classes $X$, $X'$ and $Y$ of groups where the Gorenstein module classes are perfectly behaved are closed under LH, $\Phi$, $\Phi_{\mathrm{flat}}$ and $\Phi_{\mathrm{inj}}$, and if that closure fails the orthogonality transfers in Sections 3 and 4 and hence Theorems A and B no longer follow.

Editorial extensions

If this is right

  • Every group in the closure $\overline{W}$ has a group algebra whose Gorenstein projective and Gorenstein injective orthogonal classes coincide; the paper records that this makes the associated relative derived category triangulated equivalent to homotopy categories of Gorenstein projective and Gorenstein injective modules.
  • Every group in the closure $\overline{F}$ satisfies Moore's conjecture over any commutative ring $k$, so projective-on-a-suitable-finite-index-subgroup implies projective for all $kG$-modules over such groups.
  • The flat and injective versions of Moore's conjecture also propagate: LH with $\Phi_{\mathrm{flat}}$ preserves the flat version, and LH with $\Phi_{\mathrm{inj}}$ preserves the injective version.
  • Because the closures are built by transfinite iteration, the results apply not only to LHF-groups but to every group reached by iterating the operations, including groups whose structure is far from finite.

Reading between the lines

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

  • The same Ext$^1$-orthogonality transfer is a general template: any homological property that can be stated as an equality between a module class and its Ext-orthogonal should survive the LH/Φ closure once the base class satisfies the closure axioms, so the technique may generalize to other relative homological conditions.
  • The closures $\overline{F}$ and $\overline{W}$ are likely to contain groups not of type FP$_\infty$ and not in LHF; if so, Moore's conjecture and virtual Gorensteinness are being established for genuinely new examples, though the paper itself does not exhibit such groups.
  • A direct way to test the scope of the results is to look for a group in $\overline{W}$ that is not in the closure of finite groups under the same operations; if such groups exist, the two theorems cover independent families of examples.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies two group-class operations from the literature, Kropholler's LH and Talelli's Φ (together with its flat and injective variants), and proves that certain homological properties propagate along the resulting hierarchical closures. The first main result, Theorem A, states that if kG is weakly Gorenstein regular for all groups in a starting class W, then every group in the closure of W under LH and Φ_inj has a virtually Gorenstein group algebra. The second main result, Theorem B, states that every group in the smallest class containing the finite groups and closed under LH, Φ, and Φ_flat satisfies Moore's conjecture over k. The proof of Theorem A proceeds by characterizing Ext^1-orthogonal classes of Gorenstein modules over group algebras of groups obtained by performing LH or Φ on suitable group classes (§3), then proving that the class Z of Y-groups with virtually Gorenstein group algebra is closed under LH and Φ_inj (§4). Theorem B is proved in §5 by extending the Aljadeff–Meir argument through the operations LH, Φ, and Φ_flat, together with results of Bahlekeh–Salarian and Benson–Goodearl. The paper is carefully written and the arguments in §3 and §5 are detailed, but the proof of Theorem A depends crucially on Theorem 2.1, which is imported from an unpublished preprint [23] by one of the authors.

Significance. If Theorem A and Theorem B are correct, the paper establishes that very large hierarchically defined classes of groups inherit two nontrivial homological properties: virtual Gorensteinness of the group algebra and Moore's conjecture. Theorem B is a genuine extension of the previously known LHF result, and its proof is essentially checkable from published sources. The proof of Theorem A is also conceptually interesting: it shows how cotorsion-pair completeness and orthogonality can be transported along LH and Φ operations, provided the classes X, X′, and Y have the closure properties stated in Theorem 2.1. However, the significance of Theorem A is currently conditional, because Theorem 2.1—the single most load-bearing ingredient—is cited to an unpublished preprint with no public identifier and is not proved in the manuscript. The paper's other contributions, especially the orthogonality results in §3 once Theorem 2.1 is accepted, and the Moore-conjecture results in §5, are solid and well-motivated.

major comments (2)
  1. [§2, Theorem 2.1] Theorem 2.1 is the central load-bearing result of the paper, yet it is quoted from the unpublished preprint [23] (Emmanouil and Talelli, "Total acyclicity of complexes over group algebras"). The theorem supplies the subgroup-closure and the LH-, Φ-, and Φ_flat-closure of X and the LH- and Φ_inj-closure of Y; these are used directly in Theorem 3.1, Theorem 3.4, Corollary 3.2, Corollary 3.5, and Theorem 4.2, and therefore in the proof of Theorem A. Since [23] has no public identifier and is co-authored by one of the present authors, the proof of Theorem A cannot be checked from the text alone. The authors should include a proof of Theorem 2.1 (or at least a detailed statement with full proof in an appendix), or replace the reference with a published, publicly verifiable source. Without this, Section 4, and hence Theorem A, is not self-contained.
  2. [§4, Theorem 4.2(ii)] In the proof of Theorem 4.2(ii), the assertion that Φ_inj Z ⊆ Φ_flat Z is justified by the citation "cf. [23, Lemma 1.5]". This is another dependence on the unpublished preprint [23]. Even if a reader were willing to accept Theorem 2.1 from [23], the additional fact that every Φ_inj-group is a Φ_flat-group for the class Z is needed to apply Corollary 3.2 in the Φ_inj closure argument. The paper should either prove this inclusion directly or supply a publicly available reference for [23, Lemma 1.5]. As written, the Φ_inj closure of Z is not verifiable from the manuscript.
minor comments (5)
  1. [§1.I] The name "Echmann-Shapiro" should be "Eckmann-Shapiro" in the displayed isomorphisms and in the surrounding text.
  2. [§2, properties (1')–(5')] In the proof of properties (1')–(5') for Gorenstein injective modules, the sentence "whereas (2) is an immediate consequence" should read "whereas (2') is an immediate consequence"; the numbering of the prongs is otherwise confusing.
  3. [§5, Proposition 5.2] In Proposition 5.2(i), the complex X* is right-bounded and acyclic with projective terms except possibly at the end; the contractibility argument would be clearer if the authors explicitly say that the restricted complex is a projective resolution of res_H^G M, so that Hom_{kH}(res_H^G X*, Q) is acyclic. The current wording is acceptable but slightly terse.
  4. [References] Reference [23] should include an arXiv identifier or a journal publication status; as it stands, the reader cannot locate the preprint. Similarly, [16] and [35] are arXiv preprints and could benefit from submission status information.
  5. [§1.IV] The continuity of the operations LH, Φ, Φ_flat, and Φ_inj is cited to [23, Lemmas 1.3 and 1.4]. Since these lemmas are also part of the unpublished preprint [23], they should either be proved in the paper or explicitly marked as standard results with a published reference.

Circularity Check

1 steps flagged · score 2.0 of 10

No definitional or fitted-input circularity is present; the only real issue is load-bearing self-citation for Theorem A: the closure properties of the classes X, X′, Y, and the inclusion Φ_injZ ⊆ Φ_flatZ, are quoted from the authors' unpublished preprint [23], so the main closure proof is not checkable from the text alone.

  1. self citation load bearing [Theorem 2.1 (Section 2), applied in Theorems 3.1, 3.4, 4.2 and Lemma 4.1; Theorem 4.2(ii) also uses [23, Lemma 1.5]]
    "The classes X, X′ and Y are subgroup-closed and have the following properties: (i) X = X′ ⊇ Y; cf. [23, Corollary 2.3]. (ii) X is closed under the operations LH, Φ and Φ f lat; cf. [23, Theorem 3.3]. (iii) Y is closed under the operations LH and Φ inj; cf. [23, Theorem 4.1]. ... Since G is contained in Φ injZ ⊆ Φ f latZ (cf. [23, Lemma 1.5]), we can apply Corollaries 3.2 and 3.5 in exactly the same way as in (i) above..."

    The proof of Theorem A proceeds by showing that the class Z is LH-closed and Φ_inj-closed and that W ⊆ Z. Every essential use of the operations in that proof invokes Theorem 2.1: Y is LH-closed, X is LH/Φ/Φ_flat-closed, X = X′ ⊇ Y, and the subgroup-heredity and completeness properties of Gorenstein cotorsion pairs over X- and Y-groups. Theorem 4.2(ii) additionally needs the inclusion Φ_injZ ⊆ Φ_flatZ, quoted from [23, Lemma 1.5]. All of these are taken from [23], an unpublished preprint by the first author and O. Talelli with no public identifier. Thus the central closure argument is not derivable from the text alone; it reduces to a self-citation that is not publicly checkable.

full rationale

No definitional, fitted-input, renaming, or ansatz-smuggling circularity was found. Theorem A's transfer argument (closure of Z under LH and Φ_inj, and W ⊆ Z via [19]) is structurally sound if Theorem 2.1 and [23, Lemma 1.5] are accepted. The only flagged step is that these load-bearing ingredients are imported from the authors' unpublished self-cited preprint [23], so the proof of Theorem A is not fully checkable from the present text. This is a real but minor self-citation dependency, not a construction that builds the conclusion into the assumptions. Theorem B is independent and checkable from published sources, and the countable/uncountable cardinality arguments in Theorems 3.1 and 3.4 do not introduce circularity. Accordingly the circularity score is 2.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted parameters or data. It defines several group classes (X, Y, Z, W, F, M, etc.), which are definitions rather than speculative entities. The central claims rest on imported theorems, notably the unpublished [23], and on standard results in Gorenstein homological algebra and group cohomology.

assumptions (3)
  • ad hoc to paper Theorem 2.1 (closure of X, X', Y under LH, Φ, Φ_flat, Φ_inj), imported from [23].
    Used in Sections 2-4 to transfer orthogonality and to prove Theorem A. It is not proved here and [23] is an unpublished preprint without a public identifier.
  • domain assumption Standard Gorenstein homological algebra results: completeness of cotorsion pairs (GProj,GProj⊥), PGF = GProj on X-groups, Eklof's lemma, Eckmann-Shapiro isomorphisms.
    These are invoked throughout Sections 2-3 as properties (1)-(6) and (1')-(5') and in the proofs of Theorems 3.1 and 3.4. They are standard in the field, mostly from [37], [29], [25].
  • domain assumption Chouinard's theorem: finite groups satisfy Moore's conjecture; Benson-Goodearl theorem [9, Corollary 4.8].
    Used in Section 5 (Corollary 5.8 and Proposition 5.2(iv)) as base input and descent tool for Moore's conjecture. These are published results, not proved in the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Group class operations and homological conditions." pith.science (2026). https://pith.science/paper/SZDIP2Z3

@misc{pith2026250500513,
  author       = {Pith},
  title        = {Pith review of: Group class operations and homological conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SZDIP2Z3}},
  note         = {Machine review of arXiv:2505.00513}
}
abstract

Kropholler's operation ${\scriptstyle{{\bf LH}}}$ and Talelli's operation $\Phi$ can be often used to formally enlarge the class of available examples of groups that satisfy certain homological conditions. In this paper, we employ this enlargement technique regarding two specific homological conditions. We thereby demonstrate the abundance of groups that (a) have virtually Gorenstein group algebras, as defined by Beligiannis and Reiten, and (b) satisfy Moore's conjecture on the relation between projectivity and relative projectivity, that was studied by Chouinard, Aljadeff, Cornick, Ginosar, Kropholler and Meir.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 28 canonical work pages

  1. [23]

    (preprint)

    Emmanouil, I., Talelli, O.: Total acyclicity of complexes over group a lgebras. (preprint)

    work page

  2. [1]

    Aljadeff, E., Cornick, J., Ginosar, Y., Kropholler, P.H.: On a conject ure of Moore. J. Pure Appl. Algebra 110, 109-112 (1996)

    work page 1996

  3. [2]

    Aljadeff, E., Meir, E.: Nilpotency of Bocksteins, Kropholler’s hierar chy and a conjecture of Moore. Adv. Math. 226, 4212-4224 (2011)

    work page 2011

  4. [3]

    Bahlekeh, A., Salarian, Sh.: New results related to a conjecture o f Moore. Arch. Math. 100, 231–239 (2013)

    work page 2013

  5. [4]

    Beligiannis, A.: The homological theory of contravariantly finite su bcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co-) stabilization. Commun . Algebra 28, 4547-4596 (2000)

    work page 2000

  6. [5]

    Beligiannis, A.: Cohen–Macaulay modules, (co)torsion pairs and vir tually Gorenstein algebras. J. Algebra 288, 137-211 (2005)

    work page 2005

  7. [6]

    Beligiannis, A., Reiten, I.: Homological and Homotopical Aspects of Torsion Theories. Mem. Amer. Math. Soc. 188 (2007)

    work page 2007

  8. [7]

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

    2010

Show all 41 references
  1. [8]

    Benson, D.: Complexity and varieties for infinite groups I, II. J. A lgebra 193, 260-287, 288-317 (1997)

    1997

  2. [9]

    Pacific J

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

    2000

  3. [10]

    New York J

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

    2021

  4. [11]

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

    1982

  5. [12]

    Chen, X.-W.: Homotopy equivalences induced by balanced pairs. J . Algebra 324, 2718-2731 (2010)

    2010

  6. [13]

    Forum Math

    Christensen, L.W., Estrada, S., Liang, L., Thompson, P., Wang, J .: One-sided Gorenstein rings. Forum Math. 36, 1411-1433 (2024)

    2024

  7. [14]

    Christensen, L.W., Estrada, S., Thompson, P.: Gorenstein weak global dimension is symmetric. Math. Nachr. 294, 2121-2128 (2021) 20 IOANNIS EMMANOUIL AND WEI REN

    2021

  8. [15]

    Chouinard, L.: Projectivity and relative projectivity over grou p rings. J. Pure Appl. Algebra 7, 287-302 (1976)

    1976

  9. [16]

    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

  10. [17]

    78, 235-250 (1997)

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

    1997

  11. [18]

    Cornick, J., Kropholler, P.H.: Homological finiteness conditions fo r moduels over group algebras. J. London Math. Soc. 58, 49-62 (1998)

    1998

  12. [19]

    Di, Z., Liang, L., Wang, J.: Virtually Gorenstein rings and relative ho mology of complexes. J. Pure Appl. Algebra 227, 107127 (2023)

    2023

  13. [20]

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

    2001

  14. [21]

    Algebra 372, 376-396 (2012)

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

    2012

  15. [22]

    Emmanouil, I., Talelli, O.: Gorenstein dimension and group cohomolog y with group ring coefficients. J. London Math. Soc. 97, 306-324 (2018)

    2018

  16. [24]

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

    1995

  17. [25]

    De Gruy ter Expositions in Mathematics no

    Enochs, E.E., Jenda, O.M.G.: Relative Homological Algebra. De Gruy ter Expositions in Mathematics no. 30, New York: Walter De Gruyter, 2000

    2000

  18. [26]

    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

  19. [27]

    Topology Appl

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

    1987

  20. [28]

    Rocky Mt

    Gillespie, J.: On Ding injective, Ding projective, and Ding flat module s and complexes. Rocky Mt. J. Math. 47, 2641-2673 (2017)

    2017

  21. [29]

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

    2004

  22. [30]

    Ikenaga, B.M.: Homological dimension and Farrell cohomology. J. Algebra 87, 422-457 (1984)

    1984

  23. [31]

    arXiv:2409.16094

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

    arXiv

  24. [32]

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

    1993

  25. [33]

    Lambek, J.: A module is flat if and only if its character module is injec tive. Canad. Math. Bull. 7, 237-243 (1964)

    1964

  26. [34]

    Mazza, N., Symonds, P.: The stable category and invertible modu les for infinite groups. Adv. Math. 358, 106853 (2019)

    2019

  27. [35]

    arXiv:2211.02221v2

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

    arXiv

  28. [36]

    arXiv:1412.1615

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

    arXiv

  29. [37]

    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

    2020

  30. [38]

    Talelli, O.: On groups of type Φ. Arch. Math. 89, 24-32 (2007)

    2007

  31. [39]

    Wang, J., Estrada, S.: Homotopy equivalences and Grothendiec k duality over rings with finite Gorenstein weak global dimension. J. Algebra 678, 769-808 (2025)

    2025

  32. [40]

    Wang, J., Yang, G., Shao, Q., Zhang, X.: On Gorenstein global and Gorenstein weak global dimensions. Colloq. Math. 174, 45-67 (2023)

    2023

  33. [41]

    Wang, J., Zhang, X.: On rings with finite Gorenstein weak global dim ension. J. Algebra Appl. 22, 2350111 (2023) Ioannis Emmanouil Department of Mathematics, University of Athens, Athens 15784 , Greece E-mail: emmanoui@math.uoa.gr Wei Ren School of Mathematical Sciences, Chongq...

    2023

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.