{"id":"2f18cbdb-e12b-446d-95a9-d4d308116cef","arxiv_id":"2608.11376","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"This paper computes the Cp-equivariant homotopy groups of ETHH(HF_p) for odd primes and the equivariant homotopy of ETHH(MU_G) and ETHH(MU_R), giving foundational inputs for equivariant trace methods.","lead":"Mathematicians computed equivariant topological Hochschild homology, a refined invariant used in trace methods for algebraic K-theory, for the field with p elements and for equivariant complex cobordism spectra. The odd-prime computation shows the equivariant version of Bokstedt periodicity is substantially more complex than the classical statement.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A depends on Proposition 5.11, whose E_2-formality bridge for HF_p^{ΦC_p} is the least secured external input; a failure there would collapse the Bokstedt-periodicity computation.","rationale":"The reader's weakest_assumption is exactly the formality dependency of Proposition 5.11, and I agree it is the most load-bearing step for the paper's headline result. I checked the surrounding argument for possible internal flaws: the infinite-dimensionality of the Tate homotopy groups in each degree is consistent with the claimed subring once one accounts for the b-polynomial grading and the action of q_* on products of c and d; the Mayer-Vietoris boundary maps are in fact zero for the computed q-values, even though Corollary 5.21's stated justification from injectivity alone is logically compressed; and the Bökstedt comparison in Lemma 5.27 appears to be salvageable for odd p because any 2-torsion in positive even homotopy of THH(HZ) is invisible to Tate cohomology of C_p for p odd. No contradiction internal to the paper emerged. The remaining genuine risk is external: if the E_2-formality theorems cited for F_p[y] and F_p[x]/x^2 do not hold in the exact form needed, or if the passage from CDGA formality to an E_2-algebra equivalence is not justified, then Proposition 5.11, and hence Theorem A, loses its foundation. Since the cited results are published and the paper does not misuse them in an obviously detectable way, I do not recommend changing the reader's ACCEPT verdict, but the proposed verification would raise confidence.","tokens_in":21779,"tokens_out":64552,"duration_ms":552485,"concrete_test":"Independently verify that [Hor25, Theorem 3.5] applies verbatim to F_p[y] with |y|=2 as an E_2-algebra, not merely as an E_∞-algebra over F_p, and supply an explicit construction of the E_2-HF_p-algebra map HF_p ∧ S^1_+ ∧ (ΩS^3)_+ → HF_p^{ΦC_p} in Proposition 5.11. Concretely, compute the first obstruction to E_2-formality of HF_p ∧ ΩS^3_+ (for example, the Dyer-Lashof operation Q^1 on the degree-2 class in the associated graded of the E_2-operad) and check that it vanishes; if a nontrivial obstruction appears, recompute Theorem 5.12 without this equivalence and see whether the geometric fixed point ring changes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central computation of Theorems A and B is routed through Corollary 5.7, which identifies the geometric fixed points of ETHH(HF_p) with THH(HF_p^{ΦC_p}), and then through Proposition 5.11, which asserts an equivalence of HF_p-E_2-algebras HF_p^{ΦC_p} ≃ HF_p ∧ S^1_+ ∧ (ΩS^3)_+. This equivalence is the only bridge between a geometric fixed point object and a smash product whose THH is computable. Its proof rests on two formality claims: intrinsic formality of F_p[y] (|y|=2) as an E_2-DGA, cited to [Hor25, Theorem 3.5] and [BM22, Theorem 2.1], and intrinsic formality of F_p[x]/x^2 (|x|=1) as a CDGA, asserted as 'straightforwardly checked.' The paper does not supply the explicit E_2-algebra map whose existence is needed to upgrade these formality statements to the actual equivalence of E_2-HF_p-algebras in Proposition 5.11. This is the least secured step in the chain: it is an external input, not machine-checked, and no independent verification is offered. I found no internal inconsistency in the subsequent computation, and checks of the Mayer-Vietoris and dimension-counting steps did not reveal a contradiction. But if the cited formality theorems fail, or if the upgrade from CDGA formality to E_2-algebra formality is invalid, Proposition 5.11 collapses, and with it Theorems A and B.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes equivariant topological Hochschild homology (ETHH) for several fundamental ring G-spectra. The main results are: for odd p, a complete description of the C_p-graded homotopy ring of ETHH(HF_p) (Theorem A), obtained via the Tate square together with a new computation of the C_p-geometric fixed points (Theorem B); and computations of ETHH for the equivariant complex cobordism spectra MU_G and MU_R (Theorems C–E). The proofs use the Tate square, geometric fixed points, equivariant factorization homology and Thom spectra, a new multiplicative equivariant bar spectral sequence, and explicit G-CW structures. The paper is computational in character and explicitly positions the results as input for future work on equivariant trace methods and equivariant topological cyclic homology.","tokens_in":22037,"tokens_out":32525,"duration_ms":289572,"significance":"If the computations are correct, this is a substantial contribution to a young area: it supplies the first nontrivial odd-primary equivariant Bokstedt periodicity computation and gives explicit multiplicative descriptions of ETHH for the standard equivariant cobordism spectra. The paper also develops tools of independent interest, especially the multiplicative G-equivariant bar spectral sequence and the upgrade of the equivariant Thom spectrum/factorization homology compatibility to multiplicative statements. A clear strength is that the central computations are checked against external benchmarks: Bokstedt's non-equivariant computation of THH(HF_p), the established computation of THH(MU), and the already-existing equivariant factorization homology framework. The reliance on the authors' own and collaborators' prior work is not circular, since the cited results are independent of the present computations.","major_comments":[{"comment":"This proposition is the only bridge between the geometric fixed points of ETHH(HF_p) and a smash product whose THH is computable, so Theorems A and B rest on it. The proof asserts an equivalence of E_2-HF_p-algebras HF_p^{ΦC_p} ≃ HF_p ∧ S^1_+ ∧ (ΩS^3)_+, but the argument as written does not supply the required E_2-algebra maps. For the F_p[x]/x^2 factor, the text invokes only CDGA intrinsic formality, 'straightforwardly checked from the definition,' and for F_p[y] it cites E_2 formality from [Hor25, Theorem 3.5] and [BM22, Theorem 2.1]. It is not explained why these formality statements can be upgraded to maps of E_2-HF_p-algebras with the specified effect on generators, nor why the two chosen equivalences are compatible under the smash product. The surjectivity and finite-dimensionality argument is fine once such an E_2 map exists, but the existence of the map is exactly the load-bearing point. Please provide a full proof or a direct reference that covers the E_2-algebra statement, not merely the underlying DGA or CDGA formality.","section":"§5.1, Proposition 5.11"},{"comment":"The base case of the induction is incorrect as stated: SUG(C) is SU(1), which is the trivial group, not S^1. If a degree-1 exterior generator were introduced, it would contradict the theorem's own degrees, where the first exterior generator λ_1 has degree 2·1+1 = 3. The induction should begin with n = 2, where SUG(W_2) ≃ SU(2) ≃ S^3, or the proof should explicitly use U(1) if that is the intended object. Since Theorem D and Corollary 4.12 depend on this proof, the base case needs to be fixed.","section":"§4.2, proof of Theorem 4.11"},{"comment":"The proof of the computation of H_*(LS^3; F_p) is not correct as written. In the Serre spectral sequence for ΩS^3 → LS^3 → S^3, the generator of H^*(S^3) has bidegree (3,0) and the polynomial generator of H^*(ΩS^3) has bidegree (0,2), so d_2 of the fiber generator is not a possible target; the displayed 'd_2(e) = kd' is also dimensionally inconsistent. The argument also writes LS^2 for LS^3. The statement itself is standard, and the collapse can be justified either by degree reasons or by the product splitting LS^3 ≃ S^3 × ΩS^3, but the proof should be rewritten. Since Corollary 5.15 and hence Theorem B use this computation, the correction is not purely cosmetic.","section":"§5.1, Lemma 5.14"}],"minor_comments":[{"comment":"The E_∞ page is displayed as Λ(x) ⊗ F_p[x,y,b], which is inconsistent: x cannot be both an exterior and a polynomial generator. It should presumably read Λ(x) ⊗ F_p[y^{±1}, b], matching Proposition 5.19.","section":"§5.3, Proposition 5.18"},{"comment":"The last bullet says 'where each z_i is in the image of j_i,' but no elements z_i have been defined. Please either define them or delete the phrase.","section":"§5.4, Proposition 5.23"},{"comment":"In the same lemma, the notation is inconsistent: the polynomial generator in degree 2 is called c in the statement but d in the proof, and the phrase 'determined by is d_2(e)' contains a grammatical error.","section":"§5.1, Lemma 5.14"},{"comment":"The description 'generated by b and all monomials divisible by either e or a divided power of a' should clarify that the divided powers are the positive ones γ_n(a), n ≥ 1, so that the reader does not read γ_0(a) = 1 as making the generating set trivial.","section":"§5.4, Theorem 5.29 and Theorem A"}],"recommendation":"major_revision","confidential_remarks":"The paper is mathematically rich and likely correct in its main conclusions, but the two flagged proof gaps—the E_2-algebra upgrade in Proposition 5.11 and the false base case in Theorem 4.11—are both in load-bearing positions. If the authors can supply the missing E_2-algebra arguments and correct the base case, I expect the manuscript to be suitable for publication. The proofreading issues in Lemma 5.14 and Proposition 5.18 are minor but should also be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this paper delivers the first odd-prime computations of ETHH(HF_p), and the first computations of ETHH for equivariant complex cobordism spectra. I checked the main line of argument and it holds up. The soft spot is Proposition 5.11, not because I found a gap, but because the equivalence HF_p^{Phi C_p} equiv HF_p wedge S^1_+ wedge (Omega S^3)_+ rests on cited E2-formality results rather than on a proof inside the paper. That is a normal external input, but it is load-bearing for Theorems A and B, so a referee should look at it.\\n\\nWhat is actually new: Theorem A's description of pi^{C_p}_*(ETHH(HF_p)) for odd p is new; the p=2 case was known. The MU_G and MU_R computations in Section 4 are new, and the RO(C2)-graded Green functor statement for ETHH(MU_R) is a solid result. The multiplicative equivariant bar spectral sequence of Section 3 is a useful contribution in its own right.\\n\\nThe paper does its work carefully. The Tate square, geometric fixed points, and HFPSS/Tate spectral sequence arguments are standard but applied with care. I spot-checked the generator actions in Proposition 5.23 and Corollary 5.28; they are consistent, and the final subring description in Theorem 5.29 follows from the stated images. No free parameters, no invented entities. Reliance on the authors' prior work (CGK25, HHK+24, Wis25) is normal and not circular.\\n\\nSoft spots, in order of seriousness. First, Proposition 5.11: the paper says the CDGA formality of F_p[x]/x^2 is 'straightforwardly checked' and the E2-formality of F_p[y] is cited to Hor25 and BM22. What is missing is an explicit E2-algebra map or a lemma showing the homology equivalence upgrades to an E2-equivalence. I believe it is true, but it is the part I would want a referee to verify. Second, minor typos: Lemma 5.14 has LS2 where it should be LS3 and a muddled d2(e) sentence; Proposition 5.18 should have y^{+-1}, not y, in the E_infty page. These are clearly typographical and do not affect the arguments. Third, Theorem D is stated for compact abelian Lie G and H subset G, but the ETHH corollary only applies when G is finite abelian; the paper is clear about this, so it is not a flaw, just a scope condition to keep straight.\\n\\nWho is this for: anyone working on equivariant trace methods, equivariant algebraic K-theory, or computational equivariant stable homotopy. It deserves a serious referee. I would send it to an expert in equivariant homotopy theory with a note to look at Proposition 5.11, then expect acceptance after minor revisions.","headline":"New odd-prime ETHH computations that look correct; the only real risk is the external E2-formality input behind the geometric fixed points, worth a referee's attention.","tokens_in":22683,"tokens_out":2864,"would_cite":true,"duration_ms":26393,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55P91","55N22","19D55"],"pacs":[],"model":"deepseek-v4-flash","headline":"For odd primes, the C_p-equivariant topological Hochschild homology of HF_p is a specific subring of a divided power algebra, showing equivariant periodicity is richer than the classical one.","keywords":["equivariant topological Hochschild homology","Tate square","geometric fixed points","equivariant Thom spectra","complex cobordism","divided power algebras","trace methods","equivariant algebraic K-theory"],"falsifier":"Construct an E2-DGA over \\mathbb{F}_p with homology \\mathbb{F}_p[y], |y|=2, that is not quasi-isomorphic to the formal algebra \\mathbb{F}_p[y]; its existence would invalidate Proposition 5.11 and with it the geometric-fixed-point computations of Theorems A and B.","tokens_in":21491,"feed_emoji":"🧮","tokens_out":10598,"duration_ms":83332,"temperature":0.7,"pith_summary":"This paper computes the equivariant topological Hochschild homology (ETHH) of two families of basic equivariant ring spectra: Eilenberg–MacLane spectra for F_p with a cyclic group action, for odd primes, and equivariant complex cobordism spectra. For the F_p case, the central result is a complete description of the C_p-fixed homotopy ring as a specific subring of a divided power algebra, a direct equivariant analogue of the classical periodicity theorem for THH(HF_p). The computation shows that the equivariant version of periodicity is substantially richer than the classical one: the answer involves divided powers and an exterior generator, not just a single polynomial generator. For cobordism spectra, the paper identifies ETHH(MU_G) and ETHH(MU_R) as extended Thom spectra and computes their homotopy rings. These computations are intended as input for future work on equivariant topological cyclic homology and trace methods for equivariant algebraic K-theory.","feed_headline":"For odd primes, equivariant THH of F_p is a divided-power ring","feed_subtitle":"The C_p-fixed homotopy ring contains divided powers and an exterior class, reshaping equivariant periodicity and future trace-method…","key_machinery":"Three mechanisms carry the computations. The Tate square, a homotopy pullback relating the fixed points, geometric fixed points, homotopy fixed points, and Tate fixed points of a C_p-spectrum, is used to reconstruct \\pi_*^{C_p}(\\mathrm{ETHH}(H\\mathbb{F}_p)) from the geometric and Tate sides. The geometric-fixed-point side is tamed by two facts: geometric fixed points commute with ETHH, and the geometric fixed points of H\\mathbb{F}_p are intrinsically formal as an E2-DGA, so they decompose as H\\mathbb{F}_p\\wedge $S^{1}$_+\\wedge (\\$\\Omega$ $S^{3}$)_+. For the Thom-spectrum results, the central object is equivariant factorization homology together with the equivariant Thom spectrum functor, which promotes the identification of ETHH of a Thom spectrum to an equivalence of E_\\infty ring spectra; a multiplicative equivariant bar spectral sequence and an equivariant cellular decomposition of the infinite special unitary group then yield the homotopy rings.","core_discovery":"The paper's central discovery is a complete computation of the C_p-fixed homotopy ring of equivariant topological Hochschild homology of the Eilenberg–MacLane spectrum H\\mathbb{F}_p, for odd primes p. Writing \\Gamma for divided powers and \\Lambda for exterior algebras, the answer is the subring of \\Gamma[a]\\otimes \\mathbb{F}_p[b,c]\\otimes \\Lambda[d,e] generated by b and by all monomials divisible by either e or a divided power of a, with a,b,c in degree 2, d in degree 1, and e in degree 3. In the non-equivariant limit this reduces to the classical periodicity theorem \\pi_*\\mathrm{THH}(H\\mathbb{F}_p)\\cong \\mathbb{F}_p[b], so the theorem shows that equivariant periodicity is strictly more intricate. The method is a Tate-square analysis: the geometric fixed points are computed first, as H\\mathbb{F}_p\\wedge (\\$\\Omega$ $S^{3}$)_+\\wedge $S^{1}$_+\\wedge CP^\\infty_+\\wedge ($LS^{3}$)_+, giving the full ring, and the fixed-point subring is then isolated by tracking the map from geometric to Tate fixed points on generators. For equivariant complex cobordism, the paper identifies \\mathrm{ETHH}(MU_G) with MU_G\\wedge \\Sigma^\\infty_+ SU_G and \\mathrm{ETHH}(MU_\\mathbb{R}) with MU_\\mathbb{R}\\wedge \\Sigma^\\infty_+ B(BU^\\mathbb{R}), with explicit homotopy rings.","pith_inferences":["The paper leaves the RO(C_p)-graded ring of \\mathrm{ETHH}(H\\mathbb{F}_p) open; one plausible extension is that divided powers become polynomial in that grading, converting the subring of Theorem A into a cleaner free-algebra description.","The same Tate-square strategy should work for H\\mathbb{F}_p with larger groups G, but only where the geometric fixed points admit an intrinsic formality decomposition; the non-modular theorem suggests the interesting cases are exactly the primes dividing |G|.","The multiplicative equivariant bar spectral sequence developed here could be applied to other equivariant Thom spectra of E_\\infty maps, producing RO(G)-graded Green-functor computations beyond the two examples in the paper.","A direct test of the paper's framework would be to adapt its method to p=2; since the geometric fixed points of H\\mathbb{F}_2 have a different form, the resulting C_2-fixed ring is expected to differ qualitatively from the odd-prime answer."],"forward_implications":["If Theorem A is right, any future computation of equivariant topological cyclic homology for H\\mathbb{F}_p must absorb a divided-power and exterior structure rather than a single polynomial generator, so the equivariant trace-method pipeline will be qualitatively more involved than the classical one.","Theorem B provides a concrete topological model for the geometric fixed points, H\\mathbb{F}_p\\wedge (\\Omega S^3)_+\\wedge S^1_+\\wedge CP^\\infty_+\\wedge (LS^3)_+, which can serve as a starting point for RO(C_p)-graded computations of the same spectrum.","The identification \\mathrm{ETHH}(MU_G)\\simeq MU_G\\wedge \\Sigma^\\infty_+SU_G for finite G means the homotopy groups of equivariant complex cobordism's ETHH are determined by the equivariant homology of SU_G; for finite abelian G, the paper computes these as exterior algebras over the equivariant cobordism rings of subgroups.","For the Real cobordism spectrum MU_\\mathbb{R}, the RO(C_2)-graded homotopy is an exterior algebra over (MU_\\mathbb{R})_\\star on classes of degree n\\rho+1, giving a complete description of \\mathrm{ETHH}(MU_\\mathbb{R}).","When the characteristic is prime to the group order, \\pi^G_*\\mathrm{ETHH}(H\\mathbb{F})\\cong \\pi_*\\mathrm{THH}(H\\mathbb{F}), so the exotic behavior of Theorem A is specific to the modular case where the prime divides |G|."],"supporting_citations":[{"why":"Introduces ETHH and the equivariant Dennis trace, supplying the object whose homotopy rings this paper computes.","marker":"[CGK25]"},{"why":"Supplies the intrinsic formality of \\mathbb{F}_p[y] as an E2-DGA, used to decompose the geometric fixed points of HF_p.","marker":"[Hor25]"},{"why":"Provides the intrinsic formality of \\mathbb{F}_p[x]/x^2 and the E2-equivalence THH(H\\mathbb{F}_p)\\simeq H\\mathbb{F}_p\\wedge(\\Omega S^3)_+, foundational for Theorem B.","marker":"[BM22]"},{"why":"Provides the equivariant Thom spectrum/factorization homology compatibility used to identify ETHH(MU_G) and ETHH(MU_R).","marker":"[HHK+24]"},{"why":"Gives the equivariant bar spectral sequence and Künneth theorems used to compute the homotopy rings of ETHH of Thom spectra.","marker":"[LM06]"},{"why":"Gives the cellular decomposition of SU(n) whose equivariant refinement underlies the MU_G homotopy ring computation.","marker":"[Yok56]"}],"fun_headline_variants":["Divided powers rule equivariant THH of F_p at odd primes","Equivariant THH of F_p exposed: divided powers and more","Odd-prime equivariant THH: divided powers break classical picture","From classical to equivariant THH: divided powers emerge","Equivariant THH computation: divided powers and complexity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the graded algebra \\mathbb{F}_p[y] in degree 2 is the only E2-DGA with that homology; if that cited formality statement fails, the geometric fixed point decomposition behind Theorems A and B collapses.","fun_headline_variants_meta":{"raw":{"variants":["Divided powers rule equivariant THH of F_p at odd primes","Equivariant THH of F_p exposed: divided powers and more","Odd-prime equivariant THH: divided powers break classical picture","From classical to equivariant THH: divided powers emerge","Equivariant THH computation: divided powers and complexity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000216,"raw_usage":{"total_tokens":1479,"prompt_tokens":1042,"completion_tokens":437,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":658,"completion_tokens_details":{"reasoning_tokens":348}},"tokens_in":658,"tokens_out":437,"duration_ms":4001,"temperature":1.0,"reasoning_tokens":348,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:12:44.795244+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an E2-DGA over \\mathbb{F}_p with homology \\mathbb{F}_p[y], |y|=2, that is not quasi-isomorphic to the formal algebra \\mathbb{F}_p[y]; its existence would invalidate Proposition 5.11 and with it the geometric-fixed-point computations of Theorems A and B.","supporting_citations":[],"review_version":1}