{"id":"b5af568d-b996-4b3f-a30a-bb6a33642f06","arxiv_id":"2510.24415","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A proxy-small geometric functor satisfies Grothendieck duality on all objects whose image is rigid, and the invertibility of its dualising object captures Matlis and Gorenstein duality.","lead":"This paper builds a general framework for duality theorems in tensor-triangular categories, using a weakened notion of compactness called proxy-smallness. It shows that Grothendieck, Matlis, and Gorenstein duality all arise from the same structural condition on a functor.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof gap in Prop. 3.23: Lemma 3.22 cannot yield Loc(A)=D from the module argument, but a direct retract (ε∘α=id) proves the claim; main theorems are not threatened.","rationale":"The reader's weakest-assumption flags exactly the right spot: Proposition 3.23's proof applies Lemma 3.22 in a direction that cannot produce the conclusion. I verified that no choice of X in the lemma yields 1_D∈Loc⊗(A). So the manuscript has a genuine proof gap at a load-bearing point. But the gap is cosmetic: ε∘α=id makes 1_D a retract of A in D, giving Loc⊗(A)=D without enhancement or conservativity. Therefore the central claim of the paper—Grothendieck duality for proxy-small geometric functors, and the automatic-compatibility theorem—remains correct. A conditional acceptance with a request to replace the proof of Proposition 3.23 is appropriate; no deeper flaw in Theorems 4.14/4.23/5.8/6.18 was found after checking the relevant diagrams and applications. The reader's verdict should stand, with the stated rationale adjusted to note the easy repair.","tokens_in":45673,"tokens_out":27440,"duration_ms":246035,"concrete_test":"Independent check of Prop. 3.23: delete the Mod(A)/Lemma 3.22 paragraph and prove the statement directly by showing that if A=f_*f^*1_D with unit α:1_D→A and counit ε:A→1_D satisfying ε∘α=id, then 1_D is a retract of A in D; since thick⊗ is closed under retracts, 1_D∈thick(A)⊆Loc⊗(A), so Loc⊗(A)=D. Then verify that no subsequent theorem (3.24, 4.14, 4.23, 5.8, 6.18) invokes the faulty step in any other form. If the retract membership is accepted, the conditional is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3.23 asserts Loc⊗(f_*f^*1_D)=D for enhanced f_*, and Theorem 3.24 uses it to make condition (2) of Definition 3.15 automatic. The written proof argues that 1_D lies in Loc⊗(A) inside Mod(A), A=f_*f^*1_D, and that since restriction of scalars along α:1_D→A is conservative, Lemma 3.22 transfers this to D. This is not what Lemma 3.22 says: for F_*=A⊗−:D→Mod(A), F^*=res, the lemma gives F^*(Loc(X))⊆Loc(F^*X). With X={1_D^A} this yields only 1_D∈Loc(1_D) in D, and with X={A} it is tautological; neither gives 1_D∈Loc(A). Thus the cited step does not follow. However, the proposition itself is true: the composite ε∘α is the identity on 1_D, so 1_D is a retract of A in D, hence 1_D∈thick(A)⊆Loc⊗(A) and Loc⊗(A)=D. This repair needs no enhancement. So the automatic-compatibility theorem is correct, but the manuscript's proof should be replaced. This is a genuine proof gap in a load-bearing passage; it does not invalidate the subsequent theorems once the two-line retract argument is inserted.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a tensor-triangular framework for duality phenomena, centred on the new notion of a proxy-small geometric functor. The main theorems give an automatic compatibility statement for enhanced functors (Theorem 3.24), a classification of rigid objects in the associated torsion category (Theorem 4.14), a Grothendieck duality statement on the subcategory of f_*-rigid objects without assuming that f_* preserves compact objects (Theorem 4.23), a Picard-group classification of invertible torsion objects (Theorem 5.8), a classification of Matlis dualising objects (Theorem 6.18), Morita-theoretic classification of Matlis lifts (Theorem 7.19), and a theory of Gorenstein duality for geometric functors with sections (Theorem 8.9). The paper also contains ascent and descent results and applications to commutative algebra, formal DGAs, chromatic and equivariant stable homotopy theory, and polynomial invariant theory including a new perspective on Watanabe's theorem.","tokens_in":45984,"tokens_out":6704,"duration_ms":60959,"significance":"If the results hold, this is a significant contribution: it systematically generalises and complements the duality frameworks of Balmer--Dell'Ambrogio--Sanders and Dwyer--Greenlees--Iyengar, and it replaces the restrictive compact-preservation hypothesis with the more flexible proxy-smallness condition. The paper is a long, carefully structured proof document with many detailed coherence arguments and concrete applications, and the claimed theorems are substantial and useful. However, a load-bearing proof step in Proposition 3.23 is not justified as written, although the statement itself is true and admits a short repair; this does not undermine the overall architecture of the paper but requires correction before publication.","major_comments":[{"comment":"The proof of Proposition 3.23 invokes Lemma 3.22 to transfer the conclusion 1_D ∈ Loc⊗(f_*f^*1_D) from the category of A-modules back to D, where A = f_*f^*1_D. This does not follow from Lemma 3.22. For the functor F = A⊗− : D → Mod(A), whose right adjoint is restriction of scalars and is conservative, Lemma 3.22 gives an inclusion for the left adjoint F, namely F(Loc(X)) ⊆ Loc(F X); it does not give res(Loc(A)) ⊆ Loc(res A). Thus the cited transfer in the direction used is not justified. The proposition itself is nevertheless correct: the composite ε∘α is the identity on 1_D, so 1_D is a retract of A in D; hence 1_D ∈ thick(A) ⊆ Loc⊗(A), and therefore Loc⊗(A) = D. This two-line repair does not need enhancements. Because Proposition 3.23 is used in Theorem 3.24 to make condition (2) of Definition 3.15 automatic for enhanced geometric functors, this is a genuine gap in a load-bearing passage, and the proof in the manuscript should be replaced rather than merely annotated.","section":"§3.B, Proposition 3.23"}],"minor_comments":[{"comment":"The sentence 'The Picard group of spectra is trivial, and is isomorphic to Z' is internally confusing: the second clause says the group is Z, while Definition 5.4 defines 'trivial' as surjectivity of Z → Pic(C). Please rephrase, for example 'The Picard group of spectra is Z, and hence is trivial in the sense of Definition 5.4.'","section":"Example 5.5(iii)"},{"comment":"There is a typo in 'We that the Wirthmüller isomorphism means f_* is Gorenstein'; this should read 'We note that' or 'We have'.","section":"Example 8.12"},{"comment":"The definition of the map α as f_*(η) is compressed; explicitly writing η: 1_C → f^*f_*1_C and the identification f_*1_C ≃ 1_D before defining α would make the argument easier to follow.","section":"Proposition 3.23"},{"comment":"The phrase 'Since res^G_1 k[V] is a polynomial ring' relies on an implicit identification of modules over the restriction of k[V]; adding a brief clarifying sentence would improve readability.","section":"Corollary 10.16"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read of arXiv:2510.24415.\n\nThe paper delivers what it promises: a systematic framework for duality in tt-geometry using proxy-smallness instead of compact-preservation. The new material is real—proxy-small geometric functors, Grothendieck duality for f_*-rigid objects (Thm 4.23), the classification of rigid and invertible torsion objects (Thms 4.14, 5.8), Matlis dualising objects and Gorenstein functors (Thm 6.18), plus the formal DGA and Watanabe applications. The writing is clear and the proofs are generally careful. I especially appreciate that the authors flag where proxy-smallness is used and where it fails (Remark 4.27).\n\nThe main soft spot is in Proposition 3.23, which claims Loc⊗(f_* f^* 1_D) = D for enhanced geometric functors. The written proof says that since 1_D ∈ Loc(A) in Mod(A) and restriction along α is conservative, Lemma 3.22 transfers this to D. That step doesn't follow from Lemma 3.22 as stated—the lemma gives containment in the wrong direction for this purpose. This is a genuine gap in a load-bearing passage, because Theorem 3.24 uses it to make condition (2) of Definition 3.15 automatic. The good news: the proposition itself is true, and the repair is short: ε∘α = id shows 1_D is a retract of A = f_* f^* 1_D, hence in thick(A) ⊆ Loc⊗(A). So the main theorems are not threatened, but the manuscript's proof should be replaced. I'd want that fixed before publication.\n\nOther concerns are minor. Some proofs are compressed, especially the coherence arguments in Section 4.A, but they look sound. The pure semisimple hypothesis in Section 6 is strong, but the authors use it where needed and state the unconditional parts. The examples are genuinely illustrative, not decorative.\n\nWho is this for? Anyone working in tensor-triangular geometry, duality, or stable homotopy theory. It's a serious paper that deserves a serious referee. My recommendation: send it to review, but request the Prop 3.23 repair and maybe a short remark on why the retract argument doesn't need enhancement.","headline":"A substantial, well-written framework for duality in tt-geometry via proxy-smallness, with one genuine but repairable proof gap in Proposition 3.23; the main theorems hold up and the paper deserves serious refereeing.","tokens_in":46509,"tokens_out":1927,"would_cite":true,"duration_ms":17996,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18G80","18M05","55P42"],"pacs":[],"model":"deepseek-v4-flash","headline":"A proxy-small geometric functor satisfies Grothendieck duality on all f_*-rigid objects, even when its right adjoint does not preserve compact objects.","keywords":["tensor-triangular geometry","proxy-smallness","Grothendieck duality","Gorenstein duality","Matlis dualising objects","torsion and completion functors","Picard group","rigid objects"],"falsifier":"Produce an enhanced geometric functor $f_* : C \\to D$ for which $f_* 1_D$ is proxy-small in $C$ but $\\mathrm{Loc}^{\\otimes}(f^* f_* 1_D) \\neq D$; if such a functor exists, Theorem 3.24 is false and the paper's main theorems, while conditionally true, would no longer apply to any example whose proxy-smallness was verified only through Theorem 3.24. A natural place to look is a map of commutative ring spectra where the ring is proxy-small but the induced module category is not generated by the image of the unit.","tokens_in":45482,"feed_emoji":"🔁","tokens_out":10054,"duration_ms":82877,"temperature":0.7,"pith_summary":"This paper replaces a restrictive hypothesis in tensor-triangular duality—that a geometric functor's right adjoint preserves compact objects—with the weaker condition that the functor be proxy-small. Under this condition it proves Grothendieck duality on the canonical subcategory of f_*-rigid objects, for which the image under f_* is rigid, and it classifies the rigid and invertible objects in the associated torsion category. This yields a uniform perspective on local algebra, chromatic and equivariant stable homotopy theory, and the Gorenstein property, capturing Matlis duality, Gorenstein duality, and Watanabe's theorem on invariant rings. The paper also introduces Matlis dualising objects and Gorenstein geometric functors, and relates orientability of Matlis lifts to the difference between the Gorenstein property and Gorenstein duality.","feed_headline":"Proxy-smallness yields Grothendieck duality","feed_subtitle":"A mild finiteness condition still gives the full duality formula on all f_*-rigid objects.","key_machinery":"The load-bearing device is the proxy-small geometric functor: a geometric functor $f_* : C \\to D$ such that the unit $1_D$ is proxy-small relative to $f_*$, meaning $f_* 1_D$ is proxy-small in $C$ and the localising tensor ideal generated by the image of $f_*1_D$ under $f^*$ is all of $D$. Proxy-smallness is a weak finiteness condition—an object is proxy-small when its tensor-triangular information is captured by compact objects, as with the residue field of a local ring witnessed by a Koszul complex. This hypothesis lets the paper construct the torsion category $\\Gamma_f C = \\mathrm{Loc}^{\\otimes}(f_* 1_D)$ and the completion category $\\Lambda_f C$, with the MGM equivalence between them, and to factor $f_*$ through $\\Gamma_f C$. The proof of Grothendieck duality then runs by comparing rigidity in $\\Gamma_f C$, $\\Lambda_f C$, and $D$, using the fact that strong monoidal functors preserve rigid objects and that $f_*$-isomorphisms can be detected by tensoring with a proxy-smallness witness.","core_discovery":"The central claim is Theorem 4.23: if $f_* : C \\to D$ is a proxy-small geometric functor and $M \\in C$ is such that $f_* M$ is rigid in $D$, then there is a natural isomorphism $f_! X \\otimes f_* M \\simeq f_!(X \\otimes M)$ for all $X$, and in particular $\\omega_f \\otimes f_* M \\simeq f_! M$ with $\\omega_f = f_! 1_C$. This says Grothendieck duality holds on the full subcategory of $f_*$-rigid objects even when the right adjoint does not preserve compact objects. The paper further claims that an object is $f_*$-rigid exactly when its torsion, completion, and $f_*$-image are rigid (Theorem 4.14), and that, for Gorenstein proxy-small functors with pure-semisimple target, Matlis dualising objects are exactly those whose $f_*$-image is invertible (Theorem 6.18).","pith_inferences":["The transfer principle '$\\Gamma_f X$ has property P iff $f_* X$ has property P' is shown for rigidity and invertibility but not reflexivity (Remark 5.12); a natural next step is to determine exactly which objects properties satisfy such a transfer, starting with compactness or proxy-smallness themselves.","Theorem 4.23 gives a partial answer to the question of forcing Grothendieck duality by localising the source; one could now ask whether the subcategory of $f_*$-rigid objects is the largest full subcategory on which $\\omega_f \\otimes f_* M \\simeq f_! M$ holds, and whether it has an intrinsic description in the Balmer spectrum.","The fixed-point argument in Proposition 10.20 is a template for producing new Gorenstein duality statements from equivariant ones; applying it to compact Lie groups with non-trivial adjoint action should yield non-equivariant duality statements that are not visible from the non-equivariant Gorenstein condition alone.","The paper leaves open whether relative proxy-smallness is independent of the chosen witness in general (Remark 3.10); a positive answer would simplify Definition 3.15 and would make the compatibility condition purely a statement about $f_* 1_D$."],"forward_implications":["Grothendieck duality $\\omega_f \\otimes f_* M \\simeq f_! M$ holds for every $M \\in C$ whose image $f_* M$ is rigid, so the duality formula is available without any compact-preservation assumption.","Rigidity transfers across the torsion/completion boundary: $M$ is $f_*$-rigid if and only if $\\Gamma_f M$ is rigid in $\\Gamma_f C$, if and only if $\\Lambda_f M$ is rigid in $\\Lambda_f C$, if and only if $f_* M$ is rigid in $D$.","Invertibility transfers similarly (Theorem 5.8), giving abstract criteria for when torsion or completion objects are invertible; this recovers the $K(n)$-local invertibility criterion and the local algebra dualisability results.","For a Gorenstein proxy-small functor with pure-semisimple target, the Matlis dualising objects are classified: $f_! M$ is invertible if and only if $f_* M$ is invertible, if and only if $\\Gamma_f M$ and $\\Lambda_f M$ are invertible (Theorem 6.18).","Gorenstein duality holds exactly when a specific Matlis lift of $\\omega_f$ is orientable (Proposition 8.9), and automatic orientability makes the Gorenstein condition equivalent to Gorenstein duality; this is what powers the formal-DGA and invariant-theory applications."],"supporting_citations":[{"why":"Supplies the Grothendieck–Neeman duality theorem whose compact-preservation hypothesis the paper replaces with proxy-smallness.","marker":"[4]"},{"why":"Introduces proxy-smallness and Gorenstein ring spectra, the notions the paper generalises to tensor-triangular categories.","marker":"[14]"},{"why":"Provides the local-dualisability results in local algebra that Theorem 4.14 recovers and reformulates uniformly.","marker":"[10]"},{"why":"Gives the adjointness and rigidity isomorphisms that Theorem 4.23 extends from rigid to $f_*$-rigid objects.","marker":"[16]"},{"why":"Studies the compactness locus of a geometric functor and asks whether Grothendieck duality can be forced by localising the source; Theorem B answers part of this question.","marker":"[37]"},{"why":"Describes rigid and invertible objects in $K(n)$-local spectra; Theorem 5.8 recovers this description via proxy-small geometric functors.","marker":"[27]"},{"why":"Computes Picard groups of $K(n)$-local spectra, used to illustrate the failure of injectivity and surjectivity in the Picard diagram.","marker":"[24]"},{"why":"Watanabe's invariant-ring theorem, which the paper recovers as a fixed-point shadow of an equivariant Gorenstein duality statement.","marker":"[41]"}],"fun_headline_variants":["Proxy-smallness: duality without compactness","Small probe, full Grothendieck duality","Proxy-small functors unlock duality","Rigidity from proxy-smallness","Duality freed from compact preservation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The main theorems rest on the assumption that proxy-smallness of $f_* 1_D$ automatically implies a compatibility condition (that the left-adjoint image of $f_* 1_D$ generates the target category), and the argument for this automaticity is terse; if it fails, the conditional theorems survive but the examples must be checked by hand.","fun_headline_variants_meta":{"raw":{"variants":["Proxy-smallness: duality without compactness","Small probe, full Grothendieck duality","Proxy-small functors unlock duality","Rigidity from proxy-smallness","Duality freed from compact preservation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1337,"prompt_tokens":983,"completion_tokens":354,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":599,"completion_tokens_details":{"reasoning_tokens":292}},"tokens_in":599,"tokens_out":354,"duration_ms":3540,"temperature":1.0,"reasoning_tokens":292,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:41:43.586937+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Produce an enhanced geometric functor $f_* : C \\to D$ for which $f_* 1_D$ is proxy-small in $C$ but $\\mathrm{Loc}^{\\otimes}(f^* f_* 1_D) \\neq D$; if such a functor exists, Theorem 3.24 is false and the paper's main theorems, while conditionally true, would no longer apply to any example whose proxy-smallness was verified only through Theorem 3.24. A natural place to look is a map of commutative ring spectra where the ring is proxy-small but the induced module category is not generated by the image of the unit.","supporting_citations":[{"cited_title":"Balmer, I","cited_arxiv_id":null,"evidence_quote":"Supplies the Grothendieck–Neeman duality theorem whose compact-preservation hypothesis the paper replaces with proxy-smallness."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces proxy-smallness and Gorenstein ring spectra, the notions the paper generalises to tensor-triangular categories."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the local-dualisability results in local algebra that Theorem 4.14 recovers and reformulates uniformly."},{"cited_title":"Fausk, P","cited_arxiv_id":null,"evidence_quote":"Gives the adjointness and rigidity isomorphisms that Theorem 4.23 extends from rigid to $f_*$-rigid objects."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Studies the compactness locus of a geometric functor and asks whether Grothendieck duality can be forced by localising the source; Theorem B answers part of this question."},{"cited_title":"Hovey and N","cited_arxiv_id":null,"evidence_quote":"Describes rigid and invertible objects in $K(n)$-local spectra; Theorem 5.8 recovers this description via proxy-small geometric functors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Computes Picard groups of $K(n)$-local spectra, used to illustrate the failure of injectivity and surjectivity in the Picard diagram."},{"cited_title":"Watanabe","cited_arxiv_id":null,"evidence_quote":"Watanabe's invariant-ring theorem, which the paper recovers as a fixed-point shadow of an equivariant Gorenstein duality statement."}],"review_version":2}