{"id":"5d82a99e-6615-4203-adbf-65a90c54955b","arxiv_id":"2510.23267","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any quasi-compact F_p-scheme with affine diagonal, the derived ∞-category of Frobenius modules is t-exactly equivalent to Frobenius modules on the derived ∞-category, and both satisfy Zariski descent.","lead":"The paper proves that for schemes over F_p with mild finiteness conditions, the derived ∞-category of Frobenius modules is equivalent to Frobenius modules in the derived ∞-category. This gives a clean derived-categorical framework for F_p-geometry and proves a Zariski descent property, useful for future work on local cohomology and Riemann–Hilbert correspondences.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.7 relies on t-exactness of F_*, which is never proved for geometric schemes and is contradicted by the introduction; the premise is true but left unjustified.","rationale":"The reader's weakest assumption correctly identifies a genuine gap: the proof of Theorem 5.7 requires the target t-structure, which in turn requires D(F_*) to be t-exact. The paper never proves exactness of F_* for geometric schemes, and the introduction explicitly claims the opposite. However, the underlying mathematical fact is true—F_* is exact because the absolute Frobenius is affine—so the central theorem is likely correct. The issue is an unjustified and internally contradicted premise, not a false conclusion. This merits a conditional acceptance pending an added lemma and removal of the misleading introduction sentence. I find no other independent obstruction: the affine proof via Schwede–Shipley is plausible, Zariski descent is carefully handled, and left-completeness follows if the t-structure exists. Thus the reader's CONDITIONAL verdict is appropriate, and my stress test does not change it.","tokens_in":22370,"tokens_out":9099,"duration_ms":93582,"concrete_test":"Verify the exactness of F_* on a non-affine geometric scheme X directly: choose an affine open cover {U_i}, and for each i check that F_*|_U_i is the restriction of scalars along R_i → R_i, r ↦ r^p; conclude exactness. Then re-examine Lemma 3.13, Lemma 4.15, and Corollary 5.5 to ensure every invocation of [MW24, Prop. 3.3] is justified. If this cannot be done, Theorem 5.7 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem (Thm. 5.7) is proved by combining the bounded-above equivalence (Thm. 4.16) with left-completeness (Prop. 5.2, Cor. 5.5). Both the affine case (Lemma 3.13) and the global target t-structure (Cor. 5.5) invoke [MW24, Prop. 3.3], which requires D(F_*) to be t-exact, i.e., F_* : QCoh(X) → QCoh(X) to be exact. The paper never proves this for geometric schemes. In fact, the introduction states that without regularity, 'this is no longer the case' (p.2), directly contradicting the proof's own use. The assertion is false: the absolute Frobenius F is an affine morphism, so F_* is restriction of scalars along R → R, x ↦ x^p on affine opens and is exact. But because the manuscript never says this and even denies it, the existence of the induced t-structure on Frob(D(QCoh(X)),D(F_*)) is a load-bearing premise that is left unjustified. If the premise failed, the target would have no t-structure and the left-completeness argument (Lemma 5.6) would not apply; the theorem would not be established by this proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for a quasi-compact F_p-scheme X with affine diagonal (called geometric), there is a t-exact equivalence of presentable stable ∞-categories D(Frob(QCoh(X), F_*)) ≃ Frob(D(QCoh(X)), D(F_*)). Here Frob(-,-) is the lax-equalizer construction of Frobenius modules from the authors' previous paper [MW24]. The proof has three main steps: (1) an affine reduction, where both sides are identified with module spectra over the same E_1-ring via the Schwede–Shipley theorem; (2) a Zariski-descent argument showing that the bounded-above versions of both sides satisfy descent, so the equivalence is checked on affine opens; (3) a left-completeness argument that promotes the bounded-above equivalence to the full equivalence. The paper also proves Zariski descent for the derived ∞-categories of Frobenius and Cartier modules.","tokens_in":22680,"tokens_out":12215,"duration_ms":134552,"significance":"If the proof is correct, the result is a genuine improvement over [MW24], where the Frobenius-module equivalence required X regular Noetherian. The new statement covers all quasi-compact F_p-schemes with affine diagonal, which is a broad and natural class. The strategy is clear and modular: the affine case is handled by compact projective generators and an endomorphism-ring comparison, the global case by Zariski descent, and the passage from bounded-above objects to all objects by left-completeness. As a byproduct, the paper establishes Zariski descent for derived categories of Frobenius and Cartier modules. The reliance on the companion paper [MW24] for structural facts is substantial but explicit, and the main new theorem is a genuine extension rather than a reformulation. One load-bearing premise, however, is never proved and is even contradicted in the introduction; this must be fixed before the theorem can be considered established.","major_comments":[{"comment":"The paper repeatedly uses the fact that D(F_*) is t-exact in order to form the induced t-structure on Frob(D(QCoh(X)), D(F_*)) via [MW24, Prop. 3.3]. This is used in Cor. 5.5 and hence in Thm. 5.7. For geometric X this t-exactness is never proved. Worse, the introduction asserts the opposite: 'If X is arbitrary, then this is no longer the case' (p.2). In fact the absolute Frobenius is affine, so F_*: QCoh(X)→QCoh(X) is exact for every scheme and D(F_*) is t-exact. The assertion is true but the manuscript gives no proof and its wording denies it. This is load-bearing: without exactness of F_*, the target category need not carry the induced t-structure and the left-completeness arguments collapse. Please add a proof (reduction to affine opens, where F_* is restriction of scalars along R→R) and correct the introduction.","section":"Introduction, p.2; §5, Cor. 5.5"},{"comment":"In the affine case, Lemma 3.13 defines a t-structure on Frob(D(Mod_R), D(F_*)) and says this follows from [MW24, Prop. 3.3] because D(F_*) is t-exact 'by definition of the derived functor'. This is missing the essential point that F_* must be exact. The paragraph before Prop. 3.10 asserts that F_* is exact because it has both a left and a right adjoint, but the right adjoint is not written down or referenced in enough detail. Since the global argument in §5 depends on the same property, please give an explicit proof or a precise reference for exactness of F_* both on Mod_R and on QCoh(X) for geometric X.","section":"§3, Lemma 3.13"},{"comment":"The bounded-above equivalence Φ_X is asserted to be t-exact with the target 'equipped with the induced t-structure from [MW24, Proposition 3.3]' already at the level of global geometric X, not only in the affine case. This is used to identify the sectionwise maps Φ_U with the affine equivalence of Thm. 3.28. The existence of that induced t-structure is precisely the missing exactness of D(F_*) discussed above. The statement is true, but it needs to be proved once, before Lemma 4.15, rather than being inherited from the affine case.","section":"§4, Lemma 4.15 and Thm. 4.16"}],"minor_comments":[{"comment":"The phrase 'F is flat by Kunz' theorem, and hence F_* is an exact functor' conflates F_* with F^*. Flatness of the Frobenius gives exactness of F^*, whereas F_* is exact without any flatness assumption because the Frobenius is affine. Please rephrase.","section":"Introduction, p.2"},{"comment":"The statement that F_*: Mod_R → Mod_R is exact 'as it has both a left adjoint and a right adjoint' would be clearer if the right adjoint were exhibited (e.g. Hom_R(R,-) with the appropriate module structure).","section":"§3, after Lemma 3.3"},{"comment":"The pullback defining End(-) uses '×id_dCat∞' in the top-right spot; this notation is a bit opaque. A sentence explaining the diagram or the identification of the two projections would improve readability.","section":"§2, Def. 2.1"},{"comment":"The proof says D_{≤n} ≃ D^+_{≤n}; this is correct because D^+ means bounded above, but it would help to emphasize that D_{≤n} is contained in D^+ so the reader does not confuse D^+ with the more common bounded-below convention.","section":"§5, Lemma 5.6"}],"recommendation":"major_revision","confidential_remarks":"The central claim is very likely correct and the proof strategy is sound, but the manuscript contains a real gap: it uses, without proof, that F_* is exact on QCoh(X) for geometric X, and it even asserts the opposite in the introduction. Since the missing fact is true and easy to prove, this is fixable within the scope of the paper. I would be willing to accept after the authors add the missing exactness lemma and correct the misleading introduction. The heavy reliance on the same authors' [MW24] is acceptable, but the introduction should make clear exactly which structural inputs are taken from [MW24] and which are new. The paper is a good fit for a journal in algebraic geometry or higher algebra."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: the main theorem is very likely correct and extends the authors' earlier Cartier/Frobenius result from regular Noetherian schemes to all geometric F_p-schemes, with Zariski descent as a byproduct. The proof architecture is sensible: affine case via Schwede–Shipley identification with R[F]-module spectra, then Zariski descent, then left-completeness to go from bounded-above to all objects. The descent arguments are genuinely new, and the paper is carefully organized.\n\nThe biggest soft spot is one the authors can fix in half an hour: the proof relies on the induced t-structure on Frob(D(QCoh(X)),D(F_*)) existing, which requires D(F_*) to be t-exact. The paper never proves this for geometric schemes, and the introduction actually says that for arbitrary X \"this is no longer the case.\" That statement is wrong: the absolute Frobenius is an affine morphism, so F_* is an exact endofunctor of QCoh(X) and D(F_*) is t-exact. The affine case (Lemma 3.13) gets exactness of F_* from ring-theoretic adjoints, but the global case is left hanging. This is a real gap in the written proof, but the missing lemma is true and the fix is routine. The introduction should also be reworded—what fails without regularity is flatness of F, not exactness of F_*.\n\nThe heavier external dependencies—[MW24] for the whole Frob framework, [Pos25] for the Roos axiom, [HM24] for descent—are natural but mean the paper is not self-contained. The authors are upfront about this, and the new theorem genuinely covers cases not in [MW24].\n\nOn balance: the paper deserves a serious referee. The main theorem is plausible, the proof strategy is sound, and the gap is small and repairable. I would send it out with a request that the authors state and prove the exactness lemma for F_* globally and correct the introduction. Once that is done, I expect the result to hold up.","headline":"The main theorem is very likely correct and genuinely extends the authors' earlier work from regular Noetherian schemes to all geometric F_p-schemes, but the written proof leaves a load-bearing exactness premise for F_* unstated and even contradicted in the introduction.","tokens_in":23183,"tokens_out":4530,"would_cite":true,"duration_ms":42652,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F30","18G80","18N60"],"pacs":[],"model":"deepseek-v4-flash","headline":"On any quasi-compact F_p-scheme with affine diagonal, the derived ∞-category of Frobenius modules is t-exactly equivalent to the category of Frobenius modules in the derived ∞-category of quasi-coherent sheaves.","keywords":["Frobenius modules","Cartier modules","derived ∞-categories","Zariski descent","positive characteristic","quasi-coherent sheaves","t-structures","left-complete"],"falsifier":"Take an affine singular F_p-algebra R such as R = F_p[x,y]/(xy) and compute the endomorphism ring spectrum of the rank-one Frobenius module (R, id: R → F_*R) inside D(Frob(Mod_R, F_*)); the theorem predicts this ring is the twisted polynomial ring R[F]^op, so any deviation would falsify the affine case and hence the main theorem.","tokens_in":22248,"feed_emoji":"♾️","tokens_out":12308,"duration_ms":113634,"temperature":0.7,"pith_summary":"The paper proves that for a broad class of characteristic-p schemes — those that are quasi-compact and have affine diagonal, such as quasi-compact separated schemes — two natural ways of combining 'take the derived category' and 'impose a Frobenius action' produce the same stable ∞-category. The first construction derives the abelian category of Frobenius modules; the second takes Frobenius modules inside the derived category of quasi-coherent sheaves. The canonical comparison functor between them is shown to be a t-exact equivalence, so the homological t-structures agree as well. This removes the regularity and Noetherian hypotheses of an earlier result, and, as a byproduct, the paper establishes Zariski descent for the derived ∞-categories of Frobenius and Cartier modules. The strategy reduces the statement to affine schemes via descent and then identifies both sides with module spectra over a twisted polynomial ring.","feed_headline":"Deriving and Frobenius actions commute on geometric F_p-schemes","feed_subtitle":"A t-exact equivalence without regularity or Noetherian assumptions, with Zariski descent as a byproduct.","key_machinery":"The central mechanism is the ∞-category Frob(C, G) of generalized Frobenius modules, defined as the lax equalizer of the identity functor and an endofunctor G; with C = QCoh(X) and G = F_*, the Frobenius pushforward, it recovers classical Frobenius modules. Its defining construction preserves limits, which makes Zariski descent possible. On affines, the proof relies on the module-category recognition theorem (an ∞-categorical version of Gabriel's theorem) to identify the two categories as module spectra over R[F]^op, and on a weak form of Grothendieck's AB4* axiom to obtain left-completeness of the quasi-coherent derived categories involved.","core_discovery":"The central discovery is that the canonical t-exact functor Φ_X from D(Frob(QCoh(X), F_*)) to Frob(D(QCoh(X)), D(F_*)) is an equivalence whenever X is a geometric F_p-scheme, meaning quasi-compact with affine diagonal. On affine schemes, both categories are identified with module spectra over the same E_1-ring, the twisted polynomial ring R[F]^op, using the module-category recognition theorem; the functor matches the compact projective generators. This local identity is then globalized through Zariski descent, since both the source and target presheaves are Zariski sheaves and therefore the equivalence can be checked on affine opens. Because both categories are left-complete, the equivalence","pith_inferences":["Because the equivalence is t-exact, the heart of D(Frob(QCoh(X), F_*)) is automatically equivalent to the abelian category Frob(QCoh(X), F_*) for any geometric X; the authors do not state this corollary, but it follows directly from Theorem A.","The descent proof uses a nonstandard small Zariski site built from finite disjoint unions of quasi-compact opens, precisely because affine Cech nerves fail without the affine-diagonal assumption; a natural test is whether the descent statements extend to the etale topology or to hypercovers, which the restriction to geometric schemes likely prevents.","The same combination of a limit-preserving lax-equalizer construction, descent, and left-completeness may apply to other endofunctor actions on quasi-coherent sheaves, such as actions of iterated Frobenius or of other affine endomorphisms.","The identification with R[F]^op-module spectra suggests a working definition of Frobenius modules as objects of a stable module category over a twisted polynomial ring, which may simplify concrete computations on singular rings."],"forward_implications":["The derived ∞-category of Frobenius modules satisfies Zariski descent for geometric F_p-schemes, so Frobenius-module cohomology can be computed from affine open covers.","The t-exactness of the equivalence endows D(Frob(QCoh(X), F_*)) with a t-structure whose heart is the ordinary abelian category of Frobenius modules, giving well-behaved homological algebra on singular schemes as well.","Finiteness results that previously required flatness of Frobenius (e.g., for local cohomology and F-modules) can now be phrased and proven in the derived ∞-categorical setting without regularity hypotheses.","The same descent result holds for Cartier modules, so their derived categories are also local objects, useful for duality and finiteness statements.","On affines, both sides become module spectra over R[F]^op, giving a concrete way to compute mapping spectra and Ext groups between Frobenius modules."],"fun_headline_variants":["Frobenius and derived categories: a t-exact equivalence","Derived Frobenius modules: t-exact equivalence on geometric F_p","No regularity needed: Frobenius commutes with derived functors","Zariski descent for Frobenius and Cartier modules","A t-exact match: Frobenius modules and derived categories"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof needs the Frobenius pushforward F_* to be an exact functor on quasi-coherent sheaves so that its derived functor is t-exact and the target category inherits a t-structure; this is true for the schemes under consideration because the absolute Frobenius is affine, but the paper leaves it unstated and its introduction even gestures in the opposite direction.","fun_headline_variants_meta":{"raw":{"variants":["Frobenius and derived categories: a t-exact equivalence","Derived Frobenius modules: t-exact equivalence on geometric F_p","No regularity needed: Frobenius commutes with derived functors","Zariski descent for Frobenius and Cartier modules","A t-exact match: Frobenius modules and derived categories"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000324,"raw_usage":{"total_tokens":1642,"prompt_tokens":720,"completion_tokens":922,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":464,"completion_tokens_details":{"reasoning_tokens":830}},"tokens_in":464,"tokens_out":922,"duration_ms":8058,"temperature":1.0,"reasoning_tokens":830,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T07:59:28.410493+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an affine singular F_p-algebra R such as R = F_p[x,y]/(xy) and compute the endomorphism ring spectrum of the rank-one Frobenius module (R, id: R → F_*R) inside D(Frob(Mod_R, F_*)); the theorem predicts this ring is the twisted polynomial ring R[F]^op, so any deviation would falsify the affine case and hence the main theorem.","supporting_citations":[],"review_version":1}