{"id":"6dbbe104-0fca-4b09-bac6-f96b45d2e00f","arxiv_id":"2411.18274","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Singular cohomology of Berkovich analytifications of schemes over trivially valued fields of characteristic 0 is isomorphic to cdh-cohomology, yielding vanishing of RHom for algebraic group sheaves and identifying RHom between abelian varieties with Hom_{Ab_k}(A,B).","lead":"The paper proves that the singular cohomology of the Berkovich analytic space of an algebraic variety over a trivially valued field of characteristic 0 is the same as its cdh-cohomology. It uses this to show that higher extension groups between algebraic groups vanish and that the derived Hom between abelian varieties is just the ordinary Hom, giving new integral tools in motivic homotopy theory.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.2 asserts singular-cohomology Mayer-Vietoris for a closed covering of a Berkovich space, but closed covers are not excisive without hypotheses; no excision or triangulation condition is proved, so the cdh-descent step and Theorem 4.3 rest on an unverified topological fact.","rationale":"The reader identified exactly the right pressure point. I agree that Lemma 4.2's closed-cover Mayer-Vietoris step is the weakest assumption in the central argument. The rest of Theorem 4.3 is a plausible package: Lemma 4.1 handles open Nisnevich covers correctly; Theorem 2.8 is a valid hypersheaf criterion; the computation of F^•_cdh on smooth schemes follows from Thuillier's contractibility and resolution of singularities. There is, however, a second unguarded edge: the theorem states 'locally of finite type' without the separatedness hypothesis used in Proposition 3.6(2) to identify X^ℶ with X^an. For a smooth non-separated scheme such as the affine line with doubled origin, the contractibility of X^ℶ does not automatically imply contractibility of X^an; if X^an is not contractible the proof of F^•_cdh ≃ Z on smooth schemes breaks. This supports the conditional verdict, but the decisive fix has to be Lemma 4.2. In sum, the central claims are likely right in substance, and a referee should demand a written excision argument before acceptance; I would not reject outright. I keep the reader's CONDITIONAL verdict.","tokens_in":24690,"tokens_out":26667,"duration_ms":269930,"concrete_test":"Take the nodal cubic X over an algebraically closed trivially valued field k, with X~=P^1 its normalization, Z the node, Z~ the two preimages. Compute |X^an| independently (it is a circle, so H^1_sing=Z). Then write out the closed Mayer-Vietoris sequence asserted in Lemma 4.2 for this square: if it does not produce H^1(|X^an|)=Z from H^0(|Z~^an|)=Z^2 and the restriction maps, or if no excisive-triple isomorphism H_*(|X~^an|,|Z~^an|) ≅ H_*(|X^an|,|Z^an|) can be exhibited, the lemma is false as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 4.2 is the single load-bearing step in the proof of Theorem 4.3. It claims that an abstract blow-up square induces a long exact Mayer-Vietoris sequence for C^•(|X^an|, Z), because |X^an| is 'covered by the closed subsets |X~^an|, |Z^an|' and cites [9, II.5.5]. Singular cohomology does not admit a Mayer-Vietoris sequence for arbitrary closed covers: one needs an excisive triad, for example that the two closed subspaces are subcomplexes of a triangulation of |X^an| or that |Z~^an| is a common neighborhood deformation retract in both. The paper supplies none of this. The gap is not cosmetic; if the pair fails to be excisive the sequence can be inexact, and then F^• would not satisfy cdh descent, so the quasi-isomorphism C^•(|X^an|) → RΓ_cdh(X,Z) would not follow. The later §5 acyclicity statements (Prop. 5.30, Thm. 5.31) all invoke Theorem 4.3 for symmetric-power quotients, so the defect propagates through the applications.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a comparison between the singular cohomology of the underlying space of the Berkovich analytification of a scheme over a trivially valued field and its cdh-cohomology with integer coefficients, and identifies both with weight-zero motivic cohomology. On this basis it derives vanishing statements for RHom of Nisnevich sheaves with transfers attached to commutative algebraic groups and an explicit computation of RHom between abelian varieties. The proof strategy is to define the presheaf F^•(X)=C^•(|X^an|,Z), prove Mayer-Vietoris properties for Nisnevich and abstract blow-up squares, invoke Voevodsky's cd-structure theorem to obtain cdh descent, and then identify the cdh-sheafification of F^• with the constant sheaf Z using contractibility results of Berkovich and Thuillier.","tokens_in":24936,"tokens_out":30043,"duration_ms":292994,"significance":"If Theorem 4.3 were established, it would provide a clean bridge between Berkovich geometry and cdh-motivic cohomology, and the applications to Ext vanishing for algebraic groups and abelian varieties would be interesting and non-formal. The paper is honest about its dependence on external results (Berkovich, Thuillier, Voevodsky, Mazza-Voevodsky-Weibel), and it contains no ad hoc free parameters. The main theorems are, however, conditional on a descent statement whose proof has significant gaps; the applications in Section 5 inherit those gaps, so the current version does not justify its central claims.","major_comments":[{"comment":"The proof asserts that |W^an| is homeomorphic to |U^an| ∩ |V^an|. This is false in general. Consider the elementary Nisnevich square X=A^1_k, U=G_m, V=G_m ⊔ A^1_k, with p the disjoint union of the two open inclusions, and W=G_m ⊔ G_m. The map p^an sends the A^1 component onto |X^an|, so the open subsets |U^an| and p^an(|V^an|) of |X^an| have intersection |U^an|, not |W^an|. The Mayer-Vietoris sequence for the open cover therefore involves H^*(U^an), whereas the required homotopy pullback square for F^• involves H^*(W^an). Consequently, Nisnevich descent for F^• is not established by the argument given.","section":"Lemma 4.1 (Section 4)"},{"comment":"The proof claims that |X^an| is covered by the closed subsets |X~^an| and |Z^an| with |Z~^an| ≅ |X~^an| ∩ |Z^an|. This identification is false. For X=A^2_k, Z={0}, X~=Bl_0X, one has Z~=P^1_k, and |Z~^an| maps onto the one-point space |Z^an| with large fibers, while |X~^an| maps onto all of |X^an|; hence the intersection |X~^an| ∩ |Z^an| is |Z^an|, not |Z~^an|. The total complex required for the cdh-Mayer-Vietoris property would involve C^•(|Z~^an|), whereas the sequence obtained from the stated closed cover would involve C^•(|Z^an|). Moreover, a Mayer-Vietoris sequence for closed covers requires an excisive triad or an equivalent triangulability condition; none is proved for Berkovich spaces. Thus the cdh-descent step and Theorem 4.3 rest on an invalid lemma.","section":"Lemma 4.2 (Section 4)"},{"comment":"The applications inherit the failure of the comparison theorem. Proposition 5.30 uses Theorem 4.3 to conclude that H^i_cdh((S^t)^(n-1)(G),Z)=0 and hence that the terms in the resolution are Hom-acyclic; Theorem 5.31 and Corollary 5.32 then depend on Proposition 5.30. Since the proof of Theorem 4.3 is not valid as written, the vanishing results RHom(G,Z)≃0 and the abelian-variety computation in Theorem 5.34 are unsupported.","section":"Proposition 5.30 and Theorem 5.31 (Section 5.4)"},{"comment":"The proof states that since each term (S^•)^n(G) is a disjoint union of quotients of smooth k-varieties by finite groups, Corollary 3.24 and Theorem 4.3 imply H^n_cdh((S^•)^n(G),Z)=0. Corollary 3.24 is formulated for a smooth irreducible variety with a finite group action, whereas S^•(G) is an infinite disjoint union and G need not be connected. Contractibility of each component does not imply contractibility of the disjoint union, so the cited implication is not justified without additional argument.","section":"Theorem 5.15 (Section 5.2)"}],"minor_comments":[{"comment":"There are several typos, e.g., 'underlyi ng' and 'iso morphic' in the abstract, and 'analytification a scheme' in Remark 3.21 is missing a word.","section":"Abstract and Section 3"},{"comment":"In the sentence before the displayed formula, the text refers to 'the resolution in Theorem 1.3'; this should be 'Proposition 1.3'.","section":"Introduction, Proposition 1.3"},{"comment":"The proof says 'This follows immediately from Corollary 5.31', but the statement being invoked is Theorem 5.31; the cross-reference is incorrect.","section":"Corollary 5.32"},{"comment":"The proof of Theorem 4.3 concludes 'The same holds if X is not connected' without treating the non-connected case; the identification C^•(|X^an|,Z) ≃ Z is only valid for connected X, and the argument for the cdh-sheafification should be written out for non-connected schemes.","section":"Section 4, Theorem 4.3"}],"recommendation":"major_revision","confidential_remarks":"The paper's central comparison result is potentially valuable, but the proof of Section 4 has load-bearing errors that are not merely missing details: Lemma 4.1 and Lemma 4.2 assert topological identifications that are false. A revision would need to replace the descent argument for F^• with a correct one, or prove Theorem 4.3 by a different route, before the Section 5 applications can be evaluated. The self-citation to [3] is not a concern, since the relevant statements are also attributed to the standard literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe paper has a genuinely new claim: for a trivially valued field k of characteristic 0 with resolution of singularities, singular cohomology of the Berkovich analytification of X computes cdh-cohomology with integer coefficients, and therefore weight-zero motivic cohomology. That comparison is not in the literature; the earlier results are rational and pass through étale or de Rham cohomology. The subsequent integral vanishing of RHom( G, Z) and the identification of RHom(A,B) with Hom_Ab(A,B) are also new and extend work of Orgogozo and Ancona–Enright-Ward–Huber. The paper is well-organized, the citations are appropriate, and the simplicial symmetric-power machinery is interesting.\n\nThe soft spots are real but localized. Lemma 4.2 asserts a Mayer-Vietoris long exact sequence for the closed covering |X^an| = |X~^an| ∪ |Z^an| coming from an abstract blow-up. Singular cohomology does not have a Mayer-Vietoris sequence for arbitrary closed covers; one needs an excisive triad, for instance that the two closed subspaces are subcomplexes of a triangulation or that their intersection is a neighborhood deformation retract in both. The paper cites [9, II.5.5] without verifying any such hypothesis. This is not a cosmetic issue: if the sequence can fail, the presheaf F^• need not satisfy cdh descent, and Theorem 4.3 collapses. The later applications (Prop. 5.30, Thm. 5.31) rely on Theorem 4.3 for symmetric-power quotients, so the gap propagates. A referee should ask for a proof that the pair is excisive, or a replacement argument using sheaf cohomology on the compact Hausdorff spaces involved. The same referee should ask for a fuller proof of Theorem 5.34: the step showing that all maps in the cosimplicial object become identities after identifying terms with Mor(A,B) is compressed; the rigidity argument is sketched but needs care.\n\nThe central strategy is coherent and the external theorems are used correctly. I found no hidden circularity. The paper is likely correct in substance, but as written it leaves a load-bearing topological verification to the reader.\n\nRecommendation: send to peer review, but with a clear request to fix Lemma 4.2 and expand the proof of Theorem 5.34. If the topological gap is patchable, this becomes a strong paper; until then I would not rely on Theorem 4.3 as stated.","headline":"The paper's main comparison between singular cohomology of Berkovich analytifications and cdh-cohomology is new and likely true, but the proof relies on an unjustified Mayer-Vietoris claim for closed covers.","tokens_in":25533,"tokens_out":5762,"would_cite":false,"duration_ms":54939,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F42","14G22","14K05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Berkovich singular cohomology equals weight-zero motivic cohomology","keywords":["motivic cohomology","cdh-topology","Berkovich spaces","singular cohomology","presheaves with transfers","abelian varieties","symmetric powers"],"falsifier":"For the cuspidal cubic X over a trivially valued field k, let $\\widetilde{X}$ be its normalization and Z the singular point. Compute $H^1_{\\mathrm{sing}}(|X^{\\mathrm{an}}|,\\mathbb{Z})$ directly from the closed cover $|\\widetilde{X}^{\\mathrm{an}}| \\cup |Z^{\\mathrm{an}}|$ using the purported Mayer–Vietoris sequence, and compare with $H^1_{\\mathrm{cdh}}(X,\\mathbb{Z})$ computed algebraically; any mismatch, or any failure of the Mayer–Vietoris sequence to converge, would disprove Theorem 4.3 or its proof.","tokens_in":24451,"feed_emoji":"🧮","tokens_out":13407,"duration_ms":107533,"temperature":0.7,"pith_summary":"The paper claims that for any scheme X locally of finite type over a trivially valued field of characteristic 0 admitting resolution of singularities, the singular cohomology of the Berkovich analytification |X^an| coincides with the cdh-cohomology of X with integer coefficients, and therefore with the weight-zero motivic cohomology of X. On a smooth scheme the analytification is contractible, so the comparison reduces to showing that the singular-cochain presheaf satisfies descent for the coverings that generate the cdh-topology. This topological-algebraic bridge is then used to compute Hom-groups in the category of presheaves with transfers: the representable sheaf of a commutative algebraic group G has vanishing RHom against Z, and for two abelian varieties A and B the complex RHom(A,B) is concentrated in degree zero and equals the ordinary group homomorphisms. The paper's point is that these Ext-computations, which look motivic and algebraic, can be settled by studying the topology of Berkovich spaces.","feed_headline":"Berkovich singular cohomology equals weight-zero motivic cohomology","feed_subtitle":"For trivially valued fields, analytic cohomology is algebraic; the paper computes Exts of abelian varieties.","key_machinery":"The engine is the cd-structure formalism: the cdh-topology is generated by elementary Nisnevich squares and abstract blow-up squares, and a presheaf of cochain complexes is a hypersheaf for this topology exactly when it satisfies the Mayer–Vietoris property for both classes of squares. The paper's presheaf $F^\\bullet(X)=C^\\bullet(|X^{\\mathrm{an}}|,\\mathbb{Z})$ is shown to have those Mayer–Vietoris properties, so it computes cdh-cohomology; the contractibility theorem for the analytification of smooth schemes over trivially valued fields then identifies its cdh-sheafification with the constant sheaf $\\mathbb{Z}$. On the motivic side, the computational tool is the cotriple resolution of a commutative algebraic group $G$ by iterated (reduced) symmetric powers $S^\\bullet(G)$; the correspondence between finite correspondences and maps to symmetric powers converts this into an explicit resolution of the presheaf-with-transfers $\\underline{G}$, and the group-completion theorem for simplicial monoids makes the simplicial monoids behave like their group completions. The acyclicity of the resolution against $\\mathbb{Z}$ is exactly the descent theorem from section 4.","core_discovery":"Theorem 4.3 states: for a connected scheme X locally of finite type over a trivially valued field k of characteristic 0 admitting resolution of singularities, the canonical map $C^\\bullet(|X^{\\mathrm{an}}|,\\mathbb{Z}) \\to R\\Gamma_{\\mathrm{cdh}}(X,\\mathbb{Z})$ is a quasi-isomorphism, so $H^n_{\\mathrm{sing}}(|X^{\\mathrm{an}}|,\\mathbb{Z}) \\cong H^n_{\\mathrm{cdh}}(X,\\mathbb{Z})$. Because cdh-cohomology with integer coefficients agrees with weight-zero motivic cohomology for such schemes, the singular cohomology of the analytic space is an algebraic, motivic invariant. The proof compares two presheaves: the singular cochains of the analytification and the constant sheaf $\\mathbb{Z}$, and verifies that both satisfy the same descent conditions for the cd-structure generating the cdh-topology; on smooth schemes the analytification is contractible, so the comparison is constant there and, by resolution of singularities, determines the sheaf everywhere. The paper then exploits the same descent to show that the resolution of a commutative algebraic group $G$ by (iterated) symmetric powers is acyclic against $\\mathrm{Hom}(-,\\mathbb{Z})$, yielding $R\\mathrm{Hom}_{\\mathrm{Sh}_{\\mathrm{Nis}}(\\mathrm{cor}_k)}(\\underline{G},\\mathbb{Z}) \\simeq 0$, and to prove $R\\mathrm{Hom}_{\\mathcal{PS}h_{tr}}(\\underline{A},\\underline{B}) \\simeq \\mathrm{Hom}_{\\mathbf{Ab}_k}(A,B)$ for abelian varieties $A,B$ via rigidity.","pith_inferences":["The same comparison strategy should apply to cohomology with finite or profinite coefficients, provided the corresponding coefficient presheaf satisfies the two Mayer–Vietoris properties; this would give a purely topological model for mod-$\\ell$ weight-zero motivic cohomology over trivially valued fields.","The resolution by iterated reduced symmetric powers is a general machine, not tied to $\\mathbb{Z}$ as the target: the same acyclicity question for $\\mathrm{Hom}(-,F)$ for other homotopy-invariant sheaves with transfers $F$ would be a natural next test of its reach.","The rigidity argument that forces all maps in the cosimplicial complex to be identities suggests that $R\\mathrm{Hom}$ between a semi-abelian variety and an abelian variety might also be computable, with the torus part contributing additional structure rather than vanishing.","One could try to detect the failure of the closed-cover Mayer–Vietoris hypothesis by searching for a singular X over a trivially valued field whose analytification is not locally contractible; if such a space admits a nontrivial singular cohomology class coming from the blow-up cover, the comparison theorem would need extra hypotheses."],"forward_implications":["For any X covered by the theorem, the singular cohomology groups $H^n(|X^{\\mathrm{an}}|,\\mathbb{Z})$ are actually algebraic invariants, computable from the cdh-site of X.","For every commutative algebraic group G over k, $\\mathrm{Ext}^i_{\\mathrm{Sh}_{\\mathrm{Nis}}(\\mathrm{cor}_k)}(\\underline{G},\\mathbb{Z})=0$ for all $i \\ge 1$, and for semi-abelian G the motive $M_1(G)$ has trivial RHom against $\\mathbb{Z}$ in the effective derived category.","For abelian varieties A and B, $\\mathrm{Ext}^i_{\\mathcal{PS}h_{tr}}(\\underline{A},\\underline{B})=0$ for $i \\ge 1$ and $R\\mathrm{Hom}_{\\mathcal{PS}h_{tr}}(\\underline{A},\\underline{B})$ is exactly the group of algebraic homomorphisms $\\mathrm{Hom}_{\\mathbf{Ab}_k}(A,B)$; the same holds after passing to Nisnevich sheaves with transfers.","The cdh-cohomology of singular schemes over trivially valued fields can be read off from the topology of a single analytic space, bypassing sheaf-cohomological computations."],"supporting_citations":[{"why":"Gives the analytic-space properties (pullback preservation, open/closed immersions, proper maps) used to turn Nisnevich and blow-up squares into topological covers.","marker":"[4]"},{"why":"Supplies the étale and open-mapping properties of analytic maps needed for the Nisnevich descent lemma.","marker":"[5]"},{"why":"Provides the Mayer–Vietoris sequence for singular cohomology of a closed covering, invoked in the abstract blow-up lemma.","marker":"[9]"},{"why":"States that a presheaf of cochain complexes is a hypersheaf for the cdh-topology precisely when it satisfies the two Mayer–Vietoris properties.","marker":"[7]"},{"why":"Contains the homotopy theory of simplicial sheaves for cd-structures, underlying the hypersheaf criterion.","marker":"[20]"},{"why":"Introduces the Nisnevich and cdh cd-structures and their associated topologies, the site on which the comparison is made.","marker":"[21]"},{"why":"Gives the standard identifications of cdh-cohomology with Nisnevich/Zariski cohomology on smooth schemes and the Ext-to-cohomology dictionary.","marker":"[11]"},{"why":"Contains the contractibility theorem for the analytification of smooth schemes over trivially valued fields, used to identify the sheafification with Z.","marker":"[18]"},{"why":"Establishes the correspondence between finite correspondences and maps to symmetric powers, used to turn the symmetric-power resolution into a resolution in presheaves with transfers.","marker":"[17]"},{"why":"Provides the group-completion theorem for simplicial commutative monoids, used to replace symmetric-power simplicial monoids by their group completions.","marker":"[14]"}],"fun_headline_variants":["Berkovich cohomology equals motivic weight zero","Analytic singular cohomology is algebraic","Weight-zero motivic cohomology from Berkovich spaces","Ext groups of abelian varieties now computable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that singular cohomology of the underlying topological space of a Berkovich space satisfies the Mayer–Vietoris long exact sequence for the closed covering $|X^{\\mathrm{an}}| = |\\widetilde{X}^{\\mathrm{an}}| \\cup |Z^{\\mathrm{an}}|$ induced by an abstract blow-up square, where the paper cites a textbook but never checks the topology of the covering satisfies the hypotheses.","fun_headline_variants_meta":{"raw":{"variants":["Berkovich cohomology equals motivic weight zero","Analytic singular cohomology is algebraic","Weight-zero motivic cohomology from Berkovich spaces","Ext groups of abelian varieties now computable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000413,"raw_usage":{"total_tokens":2188,"prompt_tokens":1049,"completion_tokens":1139,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":665,"completion_tokens_details":{"reasoning_tokens":1076}},"tokens_in":665,"tokens_out":1139,"duration_ms":8244,"temperature":1.0,"reasoning_tokens":1076,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:23:53.021192+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the cuspidal cubic X over a trivially valued field k, let $\\widetilde{X}$ be its normalization and Z the singular point. Compute $H^1_{\\mathrm{sing}}(|X^{\\mathrm{an}}|,\\mathbb{Z})$ directly from the closed cover $|\\widetilde{X}^{\\mathrm{an}}| \\cup |Z^{\\mathrm{an}}|$ using the purported Mayer–Vietoris sequence, and compare with $H^1_{\\mathrm{cdh}}(X,\\mathbb{Z})$ computed algebraically; any mismatch, or any failure of the Mayer–Vietoris sequence to converge, would disprove Theorem 4.3 or its proof.","supporting_citations":[{"cited_title":"Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean F ields, Surveys and Monographs 33, Amer","cited_arxiv_id":null,"evidence_quote":"Gives the analytic-space properties (pullback preservation, open/closed immersions, proper maps) used to turn Nisnevich and blow-up squares into topological covers."},{"cited_title":"Berkovich, Étale cohomology for non-Archimedean analytic spaces , Publ","cited_arxiv_id":null,"evidence_quote":"Supplies the étale and open-mapping properties of analytic maps needed for the Nisnevich descent lemma."},{"cited_title":"Iversen, Cohomology of Sheaves , Universitext, Springer-Verlag, Berlin Heidelberg, 1986","cited_arxiv_id":null,"evidence_quote":"Provides the Mayer–Vietoris sequence for singular cohomology of a closed covering, invoked in the abstract blow-up lemma."},{"cited_title":"Cortiñas, C","cited_arxiv_id":null,"evidence_quote":"States that a presheaf of cochain complexes is a hypersheaf for the cdh-topology precisely when it satisfies the two Mayer–Vietoris properties."},{"cited_title":"Voevodsky, Homotopy theory of simplicial sheaves in completely decompo sable topologies , Journal of Pure and Applied Algebra 214 (2010), no","cited_arxiv_id":null,"evidence_quote":"Contains the homotopy theory of simplicial sheaves for cd-structures, underlying the hypersheaf criterion."},{"cited_title":"Voevodsky, Unstable motivic homotopy categories in Nisnevich and cdh-t opologies, Journal of Pure and Applied Algebra 214 (2010), no","cited_arxiv_id":null,"evidence_quote":"Introduces the Nisnevich and cdh cd-structures and their associated topologies, the site on which the comparison is made."},{"cited_title":"Mazza, V","cited_arxiv_id":null,"evidence_quote":"Gives the standard identifications of cdh-cohomology with Nisnevich/Zariski cohomology on smooth schemes and the Ext-to-cohomology dictionary."},{"cited_title":"Thuillier, Géométrie toroïdale et géométrie analytique non archimédien ne","cited_arxiv_id":null,"evidence_quote":"Contains the contractibility theorem for the analytification of smooth schemes over trivially valued fields, used to identify the sheafification with Z."},{"cited_title":"Suslin and V","cited_arxiv_id":null,"evidence_quote":"Establishes the correspondence between finite correspondences and maps to symmetric powers, used to turn the symmetric-power resolution into a resolution in presheaves with transfers."},{"cited_title":"Sagave, T","cited_arxiv_id":null,"evidence_quote":"Provides the group-completion theorem for simplicial commutative monoids, used to replace symmetric-power simplicial monoids by their group completions."}],"review_version":1}