{"id":"a76d88ef-2c85-44e7-8e97-f465552f6be9","arxiv_id":"1908.10987","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Grothendieck category is n-coherent exactly when its FP_n objects are closed under kernels of epimorphisms, equivalently when FP_n equals FP_infinity.","lead":"This paper brings finiteness conditions from module theory to all Grothendieck categories. It defines n-coherent categories that generalize locally noetherian and locally coherent categories, and proves when FP_n-injective cotorsion pairs are hereditary and yield Gorenstein model structures.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 4.7 appears soundly argued; the main unresolved point is the length and subtlety of the proof of Lemma 2.6, which deserves independent verification rather than constituting a discovered flaw.","rationale":"The paper's strongest claim, Theorem 4.7, is internally consistent. I traced the equivalences and found no explicit gap: Proposition 2.8 supplies the closure properties; Lemma 2.6 supplies the needed monomorphism in the critical degree; and the hereditary cotorsion-pair implication is routed through a cited lemma whose hypotheses match condition (c). The reader's stated weakest assumption, the restriction to locally type FP_n categories, is a genuine scope condition but not a flaw, since the paper explicitly notes categories may lack nonzero type FP_n objects. The more substantive concern is verification-level: the proof of Lemma 2.6 is long, not machine-checked, and several steps are delegated to references or marked 'left to the reader' (e.g., Lemma 5.2(b)). These justify a CONDITIONAL verdict but do not amount to a demonstrated counterexample or internal inconsistency. I therefore keep the verdict UNCHANGED, while agreeing that an independent check of the appendix is worthwhile.","tokens_in":36732,"tokens_out":45252,"duration_ms":406239,"concrete_test":"Have an expert (or a proof assistant) independently reconstruct the proof of Lemma 2.6 for n=2 in a locally finitely presented Grothendieck category, checking the dimension-shift and lifting steps around diagrams (vi)-(viii) of Appendix A. If the lifting of [H] to some [H_{t0}] fails, Proposition 2.8(1) and hence Theorem 4.7 would collapse; if it succeeds, the main concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read Theorem 4.7 as the central claim: n-coherence of a locally type FP_n category is equivalent to thickness of FP_n and, for n≥1, to hereditariness of the FP_n-injective cotorsion pair. The proof depends on Proposition 2.8, whose key technical input is Lemma 2.6 (the monomorphism property for lim Ext^n(F, -)). The proof of Lemma 2.6 in Appendix A is long and uses delicate Yoneda-extension manipulations, including an appeal to [36, Lemma VII.4.1] and a dimension-shift/lifting argument. I could not locate a concrete mathematical error: the step where an element of Ext^{n-1}(F, lim W_t) is lifted to some Ext^{n-1}(F, W_{t0}) is justified by F being of type FP_n, so Ext^{n-1}(F, -) preserves direct limits, not merely by the induction hypothesis. Likewise, the implication (c)⇒(g) delegates to [26, Lemma 3.6(4)], which is plausible because condition (c) exactly says FP_n is thick. Thus the central argument appears sound, but the proof is not independently machine-checked and the appendix is the least secure part. The reader's CONDITIONAL verdict is reasonable on verification grounds, but I do not see a load-bearing flaw that would overturn the claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces objects of type FP_n in an arbitrary Grothendieck category G, defines locally type FP_n categories as those with a generating set of such objects, and defines n-coherent categories as locally type FP_n categories in which every type FP_n object is n-coherent. The central result, Theorem 4.7, gives several equivalent characterizations of n-coherence, including closure of FP_n under kernels of epimorphisms, thickness of FP_n, equality FP_n = FP_∞, and, for n ≥ 1, hereditariness of the FP_n-injective cotorsion pair and related injectivity conditions. The proofs use closure properties of FP_n developed in Section 2, whose key technical input is Lemma 2.6, a monomorphism property for the canonical map lim Ext^n(F, -) → Ext^n(F, lim -), proved in Appendix A via Yoneda n-fold extensions. Section 3 shows that FP_n-Inj is the right half of a functorially complete cotorsion pair, and Section 5 defines Gorenstein FP_n-injective objects and constructs abelian and exact model structures. The paper also gives applications to quasi-coherent sheaves and functor categories.","tokens_in":37020,"tokens_out":20866,"duration_ms":202502,"significance":"If Theorem 4.7 is correct, it unifies the module-theoretic results of Bravo–Pérez and the locally noetherian/locally coherent hierarchy in a framework that does not require enough projectives; the applications to Qcoh(X) and Fun(C^op, Ab) are natural and nontrivial. The detailed proof of Lemma 2.6, a delicate and load-bearing monomorphism property, is a genuine contribution, as are the concrete examples showing that type FP_n and n-presentability diverge without projective generators. The paper proposes new definitions with substantive consequences rather than merely reformulating known module facts. The main unresolved concerns are the amount of material left to the reader in Section 5 and the compressed presentation of a few steps of the central proof, rather than an apparent mathematical error in the main theorem.","major_comments":[{"comment":"Lemma 5.2 is described as a 'straightforward exercise' left to the reader, and Proposition 5.3 is explicitly 'left to the reader'. These results are load-bearing: Proposition 5.3's closure properties are used in Proposition 5.7 to assert that (Inj, GI) is a strong Frobenius pair, and Proposition 5.7 underpins Proposition 5.8 and Theorem 5.10. Since the abstract advertises model structures whose fibrant objects are the Gorenstein FP_n-injectives, the reader needs a complete proof or precise pointers to the literature with enough detail to verify the claims in the Grothendieck-category setting.","section":"Section 5, Lemma 5.2 and Proposition 5.3"},{"comment":"The sentence 'Construct a diagram as in the proof of Lemma 4.4' does not name the exact sequences to which Proposition 2.8(3) is applied; the diagram in Lemma 4.4 is also compressed, so the conclusion that 'both K and P are in FP_{n-1}' is not directly checkable. Please rewrite the diagram with labeled short exact sequences and state which hypothesis (B,C ∈ FP_n or ⊕C_j ∈ C_n) is used in each application of Proposition 2.8.","section":"Theorem 4.7, proof of (d) implies (b)"}],"minor_comments":[{"comment":"The step 'By the induction hypothesis, we know that Ext^{n-1}(F, lim W_t) ≅ lim Ext^{n-1}(F, W_t)' is not a consequence of Lemma 2.6 alone, which only gives a monomorphism; the isomorphism follows from the hypothesis that F is of type FP_n. The argument is sound once this is corrected, but the present wording makes the proof of the central technical lemma harder to verify.","section":"Appendix A, proof of Lemma 2.6"},{"comment":"The sentence 'the case n = 0 is well-known' about closure of FP_0-Inj under direct limits could mislead: in an arbitrary Grothendieck category, injective objects are not closed under direct limits. Please state explicitly that this is used only under the hypotheses that make the category locally noetherian, or restrict the sentence to n ≥ 1.","section":"Theorem 4.7, proof of (g) implies (e)"},{"comment":"The claim that 'any object of type FP_n is finitely presented' should explicitly be read for n ≥ 1; for n = 0 only finite generation is intended. The surrounding text already explains this, but the unqualified sentence is potentially confusing.","section":"Definition 2.1 and following paragraph"},{"comment":"The citation '[32, 31, Chapter 7]' for Hovey's stronger notion of triangulated category appears to be a typo; presumably the intended reference is [32, Chapter 7].","section":"Remark 5.12"},{"comment":"The phrase 'since FP_n(G) is a generating set' should be justified: a locally type FP_n category has some generating set of type FP_n objects, and any such generator is isomorphic to an element of FP_n(G) once a complete set of representatives is chosen.","section":"Proposition 3.5"}],"recommendation":"major_revision","confidential_remarks":"The main theorem appears sound and the paper is a substantial contribution. The length of Appendix A and the heavy reliance on the authors' prior papers and on preprints make independent verification expensive; adding full proofs in Section 5 and clarifying the two compressed parts of the proof of Theorem 4.7 would address most of my concerns. I would not recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper is worth your time. It does something real: it takes the FP_n module theory from the authors' earlier ring-level work and transplants it into arbitrary Grothendieck categories, without assuming enough projectives. The new definitions — locally type FP_n and n-coherent categories — are natural, and they recover locally finitely generated, locally finitely presented, locally noetherian, and locally coherent categories as special cases. The central result, Theorem 4.7, is a genuinely new characterization: in a locally type FP_n category, n-coherence is equivalent to thickness of FP_n, and for n≥1 also to hereditariness of the FP_n-injective cotorsion pair and to FP_n = FP_∞. That is a clean and useful theorem, and the proof strategy is sensible.\n\nThe paper earns credit for working without projectives, which lets it apply to quasi-coherent sheaves and functor categories. The examples are informative, especially the quasi-coherent sheaf case where having an n-presentation differs from being of type FP_n. The applications to Gorenstein FP_n-injectives and two model structures are a nice payoff, though they lean more heavily on prior work.\n\nSoft spots, in proportion. The proof of Lemma 2.6, which is load-bearing for the closure properties in Proposition 2.8, is long, technical, and deferred to an appendix. It relies on delicate Yoneda extension manipulations and an appeal to an external lemma. I did not find a concrete error, and the stress-test note agrees; but this is the least secure part of the paper and deserves independent checking. Also, Section 5 leaves two standard-but-unwritten proofs as exercises (Lemma 5.2 and Proposition 5.3). These are minor gaps, not fatal flaws, but a referee should ask the authors to fill them in or at least give references.\n\nOne more caveat, more of a scope note: the whole framework requires the category to be locally type FP_n, so it says nothing about Grothendieck categories with no nonzero objects of type FP_n. The authors are explicit about this, so it is not a hidden assumption, but it does limit the reach.\n\nOverall: the central claim holds up, the omissions are minor, and the citation pattern is heavy on the authors' own prior work but not in a way that manufactures the conclusions. This deserves a serious referee.\n\nRecommendation: send it to peer review.","headline":"A solid, genuinely useful generalization of FP_n finiteness to Grothendieck categories; Theorem 4.7 is the core and appears sound, with the long Lemma 2.6 as the main soft spot.","tokens_in":37558,"tokens_out":993,"would_cite":true,"duration_ms":12767,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18C35","18A25","18E15","18F20","18G15","18G25","18G55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Finiteness classes collapse in n-coherent categories","keywords":["objects of type FP_n","FP_n-injective objects","locally type FP_n categories","n-coherent categories","Gorenstein FP_n-injective objects","cotorsion pairs","Grothendieck categories","abelian and exact model structures"],"falsifier":"Exhibit a locally type $\\mathrm{FP}_2$ Grothendieck category in which the $\\mathrm{FP}_2$-injective cotorsion pair is hereditary yet $\\mathrm{FP}_2$ is strictly larger than $\\mathrm{FP}_\\infty$. Theorem 4.7 asserts these cannot coexist, so finding one settles the claim negatively.","tokens_in":36528,"feed_emoji":"🧮","tokens_out":13837,"duration_ms":114880,"temperature":0.7,"pith_summary":"Grothendieck categories—the common home of modules, chain complexes, sheaves, and functor categories—carry a natural ladder of finiteness levels: objects of type $\\mathrm{FP}_n$ are those whose $\\mathrm{Ext}^i$ into direct limits preserves direct limits for $i < n$. The paper uses this ladder to define locally type $\\mathrm{FP}_n$ categories and then $n$-coherent categories, and proves that in the locally type $\\mathrm{FP}_n$ setting the ladder collapses exactly: $\\mathrm{FP}_n = \\mathrm{FP}_\\infty$ holds precisely when $\\mathrm{FP}_n$ is closed under kernels of epimorphisms, precisely when it is thick, and (for $n \\ge 1$) precisely when the $\\mathrm{FP}_n$-injective cotorsion pair is hereditary. Thus $n$-coherent categories are the natural home for relative homological algebra with respect to $\\mathrm{FP}_n$-injectives: in them every object has $\\mathrm{FP}_n$-injective (pre)covers and $\\mathrm{Ext}^k$ against type-$\\mathrm{FP}_n$ objects can be computed from $\\mathrm{FP}_n$-injective coresolutions. The framework recovers locally noetherian ($n=0$) and locally coherent ($n=1$) categories as the first two rungs, and works without projectives, so it applies to quasi-coherent sheaves on schemes.","feed_headline":"Finiteness classes collapse in n-coherent categories","feed_subtitle":"Locally type FP_n categories are n-coherent exactly when FP_n = FP_∞ and FP_n-injectives form a hereditary cotorsion pair.","key_machinery":"The engine is the class $\\mathrm{FP}_n$ of objects of type $\\mathrm{FP}_n$—objects $F$ for which $\\mathrm{Ext}^i_{\\mathcal{G}}(F,-)$ preserves direct limits for $0 \\le i \\le n-1$—together with its Ext-orthogonal class $\\mathrm{FP}_n\\text{-Inj}$ of $\\mathrm{FP}_n$-injective objects. The proof runs through two mechanisms: the closure properties of $\\mathrm{FP}_n$ under short exact sequences (Proposition 2.8), obtained by applying the 5-lemma to the monomorphism property of Lemma 2.6; and the complete cotorsion pair $({}^\\perp_1(\\mathrm{FP}_n\\text{-Inj}), \\mathrm{FP}_n\\text{-Inj})$ cogenerated by a set (Theorem 3.6), obtained from a small-cotorsion-pair argument. Theorem 4.7 then identifies heredity of this cotorsion pair, thickness of $\\mathrm{FP}_n$, and the collapse $\\mathrm{FP}_n = \\mathrm{FP}_\\infty$ as a single condition.","core_discovery":"The central claim is Theorem 4.7: in a Grothendieck category $\\mathcal{G}$ that is locally type $\\mathrm{FP}_n$, the following are equivalent—every object of type $\\mathrm{FP}_n$ is $n$-coherent; $\\mathrm{FP}_n$ is closed under kernels of epimorphisms; $\\mathrm{FP}_n$ is thick; $\\mathcal{G}$ has a generating set of $n$-coherent objects satisfying the finite-sum condition of Lemma 4.4; and $\\mathrm{FP}_n = \\mathrm{FP}_\\infty$. For $n \\ge 1$, these are also equivalent to the $\\mathrm{FP}_n$-injective cotorsion pair $({}^\\perp_1(\\mathrm{FP}_n\\text{-Inj}), \\mathrm{FP}_n\\text{-Inj})$ being hereditary, to $\\mathrm{FP}_n\\text{-Inj}$ being closed under cokernels of monomorphisms, to $\\mathrm{FP}_n\\text{-Inj} = \\mathrm{FP}_\\infty\\text{-Inj}$ (the absolutely clean objects), and to $\\mathrm{FP}_{n+1}\\text{-Inj} \\subseteq \\mathrm{FP}_n\\text{-Inj}$. The paper's interpretation is that $n$-coherent categories are the categorical setting in which the module-theoretic theorem for $n$-coherent rings—hereditary $\\mathrm{FP}_n$-injective cotorsion pair and relative Gorenstein homological algebra—holds in full generality.","pith_inferences":["Beyond the paper: if the theorem is right, a locally type $\\mathrm{FP}_n$ category in which $\\mathrm{FP}_n\\text{-Inj}$ is covering but $\\mathrm{FP}_n \\ne \\mathrm{FP}_\\infty$ would be a genuine counterexample to the converse of Corollary 4.14; since the paper does not settle whether $n$-coherence is necessary for covering, looking for such categories is a concrete next step.","Beyond the paper: the Yoneda-extension proof of Lemma 2.6 suggests that the same monomorphism technique can push type-$\\mathrm{FP}_n$ statements into categories without projective generators, such as derived categories of sheaves or functor categories with only pseudo-kernels.","Beyond the paper: the Appendix B discussion of $n$-coherent rings suggests that a Zariski-local notion of '$n$-coherent scheme' for $n \\ge 2$ is plausible; if the product question about $n$-coherent rings is resolved affirmatively, then $\\mathrm{Qcoh}(X)$ would be $n$-coherent exactly when all local affine rings are $n$-coherent."],"forward_implications":["Every $n$-coherent category is locally type $\\mathrm{FP}_\\infty$, so the finiteness classes $\\mathrm{FP}_n$ and $\\mathrm{FP}_\\infty$ coincide throughout the category.","The $n$-coherent categories form an ascending chain: 0-coherent (locally noetherian) $\\subseteq$ 1-coherent (locally coherent) $\\subseteq \\cdots \\subseteq$ $\\infty$-coherent.","In any $n$-coherent category, $\\mathrm{FP}_n\\text{-Inj}$ is not only preenveloping but covering, generalising the existence of absolutely pure covers in locally coherent categories.","For $n \\ge 1$, whenever $F$ is of type $\\mathrm{FP}_n$, every $\\mathrm{Ext}^k(F,-)$ can be computed from $\\mathrm{FP}_n$-injective coresolutions, because $\\mathrm{FP}_n$-injectives are $\\mathrm{Hom}(F,-)$-acyclic.","If the category is additionally locally finite dimensional $n$-coherent, an abelian model structure exists whose fibrant objects are exactly the Gorenstein $\\mathrm{FP}_n$-injectives; dropping the finite-dimensional hypothesis still yields an exact model structure on the subcategory of objects of finite Gorenstein $\\mathrm{FP}_n$-injective dimension."],"supporting_citations":[{"why":"Supplies the locally presentable facts used throughout: $γ$-presented objects form a set, and direct-limit characterisations control finite presentation.","marker":"[1]"},{"why":"Gives the injective-cogenerator characterisation of type $\\mathrm{FP}_n$ objects used in Proposition 3.8 to identify $\\mathrm{FP}_n$ with $\\mathrm{FP}_{n-1} \\cap {}^\\perp_1(\\mathrm{FP}_n\\text{-Inj})$.","marker":"[10]"},{"why":"Provides the module-level theorem being generalised: $n$-coherent rings, $\\mathrm{FP}_n$ modules, and the hereditary $\\mathrm{FP}_n$-injective cotorsion pair.","marker":"[11]"},{"why":"Introduces $n$-coherent rings, whose module categories are recovered as $n$-coherent categories in Example 4.9(1).","marker":"[14]"},{"why":"Supplies the finitely-accessible cover criterion used in Corollary 4.14 to show $\\mathrm{FP}_n\\text{-Inj}$ is covering.","marker":"[15]"},{"why":"Provides locally type $\\mathrm{FP}_\\infty$ categories, absolutely clean ($\\mathrm{FP}_\\infty$-injective) objects, and the Gorenstein injective model structures adapted here.","marker":"[26]"},{"why":"Supplies the small-cotorsion-pair machinery that makes the $\\mathrm{FP}_n$-injective pair functorially complete.","marker":"[33]"},{"why":"Supplies the Grothendieck-category foundations: locally finitely generated and locally finitely presented categories, the Baer-criterion analogue, and finitely generated object facts.","marker":"[41]"}],"fun_headline_variants":["n-coherent categories collapse finiteness classes","FP_n = FP_∞: the n-coherent condition","Hereditary cotorsion pairs in n-coherent categories","The n-coherent collapse: FP_n = FP_∞"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central assumption is that the Grothendieck category is locally type $\\mathrm{FP}_n$—it has a generating set of objects of type $\\mathrm{FP}_n$, a genuine restriction since a category may contain no nonzero objects of this type—and the closure theorems further require the category to be locally finitely presented; without such generators the classes $\\mathrm{FP}_n$, $\\mathrm{FP}_n\\text{-Inj}$, and the cotorsion pairs built from them do not exist.","fun_headline_variants_meta":{"raw":{"variants":["n-coherent categories collapse finiteness classes","FP_n = FP_∞: the n-coherent condition","Hereditary cotorsion pairs in n-coherent categories","The n-coherent collapse: FP_n = FP_∞"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001031,"raw_usage":{"total_tokens":4454,"prompt_tokens":1167,"completion_tokens":3287,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":783,"completion_tokens_details":{"reasoning_tokens":3220}},"tokens_in":783,"tokens_out":3287,"duration_ms":26048,"temperature":1.0,"reasoning_tokens":3220,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:29:47.765787+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a locally type $\\mathrm{FP}_2$ Grothendieck category in which the $\\mathrm{FP}_2$-injective cotorsion pair is hereditary yet $\\mathrm{FP}_2$ is strictly larger than $\\mathrm{FP}_\\infty$. Theorem 4.7 asserts these cannot coexist, so finding one settles the claim negatively.","supporting_citations":[{"cited_title":"Adámek and J","cited_arxiv_id":null,"evidence_quote":"Supplies the locally presentable facts used throughout: $γ$-presented objects form a set, and direct-limit characterisations control finite presentation."},{"cited_title":"Bravo and C","cited_arxiv_id":null,"evidence_quote":"Gives the injective-cogenerator characterisation of type $\\mathrm{FP}_n$ objects used in Proposition 3.8 to identify $\\mathrm{FP}_n$ with $\\mathrm{FP}_{n-1} \\cap {}^\\perp_1(\\mathrm{FP}_n\\text{-Inj})$."},{"cited_title":"Bravo and M","cited_arxiv_id":null,"evidence_quote":"Provides the module-level theorem being generalised: $n$-coherent rings, $\\mathrm{FP}_n$ modules, and the hereditary $\\mathrm{FP}_n$-injective cotorsion pair."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces $n$-coherent rings, whose module categories are recovered as $n$-coherent categories in Example 4.9(1)."},{"cited_title":"Crivei, M","cited_arxiv_id":null,"evidence_quote":"Supplies the finitely-accessible cover criterion used in Corollary 4.14 to show $\\mathrm{FP}_n\\text{-Inj}$ is covering."},{"cited_title":"Gillespie","cited_arxiv_id":null,"evidence_quote":"Provides locally type $\\mathrm{FP}_\\infty$ categories, absolutely clean ($\\mathrm{FP}_\\infty$-injective) objects, and the Gorenstein injective model structures adapted here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the small-cotorsion-pair machinery that makes the $\\mathrm{FP}_n$-injective pair functorially complete."},{"cited_title":"Stenström","cited_arxiv_id":null,"evidence_quote":"Supplies the Grothendieck-category foundations: locally finitely generated and locally finitely presented categories, the Baer-criterion analogue, and finitely generated object facts."}],"review_version":1}