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Locally type $\text{FP}_n$ and $n$-coherent categories

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Pith's one-line read Finiteness classes collapse in n-coherent categories

desk verdict 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. read the letter →

arxiv 1908.10987 v1 pith:VNXG3QEL submitted 2019-08-28 math.CT math.ATmath.KT

Daniel Bravo , James Gillespie , Marco A. Pérez This is my paper · ORCID
classification math.CTmath.ATmath.KT MSC 18C3518A2518E1518F2018G1518G2518G55
keywords objectsoftypeFP_nFP_n-injectivelocallycategoriesn-coherentGorensteincotorsionpairsGrothendieckabelianandexactmodelstructures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Section 5, Lemma 5.2 and Proposition 5.3] 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.
  2. [Theorem 4.7, proof of (d) implies (b)] 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.
minor comments (5)
  1. [Appendix A, proof of Lemma 2.6] 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.
  2. [Theorem 4.7, proof of (g) implies (e)] 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.
  3. [Definition 2.1 and following paragraph] 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.
  4. [Remark 5.12] 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].
  5. [Proposition 3.5] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 4.7 is a genuine generalization whose key technical lemma is proved in the appendix, and the cited prior results are used as independent lemmas rather than as restatements of the target claims.

full rationale

The paper's central claim, Theorem 4.7, is not forced by its definitions or by fitted inputs. The notions of objects of type FP_n (Definition 2.1), locally type FP_n categories (Definition 2.3), and n-coherent categories (Definitions 4.1 and 4.6) are independent finiteness conditions, not reformulations of the equivalent properties being derived. The proof of Theorem 4.7 rests on Proposition 2.8, whose key technical ingredient is Lemma 2.6; that lemma is not assumed but proved at length in Appendix A using Yoneda extensions and standard homological facts, so the main derivation is self-contained at the level of proof obligation. The implication (c) implies (g) invokes [26, Lemma 3.6(4)], a prior published lemma whose hypothesis that the set of representatives of FP_n is thick is exactly condition (c); this is a legitimate application of an independent result, not a definitional collapse or a self-citation that smuggles in the conclusion. Similarly, Proposition 3.8 cites [10, Theorem B.1], a separate characterization of objects of type FP_n; using it to prove (i) implies (b) is a standard deduction from an earlier theorem, not a circular reduction. No fitted parameters are disguised as predictions, no uniqueness theorem is imported to forbid alternatives, and no known empirical pattern is merely renamed. Some cited results are from the authors' prior work, but they are parameter-free lemmas with stated hypotheses that do not include the target equivalence, so under the rules they count as independent support. The main unresolved point is the length and delicacy of the Appendix A proof of Lemma 2.6, which invites independent verification, but a verification burden is not circularity. Therefore the derivation chain is not circular; the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 3 invented entities

The ledger is clean: no free parameters or fitted constants. The paper's new definitions (locally type FP_n, n-coherent categories, Gorenstein FP_n-injective objects) are grounded in known special cases for n=0,1 and in prior ring-level and complex-level work, so they carry independent evidence. The main hypotheses are standard Grothendieck-category assumptions plus the local type FP_n condition.

assumptions (7)
  • standard math Every Grothendieck category is locally presentable, has exact direct limits, and has enough injectives with injective envelopes.
    Used throughout the paper; the categorical framework is set in Section 1 and cited to [41] and [1].
  • standard math The class of γ-presented objects in a locally presentable category is skeletally small.
    Used in Notation 3.3 to choose a set FP_n(G) of isomorphism representatives; cited to [1, Corollary 1.69] and [24, Appendix, Fact A.9].
  • standard math Hovey's correspondence between abelian model structures and compatible complete cotorsion pairs holds.
    Used in Section 5 to translate cotorsion pairs into model structures; cited to [33].
  • standard math Gillespie's generalization of Hovey's correspondence to exact categories holds.
    Used in Theorem 5.10 to obtain the exact model structure on GI^∨; cited to [25].
  • standard math The theory of Frobenius pairs yields complete cotorsion pairs in exact subcategories.
    Used in Proposition 5.8 and Theorem 5.10 to build the exact Gorenstein model structure; cited to [4].
  • domain assumption The ground Grothendieck category G is locally type FP_n, meaning it has a generating set of objects of type FP_n.
    Central hypothesis for Definition 2.3, Theorems 3.6 and 4.7; the authors note before Definition 2.3 that without such generators the theory may be vacuous.
  • domain assumption G is locally finitely presented for the closure properties (Proposition 2.8) and for Lemma 2.6.
    Stated explicitly in Lemma 2.6 and Proposition 2.8; implied by locally type FP_n for n≥1, but not for n=0.
invented entities (3)
  • locally type FP_n categories independent evidence
    purpose: Generalize locally finitely generated and locally finitely presented categories to arbitrary finiteness level n.
    Specializes to known notions for n=0,1 and to locally type FP_infinity of [26]; used as the framework for the whole paper.
  • n-coherent categories independent evidence
    purpose: Generalize locally noetherian and locally coherent categories and provide a setting where FP_n-injective cotorsion pairs are hereditary and covers exist.
    Recovers locally noetherian and locally coherent categories for n=0,1 (Theorem 4.7), and the ring-level version is already established in [11].
  • Gorenstein FP_n-injective objects independent evidence
    purpose: Provide relative Gorenstein injective objects and enable model structures.
    Recovers Gorenstein injectives for n=0 and Ding injectives for n=1 (Section 5).

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Pith. "Pith review of Locally type $\text{FP}_n$ and $n$-coherent categories." pith.science (2026). https://pith.science/paper/VNXG3QEL

@misc{pith2026190810987,
  author       = {Pith},
  title        = {Pith review of: Locally type $\textFP_n$ and $n$-coherent categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VNXG3QEL}},
  note         = {Machine review of arXiv:1908.10987}
}
abstract

We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type $\text{FP}_n$ and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of locally type $\text{FP}_n$ categories as a generalization of locally finitely generated and locally finitely presented categories. We also define and study the injective objects that are Ext-orthogonal to the class of objects of type $\text{FP}_n$, called $\text{FP}_n$-injective objects, which will be the right half of a complete cotorsion pair. As a generalization of the category of modules over an $n$-coherent ring, we present the concept of $n$-coherent categories, which also recovers the notions of locally noetherian and locally coherent categories for $n = 0, 1$. Such categories will provide a setting in which the $\text{FP}_n$-injective cotorsion pair is hereditary, and where it is possible to construct (pre)covers by $\text{FP}_n$-injective objects. Moreover, we see how $n$-coherent categories provide a suitable framework for a nice theory of Gorenstein homological algebra with respect to the class of $\text{FP}_n$-injective modules. We define Gorenstein $\text{FP}_n$-injective objects and construct two different model category structures (one abelian and the other one exact) in which these Gorenstein objects are the fibrant objects.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Locally finitely presented and coherent hearts

    math.CT 2019-08 accept novelty 6.0 of 10

    For large classes of Grothendieck categories, the heart of the Happel-Reiten-Smaloe t-structure is locally finitely presented exactly when the torsion pair is generated by finitely presented objects.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.