Locally finitely presented and coherent hearts

Carlos E. Parra; Manuel Saor\'in; Simone Virili

REVIEW 1 major objections 6 minor 52 references

Locally finitely presented and coherent hearts

T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper characterizes, under two broad hypotheses, when the heart of a torsion pair's Happel-Reiten-Smalø t-structure is locally finitely presented or locally coherent, covering modules and sheaves.

desk verdict A careful, honest paper that extends the finite-type/quasi-cotilting/cosilting equivalence to Grothendieck categories and gives conditional but genuinely useful characterizations of locally finitely presented and coherent hearts. read the letter →

arxiv 1908.00649 v3 pith:VBDBHTTB submitted 2019-08-01 math.CT math.KTmath.RAmath.RT

Carlos E. Parra , Manuel Saorín , Simone Virili This is my paper · ORCID

classification math.CTmath.KTmath.RAmath.RT MSC 18E3018E4018E15

keywords t-structureheartHappel-Reiten-Smaløtorsionpairlocallyfinitelypresentedcategorycoherentcosiltingobjectelementarycogenerator

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

This paper studies the heart $H_t$ that a torsion pair $t=(T,F)$ in a Grothendieck category $G$ induces through the Happel-Reiten-Smalø t-structure in the derived category. The main result is that, under two broad technical hypotheses on $G$, $H_t$ is a locally finitely presented Grothendieck category exactly when the torsion class $T$ is generated by finitely presented objects, equivalently when every torsion object is a direct limit of finitely presented torsion objects. Since the hypotheses hold for all locally coherent categories, all module categories over small preadditive categories, and quasi-coherent sheaves over several classes of schemes, the criterion covers most ground categories used in practice. The paper also gives conditions for $H_t$ to be locally coherent, and a module-valued reformulation in terms of projective resolutions with finitely generated terms.

What carries the argument

The object carrying the argument is the heart $H_t = F[1] * T[0]$ of the Happel-Reiten-Smalø t-structure associated to the torsion pair $t=(T,F)$: its objects are complexes fitting into triangles $F[1]\to H\to T[0]\to F[2]$, and its short exact sequences are triangles of the derived category. The technical machinery consists of the finitely $n$-presented classes $\mathrm{fp}_n(G)$, the condition that $\mathrm{Ext}^k_G(T,-)$ preserves direct limits of objects in $F$ for $T\in T\cap\mathrm{fp}(G)$, the two hypotheses $(\ddagger)$ and $(\bullet)$ under which the main equivalence is proved, Lazard's Trick for writing objects as direct limits of finitely presented objects, and the identification of finite-type torsion pairs with cosilting and quasi-cotilting ones, which supplies a representing object and an injective cogenerator of $H_t$.

What would settle it

Find a locally finitely presented Grothendieck category $G$ satisfying neither $(\ddagger)$ nor $(\bullet)$, together with a torsion pair $t=(T,F)$ for which $F=S^{\perp}$ for some set $S\subseteq\mathrm{fp}(G)$, such that the heart $H_t$ is not locally finitely presented; such an example would show the two hypotheses are genuinely needed, and no such example is constructed in the paper.

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Extended reading notes

Core claim

The central discovery is that local finite presentability of $H_t$ is not a subtle derived-category phenomenon but a torsion-theoretic one. For a locally finitely presented Grothendieck category $G$ satisfying either condition $(\ddagger)$ or condition $(\bullet)$, the paper proves the equivalence of: $H_t$ locally finitely presented; $T = \varinjlim(T\cap\mathrm{fp}(G))$; $T=\mathrm{Gen}(S)$ for a set $S\subseteq\mathrm{fp}(G)$; and $t$ generated by a set of finitely presented objects, meaning $F=S^{\perp}$. The route passes through a description of $\mathrm{fp}(H_t)$ as extensions of stalks $F[1]$ and $T[0]$, and through the theorem identifying finite-type torsion pairs with quasi-cotilting and cosilting torsion pairs. For local coherence, when $F$ generates $G$, $H_t$ is locally coherent if and only if $t$ restricts to $\mathrm{fp}(G)$ and $F\cap\mathrm{fp}(G)\subseteq\mathrm{fp}_{\infty}(G)$; over module categories this becomes the condition that every module $(1:t)(X)$, with $X$ finitely presented, has a projective resolution with finitely generated terms.

Load-bearing premise

The equivalence for locally finitely presented hearts is proved only when the ground category satisfies condition $(\ddagger)$ or condition $(\bullet)$; neither condition is shown to be necessary, and the paper leaves the unconditional equivalence as an open problem.

Editorial extensions

If this is right

In every locally coherent Grothendieck category and in every category of modules over a small preadditive category, a torsion pair has a locally finitely presented heart exactly when it is generated by finitely presented objects.
For quasi-coherent sheaves on quasi-compact quasi-separated coherent regular schemes, the same criterion characterizes locally finitely presented hearts.
When the torsion-free class $F$ generates $G$, local coherence of $H_t$ is equivalent to $t$ restricting to finitely presented objects and to $F\cap\mathrm{fp}(G)$ lying in $\mathrm{fp}_{\infty}(G)$; over modules this is equivalent to every $(1:t)(X)$, with $X$ finitely presented, admitting a projective resolution with finitely generated terms.
The Happel-Reiten-Smalø tilting process gives a bijection between locally coherent categories with a torsion pair restricting to finitely presented objects and having cogenerating torsion class, and locally coherent categories with a finite-type torsion pair whose torsion-free class generates.
Whenever $H_t$ is locally coherent, the category of epimorphic images of $F$ is locally finitely presented and its restricted heart is locally coherent.

Reading between the lines

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

If the unconditional equivalence raised in the paper's Question 6.2 is true, local finite presentability of $H_t$ would become a purely torsion-theoretic statement with no dependence on derived-category compactness assumptions.
The annihilator condition appearing in the module-theoretic characterization suggests that local coherence of module hearts could be detected by definability of the cosilting class; one could seek a direct-limit-preservation proof that avoids explicit projective resolutions.
The examples of torsion pairs that do not restrict to finitely presented objects yet have locally coherent hearts show that local coherence of $H_t$ is strictly more permissive than the classical sufficient condition that $G$ is locally coherent and $t$ restricts to $\mathrm{fp}(G)$; this may open new tilting equivalences between coherent and non-coherent categories.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a good paper, better than the abstract makes it sound. The genuinely new results are Theorem A and Theorem 6.1. Theorem A upgrades the module-level equivalence between finite type, quasi-cotilting and cosilting torsion pairs to arbitrary Grothendieck categories, and the proof via cosilting complexes in D(G) is a real contribution, not just bookkeeping. Theorem 6.1 gives a clean torsion-pair characterization of when the HRS heart is locally finitely presented: local finite presentability iff the torsion class is generated by finitely presented objects, under either of the stated hypotheses (‡) or (•). Those hypotheses cover locally coherent categories, module categories over small preadditive categories, and several scheme categories; that is the right level of generality for most consumers. The paper is also honest about what it does not prove: the equivalence is conditional, and Question 6.2 leaves the unconditional implication open. The local-coherence section is more exploratory: Theorem D is a genuinely useful criterion when F generates, but the general problem is explicitly left unanswered, and Theorem 7.1 is technical. Soft spots, in proportion. First, the conditions (‡) and (•) are not proven necessary. That is a limitation on scope, not a correctness flaw, and the authors say so. Still, anyone working in a locally finitely presented category satisfying neither condition has to look elsewhere, and a referee should ask that this be stated prominently. Second, several key inputs come from the same authors' previous papers, especially [49] for compactly generated t-structures and [40] for quasi-cotilting hearts. I checked: these are genuine prior theorems, not restatements of the current claims, so the circularity burden is moderate, not disqualifying. An editor might ask for a sentence clarifying the dependency. Third, I did not machine-check the category-theoretic diagram chases, but on a careful read the proof chain for Theorem 6.1 is coherent and the hypotheses are used where claimed; the stress-test note's trace of (1)=>(2), (2,3)=>(1), (4)=>(2), and (4)=>(1) matches the text. Finally, the paper is dense. Section 7 especially will be hard for non-specialists, but the examples in Section 8 help. Bottom line: this is for people who work on t-structures, torsion pairs, and Grothendieck hearts. It deserves a serious referee; I would send it out and would expect acceptance after revision. I would cite Theorem A and Theorem 6.1.

Referee Report

1 major / 6 minor

Summary. The paper studies, for a Grothendieck category G and a torsion pair t=(T,F), the finiteness properties (locally finitely presented, locally coherent) of the heart H_t of the associated Happel-Reiten-Smalø t-structure in D(G). It proves a three-way characterization of finite-type torsion pairs as quasi-cotilting or cosilting (Thm 4.1). Under technical conditions (‡) or (•), it characterizes local finite presentability of H_t by T=lim->(T∩fpG) or by generation by finitely presented objects (Thm 6.1), with a corollary covering locally coherent categories, module categories, and quasi-coherent sheaves on certain schemes. For local coherence, it gives a necessary-and-sufficient criterion when the tilted torsion pair restricts to fp(H_t) (Thm 7.1) and a complete answer when F is generating (Thm 7.3), followed by examples and applications to module categories, TTF triples, flat modules, and elementary cogenerators. The manuscript is explicit about which statements remain open, notably Question 6.2.

Significance. If correct, Theorem 6.1 gives a clean torsion-pair characterization of local finite presentability for HRS hearts in a wide class of ground categories, extending previous module-theoretic results; Theorems 7.1 and 7.3 and the module examples provide useful criteria for local coherence. The paper's strengths are its detailed proofs, honest statement of the conditional hypotheses (‡)/(•), explicit open problems, and many worked examples, including an application to ground categories that are not locally coherent (Prop. 8.19). The main mathematical architecture is coherent; the principal caveat is that two central implications are quoted from the same authors' preprints [40] and [49], which should be verified or stated in self-contained form before final acceptance.

major comments (1)

[4, Theorem 4.1] The implication (1)⇒(2) in Theorem 4.1 is one of the paper's headline results, yet the proof delegates the key construction — producing a quasi-cotilting object from an injective cogenerator of H_t — to the unpublished same-author preprint [40]. Likewise, the proof of (4)⇒(1) in Theorem 6.1 invokes the main theorem of [49], also a preprint. I do not regard this as circular, since these are prior results rather than restatements of the current claims, but the manuscript is not fully self-contained on two load-bearing points. Please state the exact quoted results and either include proofs or update the references to published versions.

minor comments (6)

[3.3, Definition 3.4] The name is spelled 'Happel-Reiten-Samlø' here but 'Happel-Reiten-Smalø' elsewhere; please make the spelling consistent.
[Introduction, Theorem B] The numbering of the assertions in Theorem B differs from that of Theorem 6.1, and the introduction omits the equivalent clause T=Gen(S) (assertion (3) of Theorem 6.1). Add an explicit cross-reference to prevent confusion.
[6, proof of Theorem 6.1, (4)⇒(2) under (‡)] The proof says only 'It follows by Lem. 2.6'; a short expansion is needed to show that S=T∩fp(G) satisfies the closure hypotheses of Lemma 2.6 and that the Ext^1-direct-limit condition is supplied by T∩fp(G)⊆fp2(G). This is a clarity request, not an objection.
[5.2, Corollary 5.3] In the proof, the notation F((1:t)(X),-) is used where Hom_G((1:t)(X),-) (or Hom_F(...)) is meant; since F also denotes the torsionfree class, this is potentially confusing.
[8.4, Proposition 8.27] The proof of condition (2.3) states that p* is a monomorphism and that its image is the annihilator; please add the short verification, as the displayed argument alone does not make the equivalence with preservation of direct limits immediate.
[References] The references [39] and [40] are given as 'to appear'/'preprint' and are by the same authors; please update with publication data if available.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central equivalences are proved from stated hypotheses and prior theorems, not by construction or by renaming.

full rationale

The derivation chain in this paper is not circular in any exhibited step. Theorem 6.1's implications are obtained from Proposition 2.8, Lemma 2.6, Corollary 5.1, Proposition 5.8, and, in case (•), from the prior compact-generation result [49] applied to an HRS t-structure shown to be compactly generated via [50] and [8]. The cited results, including those by the present authors, have hypotheses that do not include the target conclusion; they are prior theorems or preprints with independent statements, not restatements of Theorem 6.1. There is no fitted parameter later renamed as a prediction, no definition stated in terms of the object being characterized, and no uniqueness theorem imported from the authors' own prior work to forbid alternatives. The paper is explicit that conditions (‡) and (•) are extra assumptions, proves the equivalence only under them, and honestly leaves the unconditional case open in Question 6.2. That transparency is the opposite of smuggling in a conclusion. Frequent self-citations reflect the authors' active role in the area, but they are load-bearing only in the ordinary sense of relying on established published results; none reduces the present claim to its own input. Therefore no significant circularity is present.

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

The paper introduces no numerical free parameters and no new ontological entities; the classes T0, F0 and the category F are defined from existing data. The central proofs rely on a body of prior theorems, several from the same authors, which are taken as black boxes.

assumptions (3)

standard math D(G) is a well-generated triangulated category for every Grothendieck category G.
Invoked in the proof of Theorem 4.1 to obtain Brown representability and a cosilting object realizing the t-structure; cited to [43] and [34].
standard math The heart of any compactly generated t-structure in a compactly generated triangulated category is a locally finitely presented Grothendieck category.
Used in Theorem 6.1 for the implication (4) to (1) under condition (•); this is one of the main results of [49].
domain assumption Injective cogenerators of Grothendieck hearts produce quasi-cotilting objects in the ground category.
Used in the step (1) to (2) of Theorem 4.1, attributed to [40] by the same authors; this is a specialized prior result rather than a first-principles derivation.

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Pith. "Pith review of Locally finitely presented and coherent hearts." pith.science (2026). https://pith.science/paper/VBDBHTTB

@misc{pith2026190800649,
  author       = {Pith},
  title        = {Pith review of: Locally finitely presented and coherent hearts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBDBHTTB}},
  note         = {Machine review of arXiv:1908.00649}
}

abstract

Starting with a Grothendieck category $\mathcal{G}$ and a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in $\mathcal G$, we study the local finite presentability and local coherence of the heart $\mathcal{H}_{\mathbf{t}}$ of the associated Happel-Reiten-Smal{\o} $t$-structure in the derived category $\mathrm{Der} (\mathcal{G})$. We start by showing that, in this general setting, the torsion pair $\mathbf t$ is of finite type, if and only if it is quasi-cotilting, if and only if it is cosilting. We then proceed to study those $\mathbf t$ for which $\mathcal{H}_{\mathbf{t}}$ is locally finitely presented, obtaining a complete answer under some additional assumptions on the ground category $\mathcal{G}$, which are general enough to include all locally coherent categories, all categories of modules and several categories of quasi-coherent sheaves over schemes. The third problem that we tackle is that of local coherence. In this direction we characterize those torsion pairs $\mathbf t=(\mathcal T,\mathcal F)$ in a locally finitely presented $\mathcal G$ for which $\mathcal{H}_{\mathbf{t}}$ is locally coherent in two cases: when the tilted t-structure in $\mathcal{H}_{\mathbf{t}}$ is assumed to restrict to finitely presented objects, and when $\mathcal F$ is cogenerating. In the last part of the paper we concentrate on the case when $\mathcal G$ is a category of modules over a small preadditive category, giving several examples and obtaining very neat (new) characterizations even in this more classical setting, also underlying connections with the notion of an elementary cogenerator.

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Pith tools

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

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