{"id":"be77fd86-f1e0-4c0e-a738-7a91f76a6c6a","arxiv_id":"1908.00649","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"Starting from a torsion pair in a Grothendieck category, this paper determines when the associated heart category is locally finitely presented or locally coherent. It gives clean characterizations in terms of the torsion pair being generated by finitely presented objects, plus concrete module-theoretic criteria.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 6.1 is internally coherent; the only limitation is the explicitly conditional scope, which the paper states honestly.","rationale":"The reader's weakest assumption is that Theorem 6.1 is only proved under (‡) or (•), with the unconditional case left open. That is accurate and is precisely the paper's own stated limitation. However, I do not regard this as a load-bearing objection to the theorem as stated, because the theorem is explicitly conditional and the paper does not claim more. My own review of the proof found no internal inconsistency: each implication in Theorem 6.1 is supported by the cited lemmas and by the surrounding developments in Sections 4 and 5. In particular, the proof of (2,3)=>(1) under (†) is a genuine argument showing that every object of the heart is a direct limit of finitely presented objects, using the Ext^2-lifting hypothesis; the proof of (4)=>(2) under (‡) uses Lemma 2.6 in a way consistent with the closure properties of S; and the proof of (4)=>(1) under (•) relies on the established theorem of Saorin-Stovicek that hearts of compactly generated t-structures are locally finitely presented Grothendieck categories. The only mild concern is that key inputs, especially [49] and [40], come from closely related papers by the same authors and were not machine-checked; but this is a verification limitation rather than a detected error. I would keep the ACCEPT verdict unchanged.","tokens_in":61885,"tokens_out":20361,"duration_ms":216708,"concrete_test":"Re-verify the compact-generation step used in the (•) case of Theorem 6.1: take a set S of finitely presented generators of G that are compact in D(G), and prove directly that the HRS t-structure's co-aisle equals S[0]^{⊥≤0} and that D(G) is compactly generated by S[0]. If this verification fails for some G satisfying (•), then the implication (4)=>(1) would lose its stated support.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central equivalence in Theorem 6.1 is proved under explicitly stated hypotheses (‡) and (•), and the paper is transparent that the unconditional version remains open (Question 6.2). I traced the main proof chain and found no load-bearing gap: (1)=>(2) uses Coro. 5.1 together with [36, Thm. 4.8(4)]; (2,3)=>(1) under (†) uses Prop. 5.8 and a careful lifting of an Ext^2 class via condition (†); (4)=>(2) under (‡) follows from Lemma 2.6 applied to S = T∩fp(G), where the hypothesis T∩fp(G)⊆fp2(G) supplies the needed Ext^1-direct-limit preservation; (4)=>(1) under (•) reduces to compact generation of the HRS t-structure and the theorem of [49]. The conditional nature of the hypotheses is a limitation on generality, not a correctness flaw, and the paper explicitly flags it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":62066,"tokens_out":10646,"duration_ms":105894,"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":[{"comment":"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.","section":"4, Theorem 4.1"}],"minor_comments":[{"comment":"The name is spelled 'Happel-Reiten-Samlø' here but 'Happel-Reiten-Smalø' elsewhere; please make the spelling consistent.","section":"3.3, Definition 3.4"},{"comment":"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.","section":"Introduction, Theorem B"},{"comment":"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.","section":"6, proof of Theorem 6.1, (4)⇒(2) under (‡)"},{"comment":"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.","section":"5.2, Corollary 5.3"},{"comment":"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.","section":"8.4, Proposition 8.27"},{"comment":"The references [39] and [40] are given as 'to appear'/'preprint' and are by the same authors; please update with publication data if available.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"I would ask the editor to verify the publication status of [40] and, more generally, the extent to which Theorem 4.1 depends on that same-author preprint. The rest of the paper appears internally coherent, and the conditional scope of the main theorem is stated honestly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.","headline":"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.","tokens_in":815,"tokens_out":1016,"would_cite":true,"duration_ms":30683,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18E30","18E40","18E15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["t-structure","heart","Happel-Reiten-Smalø t-structure","torsion pair","locally finitely presented category","locally coherent category","cosilting object","elementary cogenerator"],"falsifier":"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.","tokens_in":61692,"feed_emoji":"📐","tokens_out":12240,"duration_ms":111406,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hearts are locally finitely presented iff torsion is f.p.-generated","feed_subtitle":"Complete for modules, locally coherent categories, and common sheaf categories; open in general.","key_machinery":"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 $H_t$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces t-structures and hearts, defining the truncation and cohomological formalism in which the whole paper works.","marker":"[6]"},{"why":"Introduces the Happel-Reiten-Smalø tilt that turns a torsion pair into a t-structure whose heart is studied.","marker":"[23]"},{"why":"Proves that the heart is Grothendieck if and only if the torsion pair is of finite type, the starting point for the local finiteness questions.","marker":"[37]"},{"why":"Describes direct limits in the heart for torsion pairs and supplies the limit computations used throughout the paper.","marker":"[36]"},{"why":"Provides the injective cogenerator of the heart and the quasi-cotilting object associated to a finite-type torsion pair.","marker":"[39]"},{"why":"States that hearts of compactly generated t-structures are locally finitely presented, which drives the implications under condition $(\\bullet)$.","marker":"[49]"},{"why":"Contains prior work on locally coherent hearts in locally coherent categories with restricting torsion pairs, which the paper extends.","marker":"[48]"},{"why":"Develops torsion pairs and trace ideals for module categories over small preadditive categories, used in the module-theoretic characterizations.","marker":"[41]"},{"why":"Lazard's Trick is used repeatedly to express objects as direct limits of finitely presented objects in key proofs.","marker":"[30]"}],"fun_headline_variants":["Torsion controls local finite presentability of hearts","Hearts locally finitely presented iff torsion is f.p.-generated","Cosilting torsion pairs and the local finite structure of hearts","Coherent hearts characterized via torsion restrictions","Torsion-theoretic criteria for heart presentability and coherence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Torsion controls local finite presentability of hearts","Hearts locally finitely presented iff torsion is f.p.-generated","Cosilting torsion pairs and the local finite structure of hearts","Coherent hearts characterized via torsion restrictions","Torsion-theoretic criteria for heart presentability and coherence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001111,"raw_usage":{"total_tokens":4722,"prompt_tokens":1129,"completion_tokens":3593,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":745,"completion_tokens_details":{"reasoning_tokens":3512}},"tokens_in":745,"tokens_out":3593,"duration_ms":26017,"temperature":1.0,"reasoning_tokens":3512,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:40:18.048348+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":": Faisceaux pervers","cited_arxiv_id":null,"evidence_quote":"Introduces t-structures and hearts, defining the truncation and cohomological formalism in which the whole paper works."},{"cited_title":": Tilting in Abelian categories and quasitilted algebras","cited_arxiv_id":null,"evidence_quote":"Introduces the Happel-Reiten-Smalø tilt that turns a torsion pair into a t-structure whose heart is studied."},{"cited_title":"Direct limits in the heart of a t-structure: the case of a torsion pair","cited_arxiv_id":null,"evidence_quote":"Proves that the heart is Grothendieck if and only if the torsion pair is of finite type, the starting point for the local finiteness questions."},{"cited_title":": Direct limits in the heart of a t-structure: the case of a torsion pair","cited_arxiv_id":null,"evidence_quote":"Describes direct limits in the heart for torsion pairs and supplies the limit computations used throughout the paper."},{"cited_title":": The HRS tilting process and Grothendieck hearts of t-struc tures","cited_arxiv_id":null,"evidence_quote":"Provides the injective cogenerator of the heart and the quasi-cotilting object associated to a finite-type torsion pair."},{"cited_title":": Locally coherent hearts","cited_arxiv_id":null,"evidence_quote":"Contains prior work on locally coherent hearts in locally coherent categories with restricting torsion pairs, which the paper extends."},{"cited_title":": Autour de la platitude","cited_arxiv_id":null,"evidence_quote":"Lazard's Trick is used repeatedly to express objects as direct limits of finitely presented objects in key proofs."}],"review_version":1}