{"id":"197a191a-131f-4ae4-b4ee-7bd7025c817d","arxiv_id":"2411.17581","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An idempotent ideal that is the trace of a projective module yields a silting module R/I and a silting TTF class; under a projective cover assumption, the converse holds.","lead":"This paper finds when a certain class of modules, built from a ring modulo an idempotent ideal, is a 'silting' class, a property with applications in tilting theory and abelian categories. The authors show that the ideal being the trace of a projective module is the key condition, and prove a converse when the quotient has a projective cover.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 18's displayed rho has a domain mismatch: Setting 16 defines sigma on P^{(P*)}, but the proof treats sigma as a map P -> R, so the proof of Theorem A is not checkable as written; the intended construction works after replacing P by P^{(P*)}.","rationale":"The reader's weakest_assumption is the projective cover hypothesis in the converse Theorem B, which is a legitimate limitation but does not affect the central sufficient condition Theorem A. My stress-test pass identified a different, more immediate issue: the statement of Theorem 18, which supplies the proof of Theorem A, is not well-formed as written because the module in the domain of rho is ambiguous. Setting 16 defines sigma on P^{(P*)}, while Theorem 18 and its proof use rho on P oplus P oplus R and write sigma : P -> R. For an arbitrary projective P with trace I, no single morphism P -> R with image I need exist, so the proof cannot be checked literally. However, replacing the domain by P^{(P*)} oplus P^{(P*)} oplus R makes the argument valid: the two inclusions D_rho = T_I follow by the same reasoning, relying on Hom(P^{(P*)}, M) vanishing for M in T_I and being forced to vanish for M in D_rho. Therefore the mathematical claim of Theorem A is supported, but the printed text requires a correction. Since this is a minor notational error rather than a mathematical gap, the reader's ACCEPT verdict remains appropriate; the paper should be accepted with a requested edit to Theorem 18.","tokens_in":13111,"tokens_out":24386,"duration_ms":202333,"concrete_test":"Re-express Theorem 18 with P' := P^{(P*)} and rho : P' oplus P' oplus R -> R oplus R given by [sigma 0 0; 0 0 1], where sigma is the evaluation map from Setting 16(d). Verify directly that (i) Coker(rho) is isomorphic to R/I, (ii) every M in T_I lies in D_rho because all morphisms P' -> M are zero, and (iii) every M in D_rho satisfies Hom(P', M) = 0 because the middle component of [0 b 0] must factor as [f g] o rho, forcing b = 0. If (i)-(iii) hold, the concern is a typographical issue; if any step fails, Theorem A needs substantive revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the main sufficient condition rests on Theorem 18, but that theorem is internally inconsistent as printed. Setting 16(d) defines P_{-1} := P^{(P*)} and sigma := iota_sigma o pi_sigma with domain P^{(P*)}. Theorem 18 then asserts that rho is the morphism [sigma 0 0; 0 0 1] : P oplus P oplus R -> R oplus R, and the proof refers to 'sigma : P -> R'. For a general projective P whose trace is I, there need not exist a single morphism P -> R with image I (for instance, if P is a direct sum of projectives with different trace ideals). Hence the displayed rho is not a well-defined morphism unless the 'P' in the statement is read as P_{-1} = P^{(P*)}. The subsequent verification that D_rho = T_I does go through with the domain P^{(P*)} oplus P^{(P*)} oplus R: for M in T_I, all morphisms P^{(P*)} -> M vanish, and for M in D_rho, the middle component of [0 b 0] forces b = 0, giving Hom(P^{(P*)}, M) = 0, hence M in P^{bot 0} = T_I. Thus the intended theorem is sound, but the printed proof cannot be checked without this correction.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies when the TTF class Gen(R/I) associated with an idempotent two-sided ideal I is silting, i.e., when R/I is a silting module. The authors prove (Theorem A) that if I is the trace of a projective module, then R/I is silting, via an explicit projective presentation (Theorem 18). They further prove a partial converse (Theorem B): if R/I admits a projective cover and is silting, then I is a trace ideal. Section 3 develops a characterization (Proposition 5) of when R/I with a given projective presentation is silting, in terms of the HRS torsion pair and recollement data; Section 4 also gives applications to semiperfect rings (Corollary 23) and an example showing that for a semiperfect ring with nonzero idempotent Jacobson radical, R/J is not silting. The paper is carefully written and the main arguments are detailed, but the proof of the central sufficient condition contains a domain mismatch that must be fixed.","tokens_in":40,"tokens_out":20654,"duration_ms":228033,"significance":"The results give a clean sufficient condition for Gen(R/I) to be silting and a converse under a projective-cover hypothesis, thereby clarifying the relationship between trace ideals and silting modules. The explicit construction in Theorem 18 is elegant and should be useful in examples; the applications to semiperfect rings and to the ring of continuous functions are instructive. The authors are careful to identify the open general converse (Question 25). The main theorems are proved from standard material, and the proofs of Propositions 5 and 14 are largely verifiable. However, because Theorem A relies on Theorem 18, the domain error in Theorem 18 affects the checkability of the paper's headline sufficient condition.","major_comments":[{"comment":"The displayed morphism ρ = [σ 0 0; 0 0 1] : P ⊕ P ⊕ R → R ⊕ R is not well-defined from Setting 16. In Setting 16(d), σ is defined as ισ ∘ πσ, where πσ: P^{(P*)} → I and ισ: I → R, so σ has domain P^{(P*)} and not P. For a general projective P with I = Tr(P), there need not be a single morphism P → R whose image is I, since I is generated by the images of all elements of P*; consequently the proof's first sentence 'Consider the morphisms σ : P → R' is unjustified. The intended theorem is sound: after replacing the first summand in the domain of ρ by P^{(P*)}, the image of ρ is I ⊕ R, so Coker(ρ) = R/I, and the verification that Dρ = T_I goes through unchanged. As printed, however, the statement and proof of Theorem A rest on an ill-defined morphism and are not checkable.","section":"Section 4, Theorem 18 and Setting 16"}],"minor_comments":[{"comment":"The phrase 'module cat egories' contains a spacing typo and should read 'module categories'.","section":"Abstract"},{"comment":"The word 'recentely' should be 'recently'.","section":"Section 2.7"},{"comment":"The word 'tht' should be 'that'.","section":"Remark 6"},{"comment":"The proof begins with 'Let us assume that I ≠ 0'; the case I = 0 is not discussed, though all equivalences are then trivial and should be stated or omitted.","section":"Corollary 23"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and is a genuine contribution to silting theory. The domain error in Theorem 18 is local and easily corrected; the intended result appears correct. I recommend major revision to fix this before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one. The main result is a clean characterization: for a two-sided idempotent ideal I, R/I is silting whenever I is the trace of a projective module, and the converse holds when R/I has a projective cover. That is a genuinely useful criterion connecting trace ideals, TTF triples, and silting theory. Theorem A is new relative to the cited literature, and the semiperfect ring equivalence in Corollary 23 packages it nicely. The paper does a lot of things right: the proofs are detailed, and I checked the Snake Lemma step in Proposition 5 and the matrix argument in Proposition 14; both look correct. The authors are also honest about the main limitation, leaving the projective-cover question open for general rings.\n\nThe soft spot is in Theorem 18, and it is not cosmetic. Setting 16 defines sigma on P_{-1} = P^{(P*)}, but the statement of Theorem 18 writes rho as a map from P ⊕ P ⊕ R to R ⊕ R, and the proof refers to a morphism 'sigma : P → R'. For a general projective P, there is no single morphism P → R with image Tr(P); the trace is a sum of images of all maps P → R. So the displayed rho is not a well-defined morphism as printed. The intended theorem is recoverable: replace the two copies of P by P^{(P*)}, let sigma be the evaluation map, and the proof then goes through. Without that replacement, the middle-component argument cannot force M ∈ P^{⊥0}. This is a fixable notation problem, not a collapsed proof, but it means the main sufficiency theorem is not checkable as written.\n\nMinor point: Proposition 14 relies on Lemma 1, which is labelled folklore, and the projective-cover assumption is essential for the converse. The authors flag this themselves, so it is a known scope restriction, not a hidden flaw.\n\nBottom line: this is a solid paper for silting theory and module theory readers. The main theorems are original, most of the arguments are careful, and the defect is localized. Send it to peer review; with Theorem 18 restated in terms of P^{(P*)}, I would accept it.","headline":"The trace-ideal criterion is real and worth knowing, but the proof of Theorem 18 as printed has a domain mismatch that needs fixing.","tokens_in":13953,"tokens_out":3273,"would_cite":true,"duration_ms":46211,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16D90","16S90","16E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that idempotent two-sided ideals that are traces of projective modules give silting quotient modules, with a converse under a projective-cover hypothesis.","keywords":["silting module","trace ideal","TTF triple","idempotent ideal","torsion pair","projective cover","module category","semiperfect ring"],"falsifier":"Find a ring $R$ and an idempotent two-sided ideal $I$ such that $\\mathrm{Gen}(R/I)$ is a silting torsion class but $I$ is not the trace of any projective module and $R/I$ has no projective cover; such a pair would show that the converse fails as soon as the projective-cover hypothesis is dropped.","tokens_in":12861,"feed_emoji":"🧮","tokens_out":8763,"duration_ms":71444,"temperature":0.7,"pith_summary":"Over any associative ring, idempotent two-sided ideals correspond exactly to torsion-torsion-free (TTF) triples in the module category: each ideal $I$ gives the torsion class $\\mathrm{Gen}(R/I)$. This paper asks when that class is silting, meaning it is generated by a module with a projective presentation that behaves like a tilting module. The main result proves that if $I$ is the trace of a projective module, then $R/I$ itself is silting, so $\\mathrm{Gen}(R/I)$ is a silting torsion class. Under the additional assumption that $R/I$ has a projective cover, the converse also holds, yielding a complete equivalence for semiperfect rings. This matters because silting torsion classes are exactly the torsion classes for which the associated abelian category has a projective generator; the paper identifies a broad family of such classes by one ideal-theoretic condition.","feed_headline":"Trace ideals make their quotient a silting module","feed_subtitle":"When an idempotent ideal is a trace, R/I generates a silting torsion class; a converse holds with projective covers.","key_machinery":"The load-bearing construction is a particular silting projective presentation of $R/I$ made from a projective module $P$ whose trace is $I$. A silting module is a module $S$ with a projective presentation $P_{-1}\\xrightarrow{\\sigma} P_0 \\twoheadrightarrow S$ such that the class $D_\\sigma = \\{X : \\mathrm{Hom}_R(\\sigma, X)\\text{ is surjective}\\}$ equals $\\mathrm{Gen}(S)$. For an idempotent ideal $I$, the class $\\mathrm{Gen}(R/I)$ is always a torsion class closed under products and subobjects, so it forms the torsion part of a TTF triple; the question is when this class is silting. Theorem 18 shows that when $I=\\mathrm{Tr}(P)$, the morphism $\\rho = \\begin{pmatrix} \\sigma & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} : P\\oplus P\\oplus R \\to R\\oplus R$ has cokernel $R/I$ and satisfies $D_\\rho = \\mathrm{Gen}(R/I)$, which is exactly the silting condition. The surrounding results (Proposition 5, Proposition 14) reduce the silting condition to three structural statements about the TTF triple and to the trace property under projective covers.","core_discovery":"Let $I$ be an idempotent two-sided ideal of a ring $R$. The central claim is that the condition 'I is the trace of a projective module' is sufficient for $R/I$ to be a silting module. Concretely, if $P$ is projective and $I=\\mathrm{Tr}(P)$, Theorem 18 builds an explicit projective presentation $P\\oplus P\\oplus R \\to R\\oplus R \\to R/I\\to 0$ and proves that the associated class $D_\\rho$ of modules for which $\\mathrm{Hom}(\\rho, -)$ is surjective coincides with $\\mathrm{Gen}(R/I)$. Hence $\\mathrm{Gen}(R/I)$ is a silting TTF class. The converse, Theorem B, states that if $R/I$ admits a projective cover and is silting, then $I$ is the trace of a projective module. For semiperfect rings the two conditions become equivalent and also match the property that $(\\mathcal{C}_I,\\mathcal{T}_I)$ is a tCG torsion pair (a torsion pair whose associated t-structure is compactly generated).","pith_inferences":["If the projective-cover hypothesis in Theorem B could be removed, the condition 'I is a trace ideal' would completely characterize when quotient-generated TTF classes are silting; the paper leaves this as an open question (Question 25).","Theorem 18 is constructive, so for any trace ideal it yields an explicit silting presentation; this could make the silting module and its associated heart computable in examples beyond the ones treated in the paper.","The semiperfect-ring equivalence suggests that for finite-dimensional algebras over a field, classifying silting modules generated by quotients $R/I$ reduces to classifying trace ideals, which may connect to known classifications of two-sided idempotent ideals."],"forward_implications":["For semiperfect rings, $R/I$ is silting if and only if $I$ is the trace of a projective module, and this is also equivalent to $(\\mathcal{C}_I,\\mathcal{T}_I)$ being a tCG torsion pair (Corollary 23).","If $I$ is an idempotent ideal contained in the Jacobson radical, then $R/I$ is automatically silting (Corollary 22).","If $I$ is finitely generated as a right ideal, then $R/I$ is a silting left module (Corollary 20).","Over a semiperfect ring, a nonzero idempotent Jacobson radical never gives a silting quotient $R/J$, by the Nakayama lemma argument in Example 24.","Trace ideals of projective modules over rings such as the continuous functions on $[0,1]$ yield explicit silting modules and silting TTF classes (Example 19)."],"supporting_citations":[{"why":"Defines silting modules and the class $D_\\sigma=\\mathrm{Gen}(S)$ that the paper uses as its criterion; the whole question is whether $\\mathrm{Gen}(R/I)$ is silting in this sense.","marker":"[4]"},{"why":"Supplies Lemma 2.1, used to prove Proposition 5, and the motivating commutative-ring example where $R/I$ is not silting for an idempotent ideal.","marker":"[3]"},{"why":"Provides the torsion-class viewpoint and the result that inspires the explicit silting presentation built in Theorem 18.","marker":"[8]"},{"why":"Gives the bijection between TTF triples in $R$-Mod and idempotent two-sided ideals, which is the paper's basic correspondence.","marker":"[13]"},{"why":"Supplies the recollement formalism and the properties of the adjoint functors used in Proposition 3 and Proposition 5.","marker":"[27]"},{"why":"Proves that an idempotent ideal finitely generated as a right ideal is the trace of a projective left module, yielding Corollary 20.","marker":"[30]"},{"why":"Provides the lemma that projective covers of modules in $\\mathcal{C}_I$ again lie in $\\mathcal{C}_I$, used in Example 13(a) and Corollary 23.","marker":"[22]"},{"why":"Connects tCG torsion pairs with classes generated by modules in $\\mathcal{C}_I$, used in the semiperfect-ring equivalence of Corollary 23.","marker":"[7]"}],"fun_headline_variants":["Trace of a projective module forces silting R/I","Silting TTF classes from trace ideals of projectives","Semiperfect rings: silting exactly when trace ideal","Idempotent trace ideals give silting module quotients","When R/I is silting: projective trace suffices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The converse theorem assumes that the quotient module $R/I$ has a projective cover, a property that holds for semiperfect rings but is not automatic for arbitrary rings; if that assumption fails, the proof's argument that $I$ is a trace ideal does not go through.","fun_headline_variants_meta":{"raw":{"variants":["Trace of a projective module forces silting R/I","Silting TTF classes from trace ideals of projectives","Semiperfect rings: silting exactly when trace ideal","Idempotent trace ideals give silting module quotients","When R/I is silting: projective trace suffices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000408,"raw_usage":{"total_tokens":2095,"prompt_tokens":898,"completion_tokens":1197,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":1116}},"tokens_in":514,"tokens_out":1197,"duration_ms":11022,"temperature":1.0,"reasoning_tokens":1116,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:57:39.122254+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a ring $R$ and an idempotent two-sided ideal $I$ such that $\\mathrm{Gen}(R/I)$ is a silting torsion class but $I$ is not the trace of any projective module and $R/I$ has no projective cover; such a pair would show that the converse fails as soon as the projective-cover hypothesis is dropped.","supporting_citations":[{"cited_title":"Angeleri Hügel, F","cited_arxiv_id":null,"evidence_quote":"Defines silting modules and the class $D_\\sigma=\\mathrm{Gen}(S)$ that the paper uses as its criterion; the whole question is whether $\\mathrm{Gen}(R/I)$ is silting in this sense."},{"cited_title":"Angeleri Hügel and M","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 2.1, used to prove Proposition 5, and the motivating commutative-ring example where $R/I$ is not silting for an idempotent ideal."},{"cited_title":"Breaz and J","cited_arxiv_id":null,"evidence_quote":"Provides the torsion-class viewpoint and the result that inspires the explicit silting presentation built in Theorem 18."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the bijection between TTF triples in $R$-Mod and idempotent two-sided ideals, which is the paper's basic correspondence."},{"cited_title":"Psaroudakis and J","cited_arxiv_id":null,"evidence_quote":"Supplies the recollement formalism and the properties of the adjoint functors used in Proposition 3 and Proposition 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that an idempotent ideal finitely generated as a right ideal is the trace of a projective left module, yielding Corollary 20."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the lemma that projective covers of modules in $\\mathcal{C}_I$ again lie in $\\mathcal{C}_I$, used in Example 13(a) and Corollary 23."},{"cited_title":"Bravo and C","cited_arxiv_id":null,"evidence_quote":"Connects tCG torsion pairs with classes generated by modules in $\\mathcal{C}_I$, used in the semiperfect-ring equivalence of Corollary 23."}],"review_version":1}