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TTF classes generated by silting modules

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Pith's one-line read 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.

desk verdict The trace-ideal criterion is real and worth knowing, but the proof of Theorem 18 as printed has a domain mismatch that needs fixing. read the letter →

arxiv 2411.17581 v1 pith:R2CHXUUV submitted 2024-11-26 math.RA

Alejandro Argudin-Monroy , Daniel Bravo , Carlos E. Parra This is my paper · ORCID
classification math.RA MSC 16D9016S9016E35
keywords siltingmoduletraceidealTTFtripleidempotenttorsionpairprojectivecovercategorysemiperfectring
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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).

Reading between the lines

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

  • 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.
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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

1 major / 4 minor

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.

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 (1)
  1. [Section 4, Theorem 18 and Setting 16] 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.
minor comments (4)
  1. [Abstract] The phrase 'module cat egories' contains a spacing typo and should read 'module categories'.
  2. [Section 2.7] The word 'recentely' should be 'recently'.
  3. [Remark 6] The word 'tht' should be 'that'.
  4. [Corollary 23] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the trace-ideal sufficiency theorem is proved by an explicit projective presentation, and the self-citations used in an application are not load-bearing for the main results.

full rationale

The main derivation chain is self-contained. Theorem 18 constructs an explicit projective presentation rho from the trace data I = Tr(P) (Setting 16) and proves D_rho = Gen(R/I) by direct Hom-matrix computations: for M in T_I, the fact that P is generated by I forces all maps P -> M (and P^(P*) -> M) to vanish, so every map P oplus P oplus R -> M factors through rho; conversely, for M in D_rho, the zero middle column of rho forces an arbitrary morphism b: P -> M to be zero, so M lies in P^bot0 = T_I. Theorem A is then an immediate corollary, and Theorem B derives the trace property from the silting conditions via Proposition 14 using projective-cover decompositions; neither theorem assumes the conclusion it proves. The self-citations [7] and [22] appear only in Corollary 23 and some examples, where they supply published characterizations of tCG torsion pairs and a lemma about projective covers; these results are not assumptions of Theorem A or Theorem B and do not smuggle the target conclusion into the proof. The skeptic's domain-mismatch point about Theorem 18 is a notational/typographical issue in the displayed domain of rho, not a circularity: the intended construction with P^(P*) in the appropriate column makes the argument checkable. Question 25 is an explicit limitation, not a circular step.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new free parameters or invented entities; it is pure mathematics. The central results rest on standard assumptions: R is an associative ring with unit; I is idempotent; the usual homological toolbox (Snake Lemma, Ext, projectivity, projective covers) is assumed. The paper also imports established theorems: the Jans correspondence between idempotent ideals and TTF triples, the Psaroudakis-Vitória recollement correspondence, and, in Corollary 23, results from the authors' own earlier papers [7] and [22]. These imported theorems are load-bearing for the equivalences in Corollary 23 but not for the main Theorems A and B, which are proved in the text.

assumptions (5)
  • domain assumption R is an associative ring with unit and R-Mod is the category of left R-modules; all statements also hold for right modules.
    Section 2.1 fixes this throughout.
  • domain assumption I is an idempotent two-sided ideal of R.
    The whole paper studies the TTF triple associated to such I, via the correspondence in Section 2.6.
  • standard math The bijection between TTF triples and idempotent ideals (Jans), and between recollements and TTF triples (Psaroudakis-Vitória), is used as a black box.
    Section 2.6 and Proposition 3 rely on [13] and [27].
  • standard math Well-known homological facts are used: Snake Lemma, properties of Ext, projectivity, projective covers, traces of projective modules.
    Used throughout Sections 3 and 4.
  • standard math Corollary 23 uses [7, Corollary 3.9] (by the same authors) and [22, Lemma 7.3] as black boxes.
    These are cited theorems, not proved in this paper; they are load-bearing for the tCG equivalences but not for Theorems A and B.

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Pith. "Pith review of TTF classes generated by silting modules." pith.science (2026). https://pith.science/paper/R2CHXUUV

@misc{pith2026241117581,
  author       = {Pith},
  title        = {Pith review of: TTF classes generated by silting modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R2CHXUUV}},
  note         = {Machine review of arXiv:2411.17581}
}
abstract

We study the conditions under which a TTF class in a module category over a ring is silting. Using the correspondence between idempotent ideals over a ring and TTF classes in the module category, we focus on finding the necessary and sufficient conditions for $R/I$ to be a silting $R$-module, and hence for the TTF class $\mathbf{Gen}(R/I)$ to be silting, where $I$ is an idempotent two-sided ideal of $R$. In our main result, we show that $R/I$ is a silting module whenever $I$ is the trace of a projective $R$-module. Furthermore, we demonstrate that the converse holds for a broad class of rings, including semiperfect rings.

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