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arxiv: 2503.20604 · v2 · submitted 2025-03-26 · 🧮 math.RT

Tilting objects in the extended heart of a t-structure

Alejandro Argudin Monroy , Octavio Mendoza , Carlos E. Parra This is my paper

Pith reviewed 2026-05-22 22:31 UTC · model grok-4.3

classification 🧮 math.RT
keywords extended tilting objectsextriangulated categoriest-structuresheartsquasi-tilting objectsprojective generatorsinterval of t-structures
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The pith

Extended tilting objects in the heart of an interval of t-structures coincide with quasi-tilting objects or projective generators depending on the interval bounds.

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

The paper characterizes extended tilting objects inside the heart H_{[t1,t2]} of an interval of t-structures, when the ambient category is extriangulated and has negative first extension. In the case t2 ≤ Σ^{-1}t1 the objects are shown to be exactly the quasi-tilting objects of the abelian heart H_{[t1,Σ^{-1}t1]}. In the case Σ^{-2}t1 < t2 they are shown to be the projective generators of the extriangulated heart H_{[t1,Σ t2]}. These two reductions let one recognize the tilting objects by passing to shorter or abelian subcategories whose properties are already understood. The results extend the classical Happel-Reiten-Smalø process and its recent extriangulated generalizations to the setting of interval hearts.

Core claim

Building on the generalization of the Happel-Reiten-Smalø tilting process to extriangulated categories, the paper gives two main characterizations. When t2 ≤ Σ^{-1}t1, the extended tilting objects of H_{[t1,t2]} coincide with the quasi-tilting objects of the abelian category H_{[t1,Σ^{-1}t1]}. When Σ^{-2}t1 < t2, they coincide with the projective generators of the extriangulated category H_{[t1,Σ t2]}. Both statements rely on the assumption that the extriangulated category has negative first extension, so that the extended tilting objects behave like 1-tilting objects in the abelian case.

What carries the argument

The extended heart H_{[t1,t2]} of an interval of t-structures inside an extriangulated category with negative first extension; the two shift conditions on t1 and t2 that reduce the extended tilting objects to quasi-tilting objects or to projective generators.

If this is right

  • Under t2 ≤ Σ^{-1}t1 the extended tilting objects inherit the defining properties of quasi-tilting objects from the shorter abelian heart.
  • Under Σ^{-2}t1 < t2 the extended tilting objects serve as projective generators for the longer extriangulated heart H_{[t1,Σ t2]}.
  • The characterizations reduce the search for extended tilting objects to verification inside already-studied abelian or shorter extriangulated subcategories.
  • The results apply whenever the ambient extriangulated category has negative first extension, recovering the classical 1-tilting case as a special instance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The two shift conditions suggest that the "length" of the interval controls whether the tilting objects reduce to an abelian or an extriangulated setting.
  • One could test the characterizations by taking a known derived category and choosing concrete t-structures whose interval satisfies one of the two inequalities.
  • The reduction technique may extend to other notions of tilting that have been generalized to extriangulated categories.
  • Applications could appear in representation theory whenever hearts of t-structures arise naturally from silting or tilting complexes.

Load-bearing premise

The extriangulated category must have negative first extension so that extended tilting objects are defined and can be shown to match 1-tilting objects from the abelian case.

What would settle it

An explicit example of an interval [t1,t2] satisfying t2 ≤ Σ^{-1}t1 in which an object that is extended tilting in H_{[t1,t2]} fails to be quasi-tilting in H_{[t1,Σ^{-1}t1]}, or an interval satisfying Σ^{-2}t1 < t2 in which an extended tilting object fails to be a projective generator in H_{[t1,Σ t2]}.

read the original abstract

Building on the recent work of Adachi, Enomoto and Tsukamoto on a generalization of the Happel-Reiten-Smal{\o} tilting process, we study extended tilting objects in extriangulated categories with negative first extension. These objects coincide with the 1-tilting objects in abelian categories as in the work of Parra, Saor{\'i}n and Virili. We will be particularly interested in the case where the extriangulated category in question is the heart $\mathcal{H}_{[\mathbf{t}_{1},\mathbf{t}_{2}]}$ of an interval of $t$-structures $[\mathbf{t}_{1},\mathbf{t}_{2}]$. Our main results consist of a characterization of the extended tilting objects of a heart $\mathcal{H}_{[\mathbf{t}_{1},\mathbf{t}_{2}]}$ for the case when $\text{\ensuremath{\mathbf{t}}}_{2}\leq\Sigma^{-1}\mathbf{t}_{1}$, and another one for the case when $\Sigma^{-2}\mathbf{t}_{1}<\mathbf{t}_{2}$. In the first one, we give conditions for these tilting objects to coincide with the quasi-tilting objects of the abelian category $\mathcal{H}_{[\mathbf{t}_{1},\Sigma^{-1}\mathbf{t}_{1}]}$. In the second one, it is given conditions for these to coincide with projective generators in the extriangulated category $\mathcal{H}_{[\mathbf{t}_{1},\Sigma\mathbf{t}_{2}]}$

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper builds on Adachi-Enomoto-Tsukamoto's generalization of the Happel-Reiten-Smalø tilting process to study extended tilting objects in extriangulated categories with negative first extension (which coincide with 1-tilting objects in the abelian sense of Parra-Saorín-Virili). It specializes to the hearts H_{[t1,t2]} of intervals of t-structures, providing two main characterizations: one for the case t2 ≤ Σ^{-1}t1 relating the extended tilting objects to quasi-tilting objects of the abelian category H_{[t1,Σ^{-1}t1]}, and one for Σ^{-2}t1 < t2 relating them to projective generators in the extriangulated category H_{[t1,Σ t2]}.

Significance. If the characterizations hold, the work extends tilting theory to extriangulated categories arising from t-structure intervals, offering concrete conditions that connect to quasi-tilting and projective generators in related abelian/extriangulated settings. This could be useful for computations in representation theory of triangulated categories when the interval conditions are satisfied.

major comments (1)
  1. [Main results (characterizations for t2 ≤ Σ^{-1}t1 and for Σ^{-2}t1 < t2)] The definitions and main characterizations of extended tilting objects presuppose that the extriangulated category has negative first extension. The manuscript applies the theory to the specific hearts H_{[t1,t2]} under the two stated inequalities on t1 and t2, but does not appear to contain an explicit verification that Ext^1 vanishes in these hearts (as opposed to inheriting it from the ambient triangulated category). This hypothesis is load-bearing for both main results.
minor comments (1)
  1. [Notation and preliminaries] The notation with boldface vectors t1, t2 for t-structures is used consistently but could be introduced more explicitly in the preliminaries to avoid any ambiguity with the interval notation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for an explicit verification of the negative first extension hypothesis in the hearts of t-structure intervals. We address the comment below.

read point-by-point responses

  1. Referee: [Main results (characterizations for t2 ≤ Σ^{-1}t1 and for Σ^{-2}t1 < t2)] The definitions and main characterizations of extended tilting objects presuppose that the extriangulated category has negative first extension. The manuscript applies the theory to the specific hearts H_{[t1,t2]} under the two stated inequalities on t1 and t2, but does not appear to contain an explicit verification that Ext^1 vanishes in these hearts (as opposed to inheriting it from the ambient triangulated category). This hypothesis is load-bearing for both main results.

    Authors: We agree that the negative first extension property is essential for applying the general results of Adachi-Enomoto-Tsukamoto, and that an explicit check for the hearts H_{[t1,t2]} under the given inequalities on t1 and t2 is required. While these hearts are subcategories of the ambient triangulated category (where the relevant Ext groups vanish by construction of the t-structure interval), we will add a short preliminary result (a lemma or proposition) that directly verifies the vanishing of the first extension functor on H_{[t1,t2]} for each of the two cases. This will be placed before the statements of the main characterizations. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to co-author's prior work; central characterizations remain independent

full rationale

The paper cites Adachi-Enomoto-Tsukamoto for the generalized tilting process and Parra-Saorín-Virili for the coincidence of extended tilting objects with 1-tilting objects in abelian categories. Although one cited author overlaps with the present authors, the reference supplies an external definition and equivalence used as input rather than a self-referential reduction of the new results. The main theorems supply explicit conditions on t1 and t2 for the characterizations in the two cases, without evidence that any prediction or uniqueness claim collapses by construction to a fit or to the cited statement itself. The derivation chain is therefore self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters, axioms, or invented entities from the proofs can be extracted. The work relies on standard axioms of extriangulated categories and t-structures from the literature.

pith-pipeline@v0.9.0 · 5807 in / 1182 out tokens · 36064 ms · 2026-05-22T22:31:35.326693+00:00 · methodology

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Reference graph

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