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Tilting objects in the extended heart of a $t$-structure

Alejandro Argudin Monroy, Octavio Mendoza, Carlos E. Parra

TL;DR

The paper develops extended tilting theory for extriangulated categories with negative first extension via the AET-tilt framework, focusing on extended hearts $\mathcal{H}_{[\boldsymbol{t}_{1},\boldsymbol{t}_{2}]}$ and their tilting objects. It establishes a bijection between t-structures in an interval and $s$-torsion pairs in the extended heart, and proves two central characterizations: when $\boldsymbol{t}_{2}\leq \Sigma^{-1}\boldsymbol{t}_{1}$ extended tilting objects coincide with quasi-tilting objects in $\mathcal{H}_{[\boldsymbol{t}_{1},\Sigma^{-1}\boldsymbol{t}_{1}]}$, and when $\Sigma^{-2}\boldsymbol{t}_{1}<\boldsymbol{t}_{2}$ they coincide with projective generators in $\mathcal{H}_{[\boldsymbol{t}_{1},\Sigma\boldsymbol{t}_{2}]}$. The paper further develops the structure of extended hearts, analyzes when they are exact or quasi-abelian, and connects extended tilting in extended or restricted hearts to classical tilting concepts like quasi-tilting and projective generation. These results yield practical criteria for identifying hearts with projective generators and provide non-abelian examples of extended tilting objects, broadening tilting theory beyond abelian settings.

Abstract

Building on the recent work of Adachi, Enomoto and Tsukamoto on a generalization of the Happel-Reiten-Smalø 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{í}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 $\text{\ensuremath{\mathbf{t}}}_{2}\leqΣ^{-1}\mathbf{t}_{1}$, and another one for the case when $Σ^{-2}\mathbf{t}_{1}<\mathbf{t}_{2}$. In the first one, we give conditions for these tilting objects to coincide with the quasi-tilting objects of the abelian category $\mathcal{H}_{[\mathbf{t}_{1},Σ^{-1}\mathbf{t}_{1}]}$. In the second one, it is given conditions for these to coincide with projective generators in the extriangulated category $\mathcal{H}_{[\mathbf{t}_{1},Σ\mathbf{t}_{2}]}$

Tilting objects in the extended heart of a $t$-structure

TL;DR

The paper develops extended tilting theory for extriangulated categories with negative first extension via the AET-tilt framework, focusing on extended hearts and their tilting objects. It establishes a bijection between t-structures in an interval and -torsion pairs in the extended heart, and proves two central characterizations: when extended tilting objects coincide with quasi-tilting objects in , and when they coincide with projective generators in . The paper further develops the structure of extended hearts, analyzes when they are exact or quasi-abelian, and connects extended tilting in extended or restricted hearts to classical tilting concepts like quasi-tilting and projective generation. These results yield practical criteria for identifying hearts with projective generators and provide non-abelian examples of extended tilting objects, broadening tilting theory beyond abelian settings.

Abstract

Building on the recent work of Adachi, Enomoto and Tsukamoto on a generalization of the Happel-Reiten-Smalø 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{í}n and Virili. We will be particularly interested in the case where the extriangulated category in question is the heart of an interval of -structures . Our main results consist of a characterization of the extended tilting objects of a heart for the case when , and another one for the case when . In the first one, we give conditions for these tilting objects to coincide with the quasi-tilting objects of the abelian category . In the second one, it is given conditions for these to coincide with projective generators in the extriangulated category

Paper Structure

This paper contains 13 sections, 35 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extriangulated categories with negative first extensions and $s$-torsion pairs
  4. $t$-structures in triangulated categories
  5. Intervals of $s$-torsion pairs
  6. Extensions of hearts and their $s$-torsion pairs
  7. Parameterizing subintervals
  8. The extended heart of a $t$-structure
  9. Extended tilting objects
  10. Projective dimension of extended tilting objects
  11. Extended tilting objects are $\mathbb{E}$-universal
  12. Extended tilting objects in restricted hearts
  13. Extended tilting objects in extended hearts

Key Result

Proposition 2.2

NP For any $\mathfrak{s}$-conflation $\delta:\:\suc[A][B][C][u][v]$ in an extriangulated category $\mathcal{D}$ and any $W\in\mathcal{D}$, the sequences are exact, where $(W,-):=\mathcal{\mathcal{D}}(W,-)$ and $(-,W):=\mathcal{\mathcal{D}}(-,W)$.

Theorems & Definitions (93)

  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Proposition 2.8
  • Proposition 2.9
  • proof
  • ...and 83 more