Tilting objects in the extended heart of a $t$-structure
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917465349423104 |
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| author | Monroy, Alejandro Argudin Mendoza, Octavio Parra, Carlos E. |
| author_facet | Monroy, Alejandro Argudin Mendoza, Octavio Parra, Carlos E. |
| contents | 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 $\mathcal{H}_{[\mathbf{t}_{1},Σ\mathbf{t}_{2}]}$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_20604 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tilting objects in the extended heart of a $t$-structure Monroy, Alejandro Argudin Mendoza, Octavio Parra, Carlos E. Representation Theory 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 $\mathcal{H}_{[\mathbf{t}_{1},Σ\mathbf{t}_{2}]}$ |
| title | Tilting objects in the extended heart of a $t$-structure |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2503.20604 |