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

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 $\hat{I}{\pounds }^{-2}{\mathbf{t}}_1<{\mathbf{t}}_2$. In the first one, we give conditions for these tilting objects to coincide with the quasi-tilting objects of the abelian category ${\mathcal{H}}_{[{\mathbf{t}}_1,\hat{I}{\pounds }^{-1}{\mathbf{t}}_1]}$. In the second one, it is given conditions for these to coincide with projective generators in the extriangulated category ${\mathcal{H}}_{[{\mathbf{t}}_1,\hat{I}\pounds {\mathbf{t}}_2]}$