Tilting objects in the extended heart of a t-structure

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, Saorín and Virili. We will be particularly interested in the case where the extriangulated category in question is the heart H [ t 1 , t 2 ] of an interval of t -structures [ t 1 , t 2 ] . Our main results consist of a characterization of the extended tilting objects of a heart H [ t 1 , t 2 ] for the case when t 2 ≤ Σ − 1 t 1 , and another one for the case when Σ − 2 t 1 ≤ t 2 . In the first one, we give conditions for these objects to coincide with the quasi-tilting objects of the abelian category H [ t 1 , Σ − 1 t 1 ] . In the second one, it is proved that the heart H [ t 1 , t 2 ] admits an extended tilting object only if it is an extended heart (i.e. t 2 = Σ − 2 t 1 ). Furthermore, we will characterize the extended tilting objects of an extended heart H [ t 1 , t 2 ] as projective generators in the abelian category H [ t 1 , Σ t 2 ] whenever H [ t 1 , Σ t 2 ] is a cogenerating class of the extended heart.