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-Smal 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\'in and Virili. We will be particularly interested in the case where the extriangulated category in question is the heart H[t1,t2] of an interval of t-structures [t1,t2]. Our main results consist of a characterization of the extended tilting objects of a heart H[t1,t2] for the case when t2≤-1t1, and another one for the case when -2t1<t2. In the first one, we give conditions for these tilting objects to coincide with the quasi-tilting objects of the abelian category H[t1,-1t1]. In the second one, it is given conditions for these to coincide with projective generators in the extriangulated category H[t1,t2]
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