TTinDmod · Tilting theory in derived module categories

Tilting theory is a set of tools and techniques used to compare and relate module categories. The development of the subject has shown wide and deep applications to representation theory, geometry and mathematical physics. The homological and combinatorial nature of these applications has led to a growing number of new approaches in the area. This project brings together some of these approaches in the setting of derived module categories. We propose to unify and reconcile views on the bounded and on the unbounded derived categories of a ring, establishing new ways to compare them. The key concepts involved range from tilting and silting objects to t-structures, infinitely generated modules, cotorsion pairs and recollements. We suggest constructions and/or classifications for some of these concepts in suitable contexts (from finite dimensional algebras to fully bounded noetherian rings), linking ring theoretical ideas with the study of derived module categories. Ultimately, we use them to investigate the structure of these derived categories and, therefore, have a better understanding of how they relate.