Injective generation of the derived category and finitistic dimension conjecture

Abstract

For a finite dimensional algebra \(\Lambda \), the problem of whether the unbounded derived category \(\mathbb {D}(\Lambda )\) is equal to its localizing subcategory generated by injective \(\Lambda \)-modules was firstly considered by Keller in 2001. If this happens to be true, it is usually said that injectives generate for \(\Lambda \). Some connections to famous homological conjectures were illuminated by Keller himself. Recently, Rickard presented several classes of rings, including particular types of finite dimensional algebras as well as commutative Noetherian rings, for which injectives generate. He also proved that if injectives generate for \(\Lambda \), then it satisfies the big finitistic dimension conjecture. The main objective of this paper is to discuss when the reverse statement also holds. We show that, under some mild condition, the injective generation phenomenon and the big finitistic dimension conjecture for \(\Lambda \) are equivalent.