Unbounded derived categories and the finitistic dimension conjecture

TL;DR

This paper investigates when injective modules generate the unbounded derived category of a ring, providing examples and conditions that relate to the finitistic dimension conjecture, especially for Noetherian and finite dimensional algebras.

Contribution

It establishes that injectives generate for Noetherian commutative rings, provides a counterexample for non-Noetherian rings, and links the generation property to the finitistic dimension conjecture in finite dimensional algebras.

Findings

Injectives generate for Noetherian commutative rings.

Counterexample for non-Noetherian commutative rings.

Generation property implies the finitistic dimension conjecture for finite dimensional algebras.

Abstract

We consider the question of whether the injective modules generate the unbounded derived category of a ring as a triangulated category with arbitrary coproducts. We give an example of a non-Noetherian commutative ring where they don't, but prove that they do for any Noetherian commutative ring. For non-commutative finite dimensional algebras the question is open, and we prove that if injectives generate for such an algebra, then the finitistic dimension conjecture holds for that algebra.