Acyclic complexes of injectives and finitistic dimensions

TL;DR

This paper investigates the relationship between acyclic complexes of injectives and finitistic dimensions of rings, establishing new conditions and characterizations for rings with finite finitistic dimension.

Contribution

It provides new necessary and sufficient conditions for a ring to have finite finitistic dimension, generalizing previous results and connecting derived categories with ring properties.

Findings

Acyclic complexes of injectives imply finite finitistic dimension under certain conditions.

Characterizations of noetherian rings satisfying the Gorenstein symmetry conjecture.

Generalizations of Rickard's results relating derived categories and finitistic dimension.

Abstract

For a ring $A$, we consider the question whether every bounded above cochain complex of injective $A$-modules which is acyclic is null-homotopic. We show that if $A$ is left and right noetherian and has a dualizing complex, then this implies that the finitistic dimension of $A$ is finite. In the appendix, Nakamura and Thompson show that the opposite holds over any ring. Our results give several new necessary and sufficient conditions for a ring to have finite finitistic dimension in a very general setting. Applications include a generalization of a recent result of Rickard about relations between unbounded derived categories and finitistic dimension, as well as several new characterizations of noetherian rings which satisfy the Gorenstein symmetry conjecture.