Acyclic complexes of injectives and finitistic dimensions

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.

18 pages, with an appendix by Tsutomu Nakamura and Peder Thompson