2023/03/15 by Shaul, Liran · 1 citation
#13D09 #16E35 #16E45 #Commutative Algebra (math.AC) #FOS: Mathematics #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2303.08756
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.