Acyclicity over local rings with radical cube zero

This paper studies infinite acyclic complexes of finitely generated free modules over a commutative noetherian local ring (R, m) with m³ = 0. Conclusive results are obtained on the growth of the ranks of the modules in acyclic complexes, and new sufficient conditions are given for total acyclicity. Results are also obtained on the structure of rings that are not Gorenstein and admit acyclic complexes; part of this structure is exhibited by every ring R that admits a non-free finitely generated module M with Ext_n ^R(M, R) = 0 for a few n > 0.