This paper studies infinite acyclic complexes of finitely
generated free modules over a commutative noetherian local
ring $(R,\fm)$ with $\fm^3=0$. 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 $n>0$.