2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103869We construct a class of Gorenstein local rings $R$ which admit minimal complete $R$-free resolutions $\bd C$ such that the sequence $\{\rank_R C_i\}$ is constant for $i< 0$, and grows exponentially for all $i>0$. Over these rings we show that there exist finitely generated $R$-modules $M$ and $N$ such that $\Ext^i_R(M,N)=0$ for all $i> 0$, but $\Ext^i_R(N,M)\ne 0$ for all $i>0$.14 pages, to appear in Internat. Math. Res. NoticesCommutative Algebratext