On modules of finite projective dimension over complete intersections

Recently Avramov and Miller proved that over a local complete intersection ring $(R,m,k)$ in characteristic $p>0$, a finitely generated module $M$ has finite projective dimension if for some $i>0$ and for some $n>0$, $\operatorname{Tor}^{R}_{i}(M,f^{n}_{R})=0-f^{n}$ being the frobenius map repeated $n$ times. They used the notion of ``complexity'' and several related theorems. Here we offer a very simple proof of the above theorem without using ``complexity'' at all.