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, Tor_i^R(M, f_Rⁿ) = 0 - fⁿ 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.