A Frobenius characterization of finite projective dimension over complete intersections

Let M be a module of finite length over a complete intersection (R,m) of characteristic p > 0. We characterize the property that M has finite projective dimension in terms of the asymptotic behavior of a certain length function defined using the Frobenius functor. This may be viewed as the converse to a theorem of S. Dutta. As a corollary we get that, in a complete intersection (R,m), an m-primary ideal I has finite projective dimension if and only if its Hilbert-Kunz multiplicity equals the length of R/I.