  Let $k$ be a field of positive characteristic $p$, $R$ be a Gorenstein graded
$k$-algebra, and $S=R/J$ be an artinian quotient of $R$ by a homogeneous ideal.
  We ask how the socle degrees of $S$ are related to the socle degrees of
$F_R^e(S)=R/J^{[q]}$. If $S$ has finite projective dimension as an $R$-module,
then the socles of $S$ and $F_R^e(S)$ have the same dimension and the socle
degrees are related by the formula: $$D_i=qd_i-(q-1)a(R),$$ where $$ d_1\le
>...\le d_{\ell}\quad\text{and}\quad D_1\le ... \le D_{\ell}$$ are the socle
degrees $S$ and $F_R^e(S)$, respectively, and $a(R)$ is the $a$-invariant of
the graded ring $R$, as introduced by Goto and Watanabe. We prove the converse
when $R$ is a complete intersection.
