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
$d_1\le \dots\le d_{\ell}$
and
$D_1\le \dots \le D_{\ell}$
are the socle degrees of $S$ 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.