Socle degrees of Frobenius powers (Q2382942)

If \(k\) is a field and \(S\) is an artinian graded \(k\)-algebra with maximal homogeneous ideal \(m\), then the socle \(\text{Soc} S=(0:_S m)\) is a finite dimensional graded \(k\)-vector space \(\bigoplus_{i=1}^l k(-d_i)\), and the numbers \( d_1 \leq \ldots \leq d_l\) are referred to as the socle degrees of \(S\); the largest degree \(d_l\) is called the top socle degree. For a field \(k\) of characteristic \(p>0\), given a noetherian graded \(k\)-algebra \(R\) with maximal homogeneous ideal \(m\) and an \(m\)-primary homogeneous ideal \(J\), the paper under review is concerned with the study of the dependence of the socle degrees of \(R/J^{[q]}\) on \(q=p^e\), where \(J^{[q]}\) is the ideal generated by all \(i^q\) with \(i \in J\). The authors prove the following result. Let \(R=P/C\) be a complete intersection ring with \(C\) generated by a homogeneous regular sequence, where \(P\) is a positively graded polynomial ring over a field \(k\) of characteristic \(p>0\). Let \(m\) be the maximal homogeneous ideal of \(R\), \(J\) be a homogeneous ideal of \(R\), and \(I\) be a lifting of \(J\) to \(P\). Let \(l\) be the dimension of \(\text{Soc} (R/J)\) and \(d_1, \ldots, d_l\) be the socle degrees of \(R/J\). For \(q=p^e\), the following are equivalent: (a) The \(R\)-module \(R/J\) has finite projective dimension. (b) \(\text{Soc}(R/J^{[q]})\) has dimension \(l\) and the socle degrees of \(R/J^{[q]}\) are \(qd_i-(q-1)a(R)\) (\(1 \leq i \leq l\)), where \(a(R)\) is the \(a\)-invariant of \(R\). (c) \((C+I)^{[q]} : (C^{[q]} : C)=C+ I^{[q]}\). (d) \(I^{[q]} \cap C=(I \cap C)^{[q]} + CI^{[q]}\). As mentioned in the paper, the implication \((a) \Rightarrow (b)\) was already known in the literature even in the more general case when \(R\) is a Gorenstein graded \(k\)-algebra.