A prime-characteristic analogue of a theorem of Hartshorne-Polini (Q2176094)

Let \(k\) be an algebraically closed field. Suppose that \(\text{ch}(k)=0\), and let \(R=k[[x_1, \dots, x_n]]\) and \({\mathfrak m}=\langle x_1, \dots, x_n \rangle\). Also, let \(\mathcal{D}=\mathcal{D}(R,k)\) be the ring of \(k\)-linear differential operators on \(R\), \(M\) a holonomic \(\mathcal{D}\)-module, and let \(\text{H}^i_{dR}(-)\) denote the \(i\)-th de Rham cohomology functor. \textit{R. Hartshorne} and \textit{C. Polini} [J. Algebra 571, 232--257 (2021; Zbl 1464.13003)], and \textit{N. Switala} [Compos. Math. 153, No. 10, 2075--2146 (2017; Zbl 1405.13033)] showed that \[\dim_k\text{H}^n_{dR}(M)=\dim_k\text{H}^0_{dR}(\text{Hom}_R(M,E_R(k)))\] \[=\dim_k \text{Hom}_{\mathcal{D}}(M,E)\] \[=\max \{t\in \mathbb{N} \mid \exists \ \text{a}\ \mathcal{D}-\text{module surjection} \ M\rightarrow E_R(k)^t \}.\] For every ideal \(I\) of \(R\) and every non-negative integer \(i\), \(\text{H}^i_I(R)\) is a holonomic \(\mathcal{D}\)-module. Also, \textit{N. Switala} and \textit{W. Zhang} [Adv. Math. 340, 1141--1165 (2018; Zbl 1425.14016)] proved an analogue of the above result for the case \(R=k[x_1, \dots x_n]\). In this paper, the latter authors prove an analogue of the above result in the case \(\text{ch}(k)\) is positive. Assume that \(\text{ch}(k)>0\), and let \(R\) be an \(F\)-finite regular ring containing \(k\). Let \(\mathfrak{m}\) be a maximal ideal of \(R\) and \(M\) an \(\mathcal{F}\)-finite \(\mathcal{F}\)-module over \(R\) in the sense of Lyubeznik. They prove that \[\dim_{k} \text{Hom}_R(M,E_R(R/{\mathfrak m}))_s=\dim_{\mathbb{F}_p}\text{Hom}_{\mathcal{F}}(M,E_R(R/\mathfrak m))\] \[=\max \{t\in \mathbb{N} \mid \exists \ \text{an} \ \mathcal{F}-\text{module surjection} \ M\rightarrow E_R(R/\mathfrak m)^t \}.\] (Recall that for a Frobenius module \((X,\varphi_X)\), the Frobenius stable part of \(X\) is the \(k\)-subspace \(X_s:=\underset{i\geq 0} \bigcap \varphi_X^i(X)\) of \(X\).) For every ideal \(I\) of \(R\) and every non-negative integer \(i\), the \(R\)-module \(\text{H}^i_I(R)\) is an \(\mathcal{F}\)-finite \(\mathcal{F}\)-module.