F-Holonomic F-Modules

Abstract

Let $R=\mathbb{F}_p[x_1,\ldots,x_n]$ and let $\mathbf{F}$ be the ring of Frobenius operators over $R$. We introduce a notion of Bernstein dimension and multiplicity for the class of finitely generated $\mathbf{F}$-modules whose structure morphism has a finite length kernel. We show that an $\mathbf{F}$-module belongs to this class if and only if it admits a great filtration with respect to the Bernstein filtration on $\mathbf{F}$. We describe the Hilbert series of these great filtrations, and prove that the dimension and multiplicity defined in terms of this Hilbert series are independent of the choice of filtration. We refer to the $\mathbf{F}$-modules of Bernstein dimension $0$ as $F$-holonomic. We show that $F$-holonomic $\mathbf{F}$-modules are a full abelian subcategory of $\text{Mod}_{\mathbf{F}}$, closed under taking extensions, on which multiplicity is an additive function. We show that Lyubeznik's finitely generated unit $\mathbf{F}$-modules are $F$-holonomic.