A prime-characteristic analogue of a theorem of Hartshorne-Polini

2019/05/29 by Nicholas Switala, Switala, Nicholas, Wenliang Zhang +1
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1905.12584

openalex publication_date 2019/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R be an F-finite Noetherian regular ring containing an algebraically closed field k of positive characteristic, and let M be an \F-finite \F-module over R in the sense of Lyubeznik (for example, any local cohomology module of R). We prove that the \mathbbF_p-dimension of the space of \F-module morphisms M → E(R/\fm) (where \fm is any maximal ideal of R and E(R/\fm) is the R-injective hull of R/\fm) is equal to the k-dimension of the Frobenius stable part of \Hom_R(M,E(R/\fm)). This is a positive-characteristic analogue of a recent result of Hartshorne and Polini for holonomic \D-modules in characteristic zero. We use this result to calculate the \F-module length of certain local cohomology modules associated with projective schemes.

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