A Note on Injectivity of Frobenius on Local Cohomology of Hypersurfaces

2015/02/11 by Eric Canton, Canton, Eric
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC

paper · pdf · doi:10.48550/arxiv.1502.03184

6 pages

arxiv created 2015/02/11 · openalex publication_date 2015/02/11 · arxiv updated 2015/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k be a field of characteristic p > 0 such that [k:k^p] < ∞ and let f ∈ R = k[x₀, ..., x_n] be homogeneous of degree d. We obtain a sharp bound on the degrees in which the Frobenius action on Hⁿ_\mathfrakm(R/fR) can be injective when R/fR has an isolated non-F-pure point at \mathfrakm. As a corollary, we show that if (R/fR)_\mathfrakm is not F-pure then R/fR has an isolated non-F-pure point at \mathfrakm if and only if the Frobenius action is injective in degrees ≤ -n(d-1).

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