A Note on Injectivity of Frobenius on Local Cohomology of Hypersurfaces

Abstract

Let $k$ be a field of characteristic $p > 0$ such that $[k:k^p] < \infty$ and let $f \in R = k[x_0, ..., x_n]$ be homogeneous of degree $d$. We obtain a sharp bound on the degrees in which the Frobenius action on $H^n_\mathfrak{m}(R/fR)$ can be injective when $R/fR$ has an isolated non-F-pure point at $\mathfrak{m}$. As a corollary, we show that if $(R/fR)_\mathfrak{m}$ is not F-pure then $R/fR$ has an isolated non-F-pure point at $\mathfrak{m}$ if and only if the Frobenius action is injective in degrees $\le -n(d-1)$.