A note on injectivity of Frobenius on local cohomology of global complete intersections

Abstract

Given a graded complete intersection ideal $J=(f_1,\dots ,f_c)\subseteq k[x_0,\dots ,x_n]=S$, where $k$ is a field of characteristic $p>0$ such that $[k:k^p]<\infty$, we show that if $S/J$ has an isolated non-F-pure point then the Frobenius action on top local cohomology $H_{\mathfrak{m}}^{n+1-c}(S/J)$ is injective in sufficiently negative degrees, and we compute the least degree of any kernel element. If $S/J$ has an isolated singularity, we are also able to give an effective bound on $p$ ensuring the Frobenius action on $H_{\mathfrak{m}}^{n+1-c}(S/J)$ is injective in all negative degrees, extending a result of Bhatt and Singh in the hypersurface case.

6 pages. To appear in the Journal of Pure and Applied Algebra