A Note on Injectivity of Frobenius on Local Cohomology of Hypersurfaces
Eric Canton

TL;DR
This paper establishes a precise bound on the degrees where Frobenius acts injectively on local cohomology of hypersurfaces, linking F-purity properties to Frobenius injectivity in algebraic geometry.
Contribution
It provides a sharp degree bound for Frobenius injectivity on local cohomology of hypersurfaces with isolated non-F-pure points, clarifying the relationship between F-purity and Frobenius action.
Findings
Frobenius action is injective in degrees ≤ -n(d-1) if and only if the hypersurface has an isolated non-F-pure point.
A sharp bound on degrees for Frobenius injectivity is established.
Characterization of non-F-purity points via Frobenius action in local cohomology.
Abstract
Let be a field of characteristic such that and let be homogeneous of degree . We obtain a sharp bound on the degrees in which the Frobenius action on can be injective when has an isolated non-F-pure point at . As a corollary, we show that if is not F-pure then has an isolated non-F-pure point at if and only if the Frobenius action is injective in degrees .
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Commutative Algebra and Its Applications · Polynomial and algebraic computation
