Let \(R\) be a regular local ring of dimension \(d\). We establish the relation between the support of the \(i-th\) local cohomology of \(R\) supported on an ideal \(I\) and the Frobenius action on the \((d-i)-th\) local cohomology of \(R/I\) supported on the maximal ideal of \(R\), for any \(0 ≤ i ≤ d\). We deduce some vanishing statements on the local cohomology modules of \(R\) supported on the ideal \(I\).
