On the relation between formal grade and depth with a view toward vanishing of Lyubeznik numbers

Abstract

Abstract. Let (R, m) be a local ring and I an ideal. The aim of the present paper is twofold. At first we continue the investigation to compare fgrade(I, R) with depth R/I and further we derive some results on the vanishing of Lyubeznik numbers.

Vanishing of local cohomology and set-theoretically Cohen-Macaulay ideals

“…The ring endomorphism ϕ : R → R induces a natural ϕ action on the local cohomology modules ϕ * : H i I (R) → H i ϕ(I)R (R) via ϕ(r)ϕ * (η) = ϕ * (rη), where r ∈ R, η ∈ H i I (R) which is an endomorphism of the underlying Abelian group (details, including notation, are given in Section 2). This action as a generalization of Frobenius action, is an effective tool in the study of local cohomology modules as it was used recently by Àlvarez Montaner in [1] and the author with his colleagues in [4], and with Boix in [3].…”