On Set Theoretically and Cohomologically Complete Intersection Ideals

Abstract

Abstract. Let (R, m) be a local ring and a be an ideal of R. The inequalitiesare known. It is an interesting and long-standing problem to find out the cases giving equality. Thanks to the formal grade we give conditions in which the above inequalities become equalities.

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

“…A natural upper bound for the min n depth(R/I n ) is the depth R/I. In an attempt to find a suitable (homological) upper bound for the min n depth(R/I n ), the third named author in [11], examined the formal grade of I, fgrade(I, R) and gave concrete examples for the equality min n depth(R/I n ) = fgrade(I, R).…”

On cohomological dimension and depth under linkage

“…Recently formal local cohomology has been used as a technical tool to solve some problems, see for instance [8]. In this paper we use it to give information on the cohomological dimension of an ideal.…”

“…j a (R) = 0 for all j < t. So by virtue of[8, Corollary 4.2], we have depth R/a = fgrade(a, R) ≥ t. Suppose that R = k[x 1 , . .…”

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

“…A natural upper bound for the min n depth(R/I n ) is the depth R/I. In an attempt to find a suitable (homological) upper bound for the min n depth(R/I n ), the third named author in[11], examined the formal grade of I, fgrade(I, R) and gave concrete examples for the equality min n depth(R/I n ) = fgrade(I, R). If (R, m) is a regular local ring, in the light of equality (1.1) and [25, Lemma 4.8] one has fgrade(I, R) = depth R/I if and only if pd R/I = cd(R, I).…”

“…Suppose k = R/m. If char(k) > 0 the result is known (cf [11,. Remark 3.1]), so we can assume char(k) = 0 (k is infinite).…”