On cohomologically complete intersections

Abstract

An ideal I of a local Gorenstein ring (R, m) is called cohomologically complete intersection whenever H i I (R) = 0 for all i = height I. Here H i I (R), i ∈ Z, denotes the local cohomology of R with respect to I. For instance, a set-theoretic complete intersection is a cohomologically complete intersection. Here we study cohomologically complete intersections from various homological points of view, in particular in terms of their Bass numbers of H c I (R), c = height I. As a main result it is shown that the vanishing H i I (R) = 0 for all i = c is completely encoded in homological properties of H c I (R), in particular in its Bass numbers.

On endomorphism rings and dimensions of local cohomology modules

“…For l ≥ dim R/I the previous result yields -as a particular case -the equivalence of the conditions (i) and (ii) of [4,Theorem 3.1]. Another Corollary is the following:…”

On Some Local Cohomology Spectral Sequences

“…Let A be a commutative Noetherian ring, and let I ⊆ A be an ideal; it is said that I is cohomologically complete intersection provided H k I (A) = 0 for any k = ht(I). Cohomologically complete intersection ideals were introduced by Hellus and Schenzel in [HS08], where the interested reader on this notion can find further details and results. This situation is achieved, among others, in the following cases:…”

Local cohomology modules and Gorenstein injectivity with respect to a semidualizing module

“…is the largest integer i for which H i a (M ) = 0. Observe that the notion of relative CohenMacaulay module is connected with the notion of cohomologicaly complete intersection ideal which has been studied in [7]. …”