Let [Formula: see text] be a unitary commutative local Noetherian ring and [Formula: see text] be an ideal. The aim of this paper is twofold: In the first part of this paper, we consider locally cohomologically complete intersection ideals of pure height [Formula: see text]. Next, we deal with cofinite modules. In particular, we investigate conditions for cofiniteness of [Formula: see text] and conditions under which, the modules [Formula: see text] are of finite length for all [Formula: see text] and all finitely generated [Formula: see text]-module [Formula: see text].