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Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] be an integer and let [Formula: see text] denote the ideals of [Formula: see text] of dimension [Formula: see text]. For an integer [Formula: see text] and [Formula: see text]-modules [Formula: see text], we define the [Formula: see text]th cohomology module [Formula: see text] and study some of its properties. Some vanishing results will be proved and the connection between the vanishing of this cohomology modules with the dimension of certain subsets of [Formula: see text] will be investigated.

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