Local cohomology based on a nonclosed support defined by a pair of ideals

We introduce a generalization of the notion of local cohomology module, which we call a local cohomology module with respect to a pair of ideals (I,J)(I,J), and study its various properties. Some vanishing and nonvanishing theorems are given for this generalized version of local cohomology. We also discuss its connection with ordinary local cohomology.