Cohen-Macaulay pairs

The main purpose of this note is to extend and establish a new approach to the concept of (relative) Cohen-Macaulayness, by investigating the cohomological dimension as well as the depth of a pair of modules over a commutative Noetherian ring. The notion also largely extends the cohomologically complete intersection property of Hellus and Schenzel. As crucial tools, we use Herzog's theory of generalized local cohomology along with spectral sequence techniques. We also provide byproducts concerning two classical problems in homological commutative algebra.