Generalized local duality, canonical modules, and prescribed bound on projective dimension

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well a general duality theorem which extends, much broader class rings, previous by Herzog-Zamani Suzuki. As an application, we establish prescribed upper bound for the projective dimension module satisfying suitable cohomological conditions, derive some freeness criteria questions Auslander-Reiten type. Along way, prove new characterization Cohen-Macaulay modules truly relies on cohomology, in addition introduce study generalization notion canonical module.