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Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module M over a (commutative) Noetherian ring R. Second, in the other direction, we investigate the impact of the finiteness of certain homological dimensions of M if R is local, mainly when R is Cohen-Macaulay and with a partial focus on duals. Along the way, we produce various freeness criteria for modules. Finally, we give applications, including characterizations of when R is Gorenstein (and other ring-theoretic properties as well, sometimes in the prime characteristic setting), particularly by means of its anticanonical module, and in addition we address special cases of some long-standing conjectures; for instance, we confirm the 1985 conjecture of Vasconcelos on normal modules in case the module of differentials is almost Cohen-Macaulay.
Dey et al. (Tue,) studied this question.