Restricted Homological Dimensions and Cohen–Macaulayness,

The classical homological dimensions—the projective, flat, and injective ones—are usually defined in terms of resolutions and then proved to be computable in terms of vanishing of appropriate derived functors. In this paper we define restricted homological dimensions in terms of vanishing of the same derived functors but over classes of test modules that are restricted to assure automatic finiteness over commutative Noetherian rings of finite Krull dimension. When the ring is local, we use a mixture of methods from classical commutative algebra and the theory of homological dimensions to show that vanishing of these functors reveals that the underlying ring is a Cohen–Macaulay ring—or at least close to being one.