Injective resolutions of some regular rings Original Research Article

Let A be a noetherian Auslander regular ring and δ the canonical dimension function on A-modules, which is defined as δ(M)=d−j(M) where d is the global dimension of A and j(M) is the grade of M. An A-module is s-pure if δ(N)=s for all its non-zero noetherian submodules N, and is essentially s-pure if it contains an essential submodule which is s-pure. Consider a minimal injective resolution of A as an A-moduleimage0→A→I0→I1→cdots, three dots, centered→Id→0.We say A has a pure (resp. essentially pure) injective resolution if Ii is (d−i)-pure (resp. essentially (d−i)-pure). We show that several classes of Auslander regular rings with global dimension at most 4 have pure or essentially pure injective resolutions.