INJECTIVE RESOLVENTS AND PREENVELOPES

Abstract

Let R be a noetherian ring, and denote the full subcategories of R-modules L such that Extⁱ(E,L)=0 for all injective R-modules E for 1⋚i⋚n and O⋚i⋚n by C_n, and C′_n respectively. Then LεC_n, if and only if every injective resolution of L is an injective resolvent of the nth cosyzygy. In this case, L is not injective if and only if its injective dimension is greater than n. If LεC′_n and idN⋚n. then Hom(N,L)=0 for all R-modules N. As an application, let K_n be the nth syzygy of an injective resolvent of the nth cosyzygy of an R-module N, then there exists a homomorphism φ:N → K such that ((φ,i_N), K_n • E(N)) and (φ,K_n) are preenvelopes of N for C_s and C′_s respectively, for s≥n. If the global dimension of R is at most 2, then C′₁ is reflective in the category of R-modules.