Weak and injective dimensional analogues of Kaplansky’s and Auslander’s lemmas for purity

Abstract

We prove a stronger form of an analogue of a Kaplansky lemma on homological dimensions by showing that in a pure-exact sequence 0 → A → B → C → 0, the weak dimensions of the modules satisfy w.d.B = max{w.d.A, w.d.C}. We also show that the same equality holds for the injective dimensions whenever the ring is noetherian. In addition, a version of Auslander’s lemma for chains of pure submodules Mρ is proved: the weak dimension of the union of the chain equals the supremum of the weak dimensions of the factor modules Mρ+1/Mρ in the chain. The same holds for injective dimensions if the ring is noetherian