Authors: LASZLO FUCHS, SANG BUM LEE

Abstract: We prove a stronger form of an analogue of a Kaplansky lemma on homological dimensions by showing that in a pure-exact sequence $0 \to A \to B \to C \to 0$, the weak dimensions of the modules satisfy $ {\rm w.d.} B = \max\{{\rm w.d.} A, {\rm 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 $M_{\rho}$ is proved: the weak dimension of the union of the chain equals the supremum of the weak dimensions of the factor modules $M_{\rho+1}/ M_{\rho}$ in the chain. The same holds for injective dimensions if the ring is noetherian.

Keywords: Tor, pure submodules, weak dimension, injective dimension

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