On the stable homology of modules

Abstract

In this paper, we study certain properties of the stable homology groups of modules over an associative ring, which were defined by Vogel [12]. We compute the kernel of the natural surjection from stable homology to complete homology, which was itself defined by Triulzi [21]. This computation may be used in order to formulate conditions under which the two theories are isomorphic. Duality considerations reveal a connection between stable homology and the complete cohomology theory defined by Nucinkis [19]. Using this connection, we show that the vanishing of the stable homology functors detects modules of finite flat or injective dimension over Noetherian rings. As another application, we characterize the coherent rings over which stable homology is balanced, in terms of the finiteness of the flat dimension of injective modules.