ON THE STABLE HOMOLOGY OF MODULES
IOANNIS EMMANOUIL AND PANAGIOTA MANOUSAKI
Abstract. In this paper, we study certain properties of the stable homology groups of mod- 
ules 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 connec- 
tion, 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.
Contents
0. Introduction 1
1. The injective completion of the contravariant Hom functor 4
2. The relation between stable and complete homology 7
3. Isomorphism criteria 10
4. Vanishing criteria 13
5. Vanishing criteria over Noetherian rings 15
6. Balance of stable homology 18
Appendix A. Hausdorff filtrations on complexes and homology 20
References 22