Abstract: Garland introduced a vanishing criterion for characteristic zero cohomology groups of locally finite and locally connected simplicial complexes based on the spectral gaps of the graph Laplacians of face links which has turned out to be effective in a wide range of examples. This talk provides a homological algebra version of this method and a class of posets which provide examples including simplicial and cubical. The former gives Garland’s original vanishing theorem while the latter gives a structure on generators rather than vanishing of cohomology. Random complexes provide examples and a conjecture.