Bulletin of the London Mathematical Society
Given a double complex X there are spectral sequences with the E2 terms being either HI (HII(X)) or HII(HI(X)). But if HI(X)=HII(X)=0, then both spectral sequences have all their terms 0. This can happen even though there is nonzero (co)homology of interest associated with X. This is frequently the case when dealing with Tate (co)homology. So, in this situation the spectral sequences may not give any information about the (co)homology of interest. In this article, we give a different way of constructing homology groups of X when HI(X)=HII(X)=0. With this result, we give a new and elementary proof of balance of Tate homology and cohomology.
Iacob A, E.E. Enochs, S. Estrada. “Balance with Unbounded Complexes”, Bulletin of the London Mathematical Society, 44(3): 439-442, 2012. doi:10.1112/blms/bdr101