Balance with Unbounded Complexes

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 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. 1.Introduction We will mainly be concerned with left R-modules over some ring R. So unless otherwise specified, the term module will mean a left R-module. By a complex (C, d) of left R-modules we mean a graded module C = (Cn)n∈Z along with a morphism d = d : C → C of graded modules of degree −1 such that d ◦ d = 0. We also use the notation C = (C) but where d is of degree +1 and where we let Cn = C . Given a complex C we let Z(C) ⊂ C be Ker(d), let B(C) = Im(d) and let H(C) = Z(C)/B(C). If M and N are modules and C = (Ci) and D = (D ) are complexes, we form complexes denotedHom(M,D) andHom(C,N) whereHom(M,D) = Hom(M,D) and where Hom(C,N) = Hom(Ci, N). By a double complex of modules X we mean a bigraded module (X)(i,j)∈Z×Z along with morphisms d and d of bidegrees (1, 0) and (0, 1) respectively such that d◦d = 0, 2010 Mathematics Subject Classification.55U15,16E05,16E30, 18G15.