Balance with Unbounded Complexes

TL;DR

This paper introduces a new method for constructing homology groups in cases where traditional spectral sequences fail, particularly in Tate (co)homology, and provides an elementary proof of the balance property.

Contribution

It presents an alternative approach to defining homology groups when spectral sequences are uninformative, specifically addressing Tate (co)homology balance.

Findings

New construction of homology groups when spectral sequences vanish.

Elementary proof of the balance property of Tate homology and cohomology.

Addresses limitations of spectral sequences in unbounded complexes.

Abstract

Given a double complex $X$ there are spectral sequences with the $E_2$ terms being either H$_I$ (H$_{II}(X))$ or H$_{II}($H$_I (X))$. But if $H_I(X)=H_{II}(X)=0$ 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 H$_I(X)=$H$_{II}(X)=0$. With this result we give a new and elementary proof of balance of Tate homology and cohomology.