Tate cohomology in commutative algebra.
J.P.C.Greenlees
Journal Pure and Applied Algebra (to appear 1994)

\begin{abstract}
We construct local Tate cohomology groups  of an $A$-module $M$
at a finitely generated ideal $I$ by splicing together the Grothendieck
local cohomology groups  with the local homology groups of
Greenlees-May. We give two quite different means of calculating them,
show they vanish on $I$-free modules and prove that many elements
of $A$ act invertibly. These local Tate groups have applications in
the study of completion theorems and their duals in equivariant
topology.
\end{abstract}