 If "G" is a finite group and "A" a "G"-module, then there is a natural map "N" from formula_1 to 
formula_2 taking a representative "a" to formula_3 (the sum over all "G"-conjugates of "a").The Tate cohomology groups formula_4 are defined by

If 
is a short exact sequence of "G"-modules, then we get the usual long exact sequence of Tate cohomology groups