Let be the field with elements, and let be a finite group. By exhibiting an –operad action on for a complete projective resolution of the trivial –module , we obtain power operations of Dyer–Lashof type on Tate cohomology . Our operations agree with the usual Steenrod operations on ordinary cohomology . We show that they are compatible (in a suitable sense) with products of groups, and (in certain cases) with the Evens norm map. These theorems provide tools for explicit computations of the operations for small groups . We also show that the operations in negative degree are nontrivial.
As an application, we prove that at the prime these operations can be used to determine whether a Tate cohomology class is productive (in the sense of Carlson) or not.
Martin Langer. "Dyer–Lashof operations on Tate cohomology of finite groups." Algebr. Geom. Topol. 12 (2) 829 - 865, 2012. https://doi.org/10.2140/agt.2012.12.829