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.
Keywords
Tate cohomology, Dyer–Lashof, cohomology operation, finite
group
