Negative cohomology and the endomorphism ring of the trivial module

Let k be a field of characteristic p>0 and let H finite group or scheme. We show that the negative Tate cohomology ring Hˆ?0(H,k) can realized as endomorphism trivial module in Verdier localization stable category kG-modules for G an extension H. This means some cases is local with infinitely generated radical square zero. stands stark contrast to known calculations which degree zero component group.