On the notion of order in the stable module category

Abstract

The notion of order in triangulated categories, as introduced by Schwede, is investigated in the case of the stable category of <i>kG</i>-modules, where <i>k</i> is a field of characteristic <i>p</i> and <i>G</i> is a finite group. For Tate cohomology classes <i>ζ</i> of even degree, we obtain bounds on the <i>ζ</i>-order which are similar to corresponding results on the <i>p</i>-order in the stable homotopy category. <br /> On our way we introduce a power operation <i>P₁</i> on Tate cohomology which serves as an obstruction for the <i>ζ</i>-order to be larger than its minimal possible value. Furthermore, it enables us to compute certain higher Massey products explicitly.