Algebra Seminar

We consider the stable category of $kG$-modules modulo projectives, where $G$ is a finite group and $k$ is a field of characteristic $p > 0$. The thick subcategory $\Cal K$ generated by the trivial module $k$ consists of all modules that can be pieced together by extension from $k$ and from $\Omega^n(k)$ where $\Omega^n(k)$ is the kernel of the $n^{th}$ boundary map in a complete resolution of $k$. There is a sense in which all cohomology takes place in $\Cal K$, and in the subcategory $\Cal K$ the module theory is reasonably well behaved. The varieties defined by ordinary cohomology measure the homological invariants of modules. A classification of the thick subcategories of $\Cal K$ can be reasonably given. Even the self-equivalences of $\Cal K$ can be characterized in a nice way. In this lecture I will try to survey some of the recent results in the area and present some examples to illustrate the points.