Abstract: 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 K
generated by the trivial module k consists of all modules that
can be pieced together by extension from k and from
O^{n}(k) where O^{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
K, and in the subcategory 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 K can be reasonably given.
Even the self-equivalences of 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.