Stable module category

In representation theory, the stable module category is a category in which projectives are "factored out."
Definition.
Let "R" be a ring. For two modules "M" and "N" over "R", define formula_1 to be the set of "R"-linear maps from "M" to "N" modulo the relation that "f" ~ "g" if "f" − "g" factors through a projective module. The stable module category is defined by setting the objects to be the "R"-modules, and the morphisms are the equivalence classes formula_1.
Given a module "M", let "P" be a projective module with a surjection formula_3. Then set formula_4 to be the kernel of "p". Suppose we are given a morphism formula_5 and a surjection formula_6 where "Q" is projective. Then one can lift "f" to a map formula_7 which maps formula_4 into formula_9. This gives a well-defined functor formula_10 from the stable module category to itself.
For certain rings, such as Frobenius algebras, formula_10 is an equivalence of categories. In this case, the inverse formula_12 can be defined as follows. Given "M", find an injective module "I" with an inclusion formula_13. Then formula_14 is defined to be the cokernel of "i". A case of particular interest is when the ring "R" is a group algebra.
The functor Ω−1 can even be defined on the module category of a general ring (without factoring out projectives), as the cokernel of the injective envelope. It need not be true in this case that the functor Ω−1 is actually an inverse to Ω. One important property of the stable module category is it allows defining the Ω functor for general rings. When "R" is perfect (or "M" is finitely generated and "R" is semiperfect), then Ω("M") can be defined as the kernel of the projective cover, giving a functor on the module category. However, in general projective covers need not exist, and so passing to the stable module category is necessary.
Connections with cohomology.
Now we suppose that "R = kG" is a group algebra for some field "k" and some group "G". One can show that there exist isomorphisms
for every positive integer "n". The group cohomology of a representation "M" is given by formula_16 where "k" has a trivial "G"-action, so in this way the stable module category gives a natural setting in which group cohomology lives. 
Furthermore, the above isomorphism suggests defining cohomology groups for negative values of "n", and in this way one recovers Tate cohomology.
Triangulated structure.
An exact sequence
in the usual module category defines an element of formula_18, and hence an element of formula_19, so that we get a sequence
Taking formula_12 to be the translation functor and such sequences as above to be exact triangles, the stable module category becomes a triangulated category.