# 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, define ${\displaystyle {\underline {\mathrm {Hom} }}(M,N)}$ 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 ${\displaystyle {\underline {\mathrm {Hom} }}(M,N)}$.

Given a module M, let P be a projective module with a surjection ${\displaystyle p\colon P\to M}$. Then set ${\displaystyle \Omega (M)}$ to be the kernel of p. Suppose we are given a morphism ${\displaystyle f\colon M\to N}$ and a surjection ${\displaystyle q\colon Q\to N}$ where Q is projective. Then one can lift f to a map ${\displaystyle P\to Q}$ which maps ${\displaystyle \Omega (M)}$ into ${\displaystyle \Omega (N)}$. This gives a well-defined functor ${\displaystyle \Omega }$ from the stable module category to itself.

For certain rings, such as Frobenius algebras, ${\displaystyle \Omega }$ is an equivalence of categories. In this case, the inverse ${\displaystyle \Omega ^{-1}}$ can be defined as follows. Given M, find an injective module I with an inclusion ${\displaystyle i\colon M\to I}$. Then ${\displaystyle \Omega ^{-1}(M)}$ 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

${\displaystyle {\underline {\mathrm {Hom} }}(\Omega ^{n}(M),N)\cong \mathrm {Ext} _{kG}^{n}(M,N)\cong {\underline {\mathrm {Hom} }}(M,\Omega ^{-n}(N))}$

for every positive integer n. The group cohomology of a representation M is given by ${\displaystyle \mathrm {H} ^{n}(G;M)=\mathrm {Ext} _{kG}^{n}(k,M)}$ 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

${\displaystyle 0\to X\to E\to Y\to 0\,}$

in the usual module category defines an element of ${\displaystyle \mathrm {Ext} _{kG}^{1}(Y,X)}$, and hence an element of ${\displaystyle {\underline {\mathrm {Hom} }}(Y,\Omega ^{-1}(X))}$, so that we get a sequence

${\displaystyle X\to E\to Y\to \Omega ^{-1}(X).\,}$

Taking ${\displaystyle \Omega ^{-1}}$ to be the translation functor and such sequences as above to be exact triangles, the stable module category becomes a triangulated category.