In mathematics, especially representation theory, the '''stable module category''' is a quotient of a module category in which projectives are "factored out."
== Definition == Let ''R'' be a ring. For two modules ''M'' and ''N'' over ''R'', define <math>\underline{\mathrm{Hom}}(M,N)</math> 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 <math>\underline{\mathrm{Hom}}(M,N)</math>.
Given a module ''M'', let ''P'' be a projective module with a surjection <math>p \colon P \to M</math>. Then set <math>\Omega(M)</math> to be the kernel of ''p''. Suppose we are given a morphism <math>f \colon M \to N</math> and a surjection <math>q \colon Q \to N</math> where ''Q'' is projective. Then one can lift ''f'' to a map <math>P \to Q</math> which maps <math>\Omega(M)</math> into <math>\Omega(N)</math>. This gives a well-defined functor <math>\Omega</math> from the stable module category to itself.
For certain rings, such as Frobenius algebras, <math>\Omega</math> is an equivalence of categories. In this case, the inverse <math>\Omega^{-1}</math> can be defined as follows. Given ''M'', find an injective module ''I'' with an inclusion <math>i \colon M \to I</math>. Then <math>\Omega^{-1}(M)</math> 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<sup>−1</sup> 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<sup>−1</sup> 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 : <math>\underline{\mathrm{Hom}}(\Omega^n(M), N) \cong \mathrm{Ext}^n_{kG}(M,N) \cong \underline{\mathrm{Hom}}(M, \Omega^{-n}(N))</math> for every positive integer ''n''. The group cohomology of a representation ''M'' is given by <math>\mathrm{H}^n(G; M) = \mathrm{Ext}^n_{kG}(k, M)</math> 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 : <math> 0 \to X \to E \to Y \to 0</math> in the usual module category defines an element of <math>\mathrm{Ext}^1_{kG}(Y,X)</math>, and hence an element of <math>\underline{\mathrm{Hom}}(Y, \Omega^{-1}(X))</math>, so that we get a sequence : <math> X \to E \to Y \to \Omega^{-1}(X).</math> Taking <math>\Omega^{-1}</math> to be the translation functor and such sequences as above to be exact triangles, the stable module category becomes a triangulated category.
== See also == * Stable homotopy theory
== References == * J. F. Carlson, Lisa Townsley, Luis Valero-Elizondo, Mucheng Zhang, ''Cohomology Rings of Finite Groups'', Springer-Verlag, 2003.
Category:Category theory Category:Representation theory Category:Homotopy theory