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In [[representation theory]], the '''stable module category''' is a category in which projectives are "factored out."
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== Definition ==
Let ''R'' be a [[ring (mathematics)|ring]]. For two [[module (mathematics)|module]]s ''M'' and ''N'', 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''&nbsp;~&nbsp;''g'' if ''f''&nbsp;&minus;&nbsp;''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 (algebra)|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 algebra]]s, <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 ring|perfect]] (or ''M'' is [[finitely generated module|finitely generated]] and ''R'' is [[semiperfect ring|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 (mathematics)|field]] ''k'' and some [[group (mathematics)|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]]

Latest revision as of 08:59, 18 June 2014

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