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 [math]\displaystyle{ \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]\displaystyle{ \underline{\mathrm{Hom}}(M,N) }[/math].
Given a module M, let P be a projective module with a surjection [math]\displaystyle{ p \colon P \to M }[/math]. Then set [math]\displaystyle{ \Omega(M) }[/math] to be the kernel of p. Suppose we are given a morphism [math]\displaystyle{ f \colon M \to N }[/math] and a surjection [math]\displaystyle{ q \colon Q \to N }[/math] where Q is projective. Then one can lift f to a map [math]\displaystyle{ P \to Q }[/math] which maps [math]\displaystyle{ \Omega(M) }[/math] into [math]\displaystyle{ \Omega(N) }[/math]. This gives a well-defined functor [math]\displaystyle{ \Omega }[/math] from the stable module category to itself.
For certain rings, such as Frobenius algebras, [math]\displaystyle{ \Omega }[/math] is an equivalence of categories. In this case, the inverse [math]\displaystyle{ \Omega^{-1} }[/math] can be defined as follows. Given M, find an injective module I with an inclusion [math]\displaystyle{ i \colon M \to I }[/math]. Then [math]\displaystyle{ \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−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
- [math]\displaystyle{ \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]\displaystyle{ \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
- [math]\displaystyle{ 0 \to X \to E \to Y \to 0 }[/math]
in the usual module category defines an element of [math]\displaystyle{ \mathrm{Ext}^1_{kG}(Y,X) }[/math], and hence an element of [math]\displaystyle{ \underline{\mathrm{Hom}}(Y, \Omega^{-1}(X)) }[/math], so that we get a sequence
- [math]\displaystyle{ X \to E \to Y \to \Omega^{-1}(X). }[/math]
Taking [math]\displaystyle{ \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
References
- J. F. Carlson, Lisa Townsley, Luis Valero-Elizondo, Mucheng Zhang, Cohomology Rings of Finite Groups, Springer-Verlag, 2003.
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