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In group representation theory, one attempts to explain and classify (some of) the modules over the group ring $k[G]$, for some field $k$. In group cohomology, one develops the machinery of the cohomology of groups out of the category of modules over the ring $\mathbb{Z}[G]$. In both cases, one has tensor products, restriction maps, induced modules, etc, and all these constructions look very similar in both cases, with $k$ replaced by $\mathbb{Z}$ in group cohomology.

My question is, might one develop a theory of cohomology of $G$-representations over a field $k$ analogous to the theory of cohomology of $G$-modules? Essentially, one would take a representation $M$, take an injective resolution of $M$ in the category of $k[G]$-modules, then find the fixed elements to get the cohomology. I believe this would be $\mathrm{Ext}_{k[G]}(k,M)$, where $k$ is given the trivial $G$-action.

Given that we know a lot about the structure of finite-dimensional representations of finite groups (character tables), what would the character table tell us about the cohomology? What might the cohomology tell us about the character table? What happens if we consider representations of infinite groups, such as Lie groups, algebraic groups, or Galois groups?

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This only makes sense for modular representation theory, because otherwise you have a semisimple category. Some googling suggests that this has been studied, and is in fact an important tool; I found www.math.uchicago.edu/~allanaa/topic.pdf. –  Akhil Mathew Aug 23 '10 at 15:48
This is just group cohomology with fixed "coefficients", so it is perfectly interesting, even in characteristic 0 with various infinite groups, especially when topology of field and group interact (such as Galois groups). It is ubiquitous for Galois deformation theory (for characteristic 0, it is useful when relating $p$-adic Hodge theory to deformation theory), and shows up everywhere that you'd expect to meet non-semisimple objects. –  BCnrd Aug 23 '10 at 16:06
Dear David: this is a topic about which your professors (you can guess which ones) will be able to tell you a lot in person when you return to campus in the fall. –  BCnrd Aug 23 '10 at 16:09
As Brian says, one routinely replaces $\mathbb Z$ by $k$ in group cohomology calculations, and uses these calculations as a tool in many different representation-theoretic contexts. Akhil Mathew's comment is relevant when studying reps. of finite groups in char. zero, but there are lots of contexts in which one studies the representation theory of infinite groups, in char. zero and in positive char., in which group cohomology (along with its close cousins, such as Lie algebra cohomology) is an indispensable tool. –  Emerton Aug 23 '10 at 16:36
@Akhil: I kind of want to accept your answer for its link, though unfortunately you can't accept comments :\ –  David Corwin Aug 23 '10 at 17:49
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1 Answer

You have already used the following fact in your statement:

For every representation $\rho: G\to \mathrm{Aut}_R(A)$ ($A$ an $R$-module) we can give $A$ the structure of an $R[G]$-module by $g\cdot a = \rho_g(a)$. Conversely, any torsion free $R[G]$-module $A$ admits a representation $\rho: G \to \mathrm{ Aut}_R(A)$ given by $\rho_g(a) := g\cdot a$.

So studying $R[G]$-modules IS studying representations of $G$. My answer would be if you want to study representations study $R[G]$-modules. $R[G]$-modules aren't like a cure-all, because they don't give you an action to start with. Does anyone know how to classify actions?

  1. Studying a fixed $R[G]$-module is the same as studying a fixed representation. So it won't tell you anything about the character tables which involves different representations.
  2. You can still use $R[G]$-modules to study representations of infinite groups and galois groups. You can use it to derive properties about galois groups acting on roots of unity or torsion points of elliptic curves. http://en.wikipedia.org/wiki/Kummer_theory
  3. The wikipedia article http://en.wikipedia.org/wiki/Group_cohomology tells you that the Ext construction from resolutions is the same as the "standard" construction of group cohomology by cochain complexes.

I'm not sure if my answer was helpful. Hopefully it saved you some time. I don't think I understood what you meant about "taking the fixed elements to get the cohomology".

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a. I know that a representation over $R$ is an $R[G]$-module - my main question was $\mathbb{Z}[G]$ (which comes up in discussing group cohomology, such as in class field theory) versus $k[G]$ (the ring studied in representation theory - at least in representation theory over vector spaces!). re 2: How does representation theory come up there? I know how group cohomology applies to Kummer theory and elliptic curves (or maybe this is simply what you mean) –  David Corwin Aug 23 '10 at 17:42
re 3: I know this and mentioned it in my question. "Taking fixed elements": I mean take a representation $V$, then take $V^G := \{v \in V: gv=v \forall g \in G\}$. Note that this gives a functor from $k[G]$-mod to $k$-mod. –  David Corwin Aug 23 '10 at 17:42
Dear Davidac897: not knowing your background in homological algebra, do you know that for a $k[G]$-module, its cohomology as a $\mathbf{Z}[G]$-module naturally coincides with its cohomology as a $k[G]$-module (in the sense of isomorphism of $\delta$-functors initially defined on different categories)? It is analogous to how the sheaf cohomology on a ringed space $(X,A)$ can't tell if you use derived functors on the abelian category of $A$-modules or underlying abelian sheaf category. –  BCnrd Aug 23 '10 at 23:10
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