Application of Nakayama's Lemma

enigma182 · science forum beginner Joined: 25 Apr 2006 Posts: 5

Hi there,

I've been stuck with this problem for days now, and I just can't figure it out, maybe someone out there can help me. You can find the problem in Fernando Gouvea's lecture "Deformations of Galois Representations".

First of all, we fix a finite field k and a complete noetherian local ring A with residue field k. C shall be the category of complete noetherian local A-algebras with residue field k. Morphisms in this category are local homomorphisms, which induce the identity on k.

So far so good, now we take a representation of a profinite group G, let's say rho: G -> GL_n(k). If for an object R of C, we define pi: GL_n(R) -> GL_n(k) the induced map, we can consider possible liftings of rho to GL_n(R).

Now I define

C(rho) = {P in M_n(k) | P rho(g) = rho(g)P forall g in G}

and for a lifting rho':G -> GL_n(R)

C_R(rho') = {P in M_n(R) | P rho'(g) = P rho'(g) forall g in G}

Now Gouvea says, in the case that C(rho) = k, that by applying Nakayama's Lemma and considering the inclusion of R-modules R \subset C_R(rho'), you can find C_R(rho') = R.

I've really been trying and now I'm desperate for help.

I hope this post is understandable and somebody can help me.

Thanks anyway

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