Let $p,\ell$ be two different primes. Let $K$ be a finite field extension of $\mathbb{Q}_{\ell}$ and $ \bar{\rho}:G_{K}\to {\rm GL}_{n}(\mathbb{F}_p) $ be a continuous mod $p$ representation of the absolute Galois group $G_K$ of $K$. I believe it's well-known for experts that the corresponding framed deformation ring $ R^{\square}$ of $\bar{\rho}$ is flat over the ring $\mathbb{Z}_p$ of $p$-adic integers and of relative dimension $ n^{2} $. Can anyone tell me the specific literature that contains the details of the proof of this fact?
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2$\begingroup$ This is proved in Helm's paper arxiv.org/pdf/1605.00487. $\endgroup$– Satan's MinionCommented Sep 11 at 17:31
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As it is commented, this is essentially due to Helm (see proposition 4.2 and corollary 7.3 of this). Note that the ring $R_{q,n}$ in his paper is not the deformation ring. Let $I_K^{(\ell)}$ be the prime to $\ell$ part of the inertia. Then in proposition 4.2 he proves that the scheme parametrizing representations of (a discrete version of) the Weil group that are trivial on $I_K^{(\ell)}$ is flat, l.c.i and of the right dimension. Any such residual representation gives a closed point of this scheme and the completion of the local ring at this point gives the deformation ring. To reduce the general case to this case (of being trivial on $I_K^{(\ell)}$) you need to use a result of Choi, proposition 2.6 of this.
You can also look at theorem 2.5 of this paper of Shotton. Also, I think in the two dimensional case, everything should be contained in this paper of Pilloni. It is worth mentioning that in the $\ell=p$ case one still has flatness and lci but the right dimension is $n^2(1+[K:\mathbb{Q}_p])$. This is due to Böckle-Iyengar-Paškūnas.
