I've been working on understanding the proof of Fermat's last theorem and now focusing on the patching technique for modularity lifting. I found that the patching technique described in the paper Modularity lifting beyond the Taylor–Wiles method, i.e. Proposition 2.3 is helpful. However, I have come up with some difficulties on understanding certain parts of the proof.
My confusions mainly come from the following paragraph (At the bottom of Page 13 of the linked paper above):
Question 1: Why the image of $S_{\infty}$ in $\mathrm{End}_{\mathcal{O}}(X_{\infty})$ is contained in the image of $R_{\infty}$? Though the authors explained in the parenthese, I still found that hard to understand, and hoping for more details.
Question 2: Why does the inclusion of image (as in the Question 1) implies that $X_{\infty}$ is a finite $R_{\infty}$-module?
Question 3: Why the fact that "$S_{\infty}$ is formally smooth over $\mathcal{O}$" implies that we can choose a homomorphism $\iota: S_{\infty} \rightarrow R_{\infty}$ in $\mathsf{C}_{\mathcal{O}}$? Where does such a homomorphism come from. I have gone through Chapter 11 (Formal Smoothness) of Matsumura's book Commutative Algebra, yet haven't found similar arguments.
Sorry for such a post with so detailed questions and the lack of explanations in the post (although the contexts are clear in the linked paper). Thank you all for answering and commenting!
Any more detailed references are welcome as well!