Let $(A,\mathfrak{m},k)$ be a Noetherian local ring. And $\hat{A}$ be the completion of $A$ with respect to $\mathfrak{m}$. It's known that $A\to \hat{A}$ is faithfully flat and $\hat{A}$ is $\mathfrak{m}$-adically complete.(can fail if $A$ is not Noetherian).
Question 1: The Kahler differential $\Omega^1_{\hat{A}/A}=0$?
Motivation: In the structure theorems of complete local rings, the complete local ring $\hat{A}$ is always a quotient of a formal power series over some ring/field $R[[x_1,...,x_n]]$. Say $\hat{A}=\frac{R[[x_1,...,x_n]]}{I}$. In $\Omega^1_{\hat{A}/A}$, $df=\sum\partial f_i dx_i$ and I believe that each $x_i+I$ is an image of $A\to \hat{A}$, so $dx_i=0$ and $\Omega^1_{\hat{A}/A}=0$.
Question 2: The diagonal map $\Delta_{\hat{A}/A}:\hat{A}\otimes_{A}\hat{A}\to \hat{A}$ agrees with the canonical map $M\otimes_A \hat{A}\to \hat{M}$ when $M=\hat{A}$? (note that $\hat{\hat{A}}=\hat{A}$)
Motivation: This is porbably ture, just need a confirmation from someone else. The definition of weakly étale morphism $f$ is both $f$ and $\Delta_f$ are flat, so I need to study diagonal morphism.
Question 3: Is the above diagonal morphism isomorphism?
Motivation: We know if $M$ is a finite $A$-module then $M\otimes_A \hat{A}\cong\hat{M}$. I can't tell if $A\to \hat{A}$ is finite in general, probabaly not. But there is a chance that $\hat{A}\otimes_{A}\hat{A}\cong\hat{\hat{A}}=\hat{A}$ and if the above conjecture is correct, then the diagonal morphism is isomorphic.
Question 4 (This is my ultimate goal): If $A$ is of characteristic $p$ and write Frobenius map $F_A:A\to A'$, further assume $F_A$ is finite free, is the relative Frobenius morphism $F_{\hat{A}/A}:A'\otimes_A \hat{A}\to \hat{A}'$ isomorphic?
Motivation: If $A\to\hat{A}$ is weakly étale, then it's true see tag 0F6W. As $A\to\hat{A}$ is already flat, it suffices to show $\Delta_{\hat{A}/A}=0$. Note that if the relative Frobenius is isomorphic (or just surjective), then $\Omega^1_{F_{\hat{A}/A}}=0$ and the map $\Omega^1_{A'\otimes_A \hat{A}/A'}\otimes \hat{A}'\to \Omega^1_{\hat{A}'/A'}$ is surjective, but it sends $d(a\otimes b)\otimes 1$ to $dab^p=0$, then we necessarily have $\Omega^1_{\hat{A}/A}=0$ . For another direction, I suspect that it agrees with the caonical map $M\otimes \hat{A}\to \hat{M}$ when $M=A'$.