Let $k$ be a field. I am unable to find a precise definition of essentially smooth $k$ schemes, but I will stick to this definition below, since this is exactly what I need:


Definition: A $k$-scheme $X$ is called essentially smooth if it is a filtered projective limit $X=\underset{\longleftarrow}{\rm lim}\ X\_i $ of **smooth** $k$-schemes $X\_i$ such that all transition maps $\phi\_{ij}:X\_i\to X\_j$ are affine and **etale**. 


**Q1**: Are essentially smooth schemes is noetherian? (just like Henselisation of the local ring of a point in a variety). 


**Q2**:What if one drops conditions '$X\_i$ smooth' and/or '$\phi\_{ij}$ etale' from the above definition? 



Are essentially smooth schemes noetherian?
This answer is essentially the same as the comment above. Let $d$ be any positive integer that is prime to the characteristic of $k$. Let $R$ be the $k$-algebra,
$$
R=k[x\_0,x\_1,x\_2,\dots,x\_n,\dots]/\langle x\_0x\_1 - 1, x\_2^d - x\_1, x\_3^d - x\_2,\dots, x\_{n+1}^d-x\_n,\dots \rangle.
$$
For every integer $n$, define $R\_n\subset R$ to be the $k$-subalgebra generated by $x\_0,x\_1,\dots,x\_n$. Then $R\_n$ is isomorphic to $k[x\_n,x\_n^{-1}]$, which is a smooth $k$-algebra. Moreover, the transition map $R\_n\to R\_{n+1}$ is the same as
$$
f\_n : k[x\_n,x\_n^{-1}] \to k[x\_{n+1},x\_{n+1}^{-1}], \ \ f(x\_n) = x\_{n+1}^d.
$$
This is étale since $d$ is prime to the characteristic. The ring $R$ is not Noetherian


