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?