Let $f: X \to Y$ be a morphism of schemes. Then $f$ is called (EGA IV.17) formally smooth if whenever $T$ is an affine $Y$-scheme and $T'$ a closed subscheme of $T$ defined by a nilpotent ideal (it's sufficient to consider ideals of square zero), any $Y$-morphism $T' \to X$ can be extended to $T \to X$. $f$ is called formally étale if such a lifting always exists and is unique. $f$ is called formally unramified if such a lifting is unique. I am trying to understand what this should mean geometrically. I don't have a specific question in mind, but I don't see much geometric intuition in the EGA definition, and would appreciate any explanation.
For the geometric intuition that I see, suppose $X,Y$ are algebraic varieties over an algebraically closed field $k$. Let $y \in Y$ be the image of $x \in X$ (consider only closed points for simplicity). Then, formal smoothness states that any tangent vector (i.e. a map $\mathrm{Spec} k[\epsilon]/\epsilon^2 \to Y$) to $y$ lifts to a tangent vector of $x$. Formal unramifiedness states that any tangent vector can lift in only one way to a tangent vector of $x$. Formal étaleness implies that the map on tangent spaces is an isomorphism.
I don't know whether these tangent space conditions are equivalent to the formal definitions for varieties. I believe it is true for formal unramifiedness, at least.
(For schemes of finite type over a noetherian scheme, formal unramifiedness is equivalent to the relative sheaf of differentials being zero. This can be checked on the fibers. So let $f: X \to Y$ be a morphism of $k$-varieties. Then $f$ is formally unramified if $\Omega_{X/Y} = 0$. The exact sequence $f^*{\Omega_{Y/k}} \to \Omega_{X/k} \to \Omega_{X/Y} \to 0$ shows that formal unramifiedness holds iff the map on the cotangent spaces is surjective, i.e. the map on the tangent spaces is injective.)
Question: Am I correct in thinking of a formally smooth map as a submersion, a formally unramified map as an immersion (in the sense of differential geometry), and a formally étale morphism as a local isomorphism (again, using neighborhoods smaller than the Zariski neighborhoods---I understand when the residue fields are the same étaleness implies that the map on the completions of the local rings are the same)? Is this geometric intuition reasonable?
(This intuition will, of course, ignore the distinction between plain unramifiedness/smoothness/étaleness and the formal analog, which only exists when one leaves noetherian and finite-type hypotheses.)
Question' Are the tangent space remarks sufficient for etaleness/smoothness/unramifiedness in the variety case?
