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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?

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Try to solve finite system of formal power series eqns in several formal power series unknowns (with constant term 0, over a field) via succ. approx. That's where square-zero business comes from, and linear part matrix (= Jacobian) being injective, surjective, or isom is what controls whether can lift solns from each stage to next and whether uniquely or not. In view of submersion/immersion/inv.fn. thms in diff. geom., one arrives at Grothendieck's notions. For varieties over $k$, tangential criteria OK in smooth case at $k$-points (as then complete local ring is power series ring over $k$). –  BCnrd Nov 14 '10 at 18:57
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In the final sentence of previous comment, "smooth case" meant "for smooth varieties". Anyway, real content is not the definitions, but rather that for locally finitely presented maps these formal criteria imply Zariski-local descriptions which look exactly like differential geometry (formally etale implies Zariski-locally looks like open in Spec(($R[t]/(f))[1/f']) \rightarrow $ Spec($R$) for some monic $f \in R[t]$ and hence flat by inspection, formally smooth implies Zariski-locally etale over an affine space, and formally unramified implies "etale-locally" a closed immersion). –  BCnrd Nov 14 '10 at 19:04
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Dear Akhil: Solving formal power series by succ. approx. uses square-zero steps, so saying "Hensel's Lemma" is too fancy. Also, "Neron Models" was written by 3 people, not 1. Equivalence with "primitive element thm" formulation is deep: requires Zariski's Main Thm. Need ZMT to get to open in something finite in order to "lift" (via Nakayama) the "primitive element thm" description at a point in the fiber to a description on a Zariski-neighborhood in the total space. See EGA IV$_4$, 18.4.6(ii) (beware (i) is false in formally etale case, as I noted elsewhere on MO with counterexample). –  BCnrd Nov 14 '10 at 20:34
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Dear Akhil: It should also be emphasized that this "primitive element theorem" description of etale maps is the only way (that I can think of offhand) to prove that locally finitely presented and formally etale morphisms are actually flat, or even that etale maps as defined via the Jacobian criterion in "Neron Models" are flat. Of course one can cheat and impose flatness in the definition, but that just makes the carpet pop up in another corner of the room when it comes to actually verifying the definition in abstract situations. –  BCnrd Nov 14 '10 at 20:39
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Akhil, the 1st paragraph of their pf of flatness of smooth maps is exactly what I said: smooth is etale over affine space, so one reduces to etale case, and in etale case one stares at local structure thm for etale maps (2.3/3), which is exactly 18.4.6(ii) in EGA IV$_4$, whose proof absolutely uses ZMT (in the finitely presented case). In "Neron Models" they state general finite type case of ZMT (18.12.13 in EGA IV$_4$), but beware that the pf in this generality uses the local structure thm for etale maps. Stick to f. presented ZMT to avoid circularity. Cannot avoid ZMT. That is the engine. –  BCnrd Nov 14 '10 at 23:59
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