I was recently watching the YouTube video lectures of Richard Borcherds about the Weil conjectures. In Lecture 7 (https://www.youtube.com/watch?v=yDR7_d5hI5Q) he introduces étale morphisms of schemes $f:X\rightarrow Y$. There, he first introduces these morphisms for affine varieties over algebraically closed fields as those morphisms "whose induced maps on completions of local rings $\widehat{O}_{Y,f(x)}\rightarrow\widehat{O}_{X,x}$ are isomorphisms."
To move on to general schemes, he first introduces formally étale morphisms. This he does by mentioning that characterizing the map of completions of the local rings involves a bunch of lifting procedures, since ring completions are inverse limits. Then he simply states the formal dual of the definition of formally étale morphisms of rings, which indeed consists of a lifting problem.
However, now comes my question, the definition of these formally étale morphisms of rings involves nilpotent ideals. But the lifting of the morphisms in the characterization of the maps of ring completions involves the maximal ideal, which I cannot see to be nilpotent (unless one is working over Artin rings, which in general is not the case). So how do these ideas concretely line up?
Note: I understand the definition in terms of nilpotent ideals when starting from the formal geometry point-of-view (with tangent cones and infinitesimal thickenings).
Best Answer
Given a pair $(R,I)$ of a ring and an ideal, the completion of R with respect to 𝐼 is an inverse limit over the quotients $R/I^n$ with the usual projection morphisms $R/I^{n+1}\to R/I^n$ giving the directed system. Note that the image of the ideal 𝐼 in each of these quotient rings is indeed nilpotent, even though it may not be in the final completion. Even more, the ideal $I^n$ is a square-zero ideal in $R/I^{n+1}$. Think of the formal methods as trying to inductively work with these intermediate steps in building the completion of the rings.