ac.commutative algebra - Formally étale at all primes does not imply formally étale.

Using the module of Kähler differentials, it is easy to show that $Rto S$ is formally unramified if and only if the induced maps $Rto S_{mathfrak{p}}$ are formally unramified for all primes $mathfrak{p}subset S$.

Consider a presentation of $S$ over $R$ as $R[X]/I$ in generators and relations, where $R[X]:=R[X_m]_{min M}$ is a polynomial ring in a possibly infinite family of indeterminates indexed by $M$, and $Isubset R[X]$ is an ideal. Fix a family of generators of $I=(F_j)_{jin J}$ indexed by $J$, again not necessarily finite.

It is enough to show that $Rto S$ is formally smooth. This is equivalent to showing that there exists a morphism of $R$-algebras that is a splitting for the canonical projection $pi:R[X]/I^2 to R[X]/I=S$, which will necessarily be unique because $Rto S$ is formally unramified.

Let $overline{X}_m$ denote the image of $X_m$ in $R[X]/I^2$. We must find elements $delta_min I/I^2$ such that $(forall jin J)F_j(X_m + delta_m)=0$. We rewrite this using Taylor's formula as $$bar{F}_j+ sum_{min M}overline{frac{partial F_j}{partial X_m}}delta_m=0.$$

Rearranging, we get a system of equations indexed by $J$
$$(*)_{jin J} qquad sum_{min M}overline{frac{partial F_j}{partial X_m}}delta_m=-overline{F}_j.$$

We wish to find a unique solution for this system in the $delta_m$. Since $Omega_{S/R}=0$, each $dX_min Omega_{R[X]/R}$ is an $S$-linear combination $dX_m=s_{m,1}dF_{j_{m,1}}+cdots + s_{m,h_m}dF_{j_{m,h_m}}$. If we use the $s_{m,k}$ as coefficients to form $S$-linear combinations of the equations $(*)_{j_k}$, for each $m$, we get an equation of the form $$(**)_m qquad delta_m=-(s_{m,1}overline{F}_{j_{m,1}}+cdots + s_{m,h_m}overline{F}_{j_{m,h_m}}).$$

Showing that these define solutions for all of the equations $(*)_j$ is not immediate, but it is a local question on $S$. However, our local rings $S_{mathfrak{p}}$ are all formally étale, so the local conditions are satisfied. Then this proves the global claim.

(Note: This is not my proof. I've paraphrased the proof communicated to me by Mel Hochster.)

Edit: Fixed LaTeX using Scott's suggestion.