Timeline for $\Omega^1_{B/A}=0$ implies $A\subset B$ unramified

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Feb 1, 2018 at 19:59	comment	added	nfdc23		Apply scalar extension by $A\to A/\mathfrak{p}$ to reduce to the case $A=k$ is a field. Now $B$ is a finite local $k$-algebra. Dropping the locality assumption, $B$ is finite over $k$ with $\Omega^1_{B/k}=0$ and you want to conclude that the local factor rings of $B$ are fields separable over $k$. For this purpose you can extend scalars to $\overline{k}$ so $k$ is alg. closed, and then pass back to the local case. You want $B=k$. If not, it admits as a quotient $B'=k[\epsilon]/(\epsilon)^2$, so $\Omega^1_{B'/k}\ne 0$ by hand, but this is a quotient of $\Omega^1_{B/k}=0$, contradiction.
Feb 1, 2018 at 19:06	history	asked	Vincenzo Zaccaro	CC BY-SA 3.0