Hi,


Let $S = R[T\_1,\dots,T\_n]/(f\_1,\dots,f\_r)$ where $\det(\partial f\_i/\partial T\_j)\_{i,j=1,\dots,r}\in S^\times$. Then $S$ is flat over $R$. How to prove it?
I am not looking for an answer like: "$Spec(S)$ is smooth over $Spec(R)$, hence flat.".


Thanks!



How to prove this algebra is flat?
The algebra $S$ is etale, i.e., smooth of relative dimension $0$. A proof, that $S$ is
flat over $R$ can be found in


www.math.purdue.edu/~dvb/preprints/etale.pdf‎,


see Proposition $1.3.5$ on page $17$. The proof uses induction, and the
following lemma: Let $S$ be an $R$-algebra. Suppose that $M$ is an $S$-module,
$f\in S$ an element such that multiplication by f is injective on $M \otimes k(m)$ for all
$m \in Max (R)$, and $M$ is flat over $R$. Then $M/fM$ is flat over $R$.


I hope this is more than just $Spec(S)$ is smooth over $Spec(R)$, hence flat.


