# How to prove this algebra is flat?

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!

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