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In order to show that two definitions of smoothness in algebraic geometry coincide I would like to see a direct proof of the following fact:

Let $R$ be a commutative ring and $f_1,\dots,f_k\in R[x_1,\dots,x_n]$ polynomials such that the Jacobian $J=(\partial f_{j}/\partial x_i)_{i,j}$ has total rank in every residue field $k(P)$ for $P\in\text{Spec}(R)$.

Then the following $R$-algebra is flat $$A=R[x_1,\dots,x_n]/(f_1,\dots,f_k).$$

The idea is that $\text{Spec }A\rightarrow \text{Spec }R$ should be a smooth morphism under this hypothesis and smooth morphisms are flat by definition.

Edit: The two definitions that I am trying to compare are the following

  1. (Gortz and Wedhorne Algebraic Geometry I)

    A morphism $f:X\rightarrow Y$ of schemes is smooth of relative dimension $d$ if $\forall x\in X$ there exists an open neighborhoods $U$ of $x$ such that $f(U)\subseteq V$, $V=\text{Spec}(R)$ and an open immersion $$j:U\rightarrow R[T_1,\dots,T_n]/(f_1,\dots, f_{n-d})$$ of $R$-schemes for suitable $n$ and $f_i$ such that the Jacobian matrix $$J_{f_1,\dots,f_{n-d}(x)}=\left ( \frac{\partial f_i}{\partial T_j}(x)\right )\in M_{(n-d)\times n}(\kappa(x))$$ has rank $n-d$.

  2. (Qing Liu, Hartshorne, Wikipedia)

    A morphism $f:X\rightarrow Y$ is called smooth if it is locally of finite presentation, flat and for each $y\in Y$ the fiber $X_y$ is a smooth $\kappa(y)$-variety.

Even though this definitions are stated in full generality I think it can be wise to restrict them to the locally noetherian case (otherwise we can't speak about regular points for example). Maybe later the locally noetherian case implies the finite presentation case by some base change trick.

The implication 2.$\implies$ 1. is a bit hard but is based in the fact that if $f:X\rightarrow Y$ is a smooth morphism then it is locally a complete intersection, that is, locally in $x$ we can decompose $f$ as a regular immersion $X\rightarrow \mathbb{A}^{n+1}_Y$ followed by a projection $\mathbb{A}^{n+1}_Y\rightarrow Y$ (the sketch is in Liu Remark 3.19 and the references therein).

Now for 1.$\implies$ 2., it is clear that 1. implies locally of finite presentation and smooth fibers. The problem was the flatness part and that's the reason of the question.

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Walter Simon
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    Have you made any attempts of your own at showing this, or done any research about it? If so, please include them, and if not, please consider doing so. Questions without demonstrated effort from the asker usually do not receive much engagement here. One place you may wish to look for information about this is Stacksproject chaper 10 - lemmas tagged 00SV and 00T7 probably combine to give you what you want. – KReiser Feb 04 '19 at 21:20
  • @KReiser Thank you for the references! I check them and indeed the proof is contained in there. I was expecting it to have an easier or more direct proof but at least it reassures me that there is a reference. – Walter Simon Feb 07 '19 at 14:04
  • @WalterSimon, please, is there any restriction on $k$ (= the number of polynomials)? I mean, is it ok to have $k > n$? I think yes, but wish to be sure. – user237522 Feb 07 '20 at 02:11
  • @KReiser, if I am not wrong, it is possible to take $k \leq n$ only, so the case $k > n$ is not relevant; am I right? Thank you. – user237522 Feb 09 '20 at 11:43
  • @KReiser, I really apologize for my trivial question (trivial for you, but not trivial for me. Unfortunately, I am still not familiar with algebraic geometry). I am confused, since in the above question it seems ok to allow $k > n$, while in your references it seems (if I am not wrong) that necessarily $k \leq n$. Thank you very much. – user237522 Feb 09 '20 at 12:11

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