Vakil's definition of smoothness -- what happens at non-closed points?

I cite 12.2.5 from FOAG of Vakil:

[...] For finite type schemes over $\displaystyle\overline{\mathbb{K}}$ (an algebraically closed field), the criterion gives a necessary condition for regularity, but it is not obviously sufficient, as we need to check regularity at non-closed points as well. [...]

so I undestand (and I agree) that the scheme-theoretic definition of smooth point, into the chapter 12, is bounded to closed points of schemes over a (generic) field $\mathbb{K}$.

But, by exercise 12.2.H, given an affine $\mathbb{K}$-scheme $X=\operatorname{Spec}\mathbb{K}[x_1,\dots,x_n]_{\displaystyle/(f_1,\dots,f_r)}$ of finite type, if the Jacobian matrix has maximal corank at all closed point of $X$ then it has maximal corank at all points; that is the definition 12.2.6 (and nextly the definition 12.6.2) must be meant restricting our attention to the closed points of the affine neighbourhoods (if exist) of $X$ (generic $\mathbb{K}$-scheme) of finite type.

If you are unsatisfactory of the previous definition(s): you can jump to chapter 21, paragraphs 1, 2 and 3!

Are you agree? Is it all clear?

EDIT: The numeration is refered to December 29 2015 FOAG version!

The Jacobian criterion does not work on non-closed $\mathbb{K}$-points of a $\mathbb{K}$-scheme locally of finite type; or it does not work as we know it!

Let $X$ be a scheme over a field $\mathbb{K}$, locally of finite type; that is: \begin{gather} \forall P\in X,\,\exists U\subseteq X\,\text{open,}\,t_1,\dots,t_n\,\text{indeterminates,}\,f_1,\dots,f_r\in\mathbb{K}[t_1,\dots,t_n]=R,\\(f_1,\dots,f_r)=I: P\in U\simeq\operatorname{Spec}R_{\displaystyle/I}\cong V(I)\subseteq\mathbb{A}^n_{\mathbb{K}}\,\text{affine closed subscheme} \end{gather} therefore \begin{equation} \forall P\in U\subseteq X,\,T_PX\cong T_PU\cong T_PV(I)\leq T_P\mathbb{A}^n_{\mathbb{K}}=T_{\mathfrak{m}_P}\mathcal{O}_{\mathbb{A}^n_{\mathbb{K}},P}=\left(\mathfrak{m}_{P\displaystyle/\mathfrak{m}_P^2}\right)^{\vee}\cong\kappa(P)^n \end{equation} where $\mathfrak{m}_P$ is the maximal ideal of the local ring $\mathcal{O}_{\mathbb{A}^n_{\mathbb{K}},P}$ and $\kappa(P)$ is the relavant residue field.

If $P$ is a closed point of $X$, then $P$ is a closed point in $U$; let $\mathfrak{p}$ be the maximal ideal of $R$ corresponding to $P$, and let $\varphi:\mathbb{K}[s_1,\dots,s_r]\to\mathbb{K}[t_1,\dots,t_n]$ the morphism of $\mathbb{K}$-algebras such that \begin{equation} \forall i\in\{1,\dots,r\},\,\varphi(s_i)=f_i; \end{equation} then: \begin{gather*} \operatorname{coker}\varphi=R_{\displaystyle/I},\\ \varphi^{*}:\mathbb{A}^n_{\mathbb{K}}\to\mathbb{A}^r_{\mathbb{K}},\\ \ker\varphi^{*}=V(I); \end{gather*} let $\varphi_0=\varphi_{\varphi^{*}(P)}:\mathbb{K}[s_1,\dots,_r]_{(s_1,\dots,s_r)}\to\mathbb{K}[t_1,\dots,t_n]_{\mathfrak{p}}$, by definition: \begin{equation} (d_P\varphi_0)^{\vee}:\left(T_O\mathbb{A}^r_{\mathbb{K}}\right)^{\vee}\to \left(T_P\mathbb{A}^n_{\mathbb{K}}\right)^{\vee} \end{equation} and in particular: \begin{equation} \left(T_PV(I)\right)^{\vee}=\operatorname{coker}(d_P\varphi_0)^{\vee}\Rightarrow T_PV(I)=\ker d_P\varphi_0; \end{equation} where $(d_P\varphi_0)^{\vee}$ is a $\mathbb{K}$-linear map and $T_P\mathbb{A}^n_{\mathbb{K}}\cong\kappa(P)^n\cong\mathbb{K}^m$ by Hilbert's (Strong) Nullstellensatz (see also the REMARK1).

By Hilbert's Basis Theorem, $\mathfrak{p}$ is a finite generated $\mathbb{K}$-vector space, therefore: \begin{gather} \mathfrak{p}=(e_1,\dots,e_m)\\ \forall i\in\{1,\dots,r\},\,(d_P\varphi_0)^{\vee}(\overline{s_i})=\overline{f_i}=\sum_{j=1}^ma_i^j\overline{e_j},\,\text{where:}\,a_i^j\in\mathbb{K}; \end{gather} but every $a_i^j$ no makes sense as the formal derivation of $f_i$ with respect to the element $e_j$ computed at $P$, unless $P$ is a closed $\mathbb{K}$-point of $X$! (Jump to UPDATE.)

Again: is it all clear?

I repeat, the definition 12.2.6 and it is completed by exercise 12.2.H, it is true that this definition is intricated and in apparence is unsatisfactory; but this definition is completely right and it works on $\mathbb{K}$-schemes locally of finite type.

REMARK1: If $P$ is not a closed point of $X$, that is a closed point of $\mathbb{A}^n_{\mathbb{K}}$, we can't apply the Hilbert (Strong) Nullstellensatz.

UPDATE: By a base change, we can define $\overline{\varphi}:\mathbb{F}[s_1,\dots,s_r]\to\mathbb{F}[t_1,\dots,t_n]$ where $\mathbb{F}=\kappa(P)$, we can repeat the same reasoning described for $\varphi$ and we can prove that: \begin{equation} T_P(V(I)/\operatorname{Spec}\mathbb{F})=\ker d_P\overline{\varphi}_0 \end{equation} where the notation is clear.

Because $P$ is a closed $\mathbb{F}$-point of $\mathbb{A}^n_{\mathbb{F}}$, then: \begin{equation} \exists\alpha_1,\dots,\alpha_n\in\mathbb{F}\mid\mathfrak{p}=(t_1-\alpha_1,\dots,t_n-\alpha_n) \end{equation} and therefore, via a formal Taylor series of the $f_i$'s \begin{gather} \forall i\in\{1,\dots,r\},\,(d_P\varphi_0)^{\vee}(\overline{s_i})=\overline{f_i}=\dots=\sum_{j=1}^n\frac{\partial f_i}{\partial t_j}\bigg|_{(t_1-\alpha_1,\dots,t_n-\alpha_n)}\left(\overline{t_j-\alpha_j}\right) \end{gather} that is $T_P(V(I)/\operatorname{Spec}\mathbb{F})$ is the kernel of the linear map from $\mathbb{F}^r$ to $\mathbb{F}^n$ described from the Jacobian matrix of the $f_i$'s with respect to $t_j$'s, with entries in $\mathbb{F}$ and valued in $P$.

REMARK2: In general $T_PV(I)$ and $T_P(V(I)/\operatorname{Spec}\mathbb{F})$ are not isomorphic as $\mathbb{F}$-vector spaces; see exercise 6.3 from Görtz and Wedhorn - Algebraic Geometry I, Schemes With Examples and Exercises.

EDIT: The enumeration is refered to December 29 2015 FOAG version.