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The following is definition 12.2.6 in Vakil's notes.

A $k$-scheme is $k$-smooth of dimension $d$, or smooth of dimension $d$ over $k$, if it is pure dimension $d$, and there exists a cover by affine open sets $\operatorname{Spec} k[x_1,\dots , x_n]/(f_1,\dots, f_r)$ where the Jacobian matrix has corank $d$ at all points.

The Jacobian matrix at a point $p$ is the usual matrix of partials evaluated at $p$.

At closed points, the quotient ring is a field, so we have a map of vector spaces and the usual definition of rank applies. For a general prime $P$, what is meant here? For Vakil, a ring element $f\in R$ has the value $f\in R/P$ at the prime $P$. But in this general case we have only a map of domains, not fields, and its not clear to me what is meant by corank here.

One way to interpret this is that certain minors vanish, but I'm not sure if that was what Vakil intended. He explicitly defines "corank" as dimension of the cokernel.

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    $\begingroup$ Maybe he considers the field $\kappa(P) = Q(R/P) = R_P/P R_P$ and the imbedding $R/P \to \kappa(P)$ $\endgroup$ Commented Feb 13, 2016 at 0:02
  • $\begingroup$ @JürgenBöhm I think you're right. This is the definition I found in Gortz and Wedhorn's book. $\endgroup$
    – Potato
    Commented Feb 13, 2016 at 0:05
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    $\begingroup$ the "value" f(p) of a "regular function" f at a point p is usually defined to be its image in the residue field, so yea what @JürgenBöhm said $\endgroup$
    – user304022
    Commented Feb 13, 2016 at 11:14
  • $\begingroup$ it is so sad that students sometimes have to follow this text where vakil is always so vague. $\endgroup$ Commented Jun 4, 2023 at 12:51

2 Answers 2

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The Jacobian criterion does not work on non-closed 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} By hypothesis \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 $\varphi_0=\varphi_{\varphi^{*}(P)}:\mathbb{K}[s_1,\dots,s_r]_{(s_1,\dots,s_r)}\to\mathbb{K}[t_1,\dots,t_n]_{\mathfrak{p}}$, 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.

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.

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 $X$! 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 $P$. 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.

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.

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  • $\begingroup$ So, I think you're right that there's no necessary connection to regularity, but regularity and smoothness are different concepts, and it seems that Gortz and Wedhorn (page 153) define smoothness in terms of the Jacobian considered as a linear transformation on a $k(\mathfrak p)$ vector space. Indeed, I think it has to be this way for the usual connections with the relative contangent sheaf to work out. $\endgroup$
    – Potato
    Commented Feb 20, 2016 at 19:26
  • $\begingroup$ Now I had read the Görtz and Wedhorn's book: they are used a trick, or you would like, they use the definition 12.2.6 and the exercise 12.2.G of Vakil's FOAG jointly. For clarity, I'll update my second answer! P.S.: Görtz and Wedhorn's book is intersting; thank you, in the future I'll consult it. $\endgroup$ Commented Feb 21, 2016 at 15:43
  • $\begingroup$ Great, thank you for untangling that! Also I think your comment about base-changing using 12.2.G is very enlightening and should perhaps be added to the answer. So in conclusion, we have determined that one way to check smoothness at non-closed points is to "base change" to the residue field and use the Jacobian criterion? $\endgroup$
    – Potato
    Commented Feb 21, 2016 at 19:23
  • $\begingroup$ Yes it is, if $P$ is a closed $\kappa(P)$-point of $X$ (the notation is clear); otherwise you can't use the "coordinate" of $P$ in $\mathbb{A}^n_{\kappa(P)}$. Do you understand this? $\endgroup$ Commented Feb 22, 2016 at 12:03
  • $\begingroup$ Yes, thank you! $\endgroup$
    – Potato
    Commented Feb 22, 2016 at 12:27
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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!

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  • $\begingroup$ You are completely right, but do you agree there's something unsatisfactory about the definition he gives? It seems that really we must consider all points, no necessary closed, and consider the Jacobian as acting on vector spaces over the residue fields. $\endgroup$
    – Potato
    Commented Feb 19, 2016 at 18:51
  • $\begingroup$ The previous reasonig holds for the Zariski (co)tangent space at closed points $P$ of a $\mathbb{K}$-schemes $X$ locally of finite type, such that the residue field $\kappa(P)$ is $\mathbb{K}$ (see exercise 12.1.G) or it is a separable extension of $\mathbb{K}$ (see remark 12.1.7); in particular, by Hilbert's Nullstellensatz, $\kappa(P)$ is a finite extension of $\mathbb{K}$. I remember that the Jacobian criterion doesn't work on non-closed points... I'll check it! $\endgroup$ Commented Feb 19, 2016 at 20:50

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