Suppose I have an affine variety $V \in k^n$ and $I := I(V)$. Let $f_1,\dots,f_t$ be generators of I. For a point $P \in V$, define the jacobian $J_P(I) = \begin{pmatrix} \frac{\partial f_1}{\partial x_1}(P) & \dots &\frac{\partial f_1}{\partial x_n}(P)\\ \vdots & \dots & \vdots\\ \frac{\partial f_t}{\partial x_1}(P) & \dots &\frac{\partial f_t}{\partial x_n}(P) \end{pmatrix}$ .
Is it always true that $\dim V \leq n -$ rank $J_p(I)$?
Edit: Let me briefly explain why I think this is true. If $T_P(V)$ is the tangent space of V at P, then I think the following two statements are true: $\dim V \leq \dim T_P(V)$ and $\dim T_P(V) = n - $rank $J_P(A)$. They immediately imply what I stated. Am I correct?
Question: "Is it always true that $dim(V)≤n− rank (J_p(I))$?"
Answer: I use the notation of Hartshorne, Chapter I. If $(A, \mathfrak{m})$ is a noetherian local ring with residue field $k$ it follows (HH.I.Prop.5.2A)
$$krdim(A) \leq dim_k(\mathfrak{m}/\mathfrak{m}^2).$$
There is a formula (see the proof of HH.Thm.I.5.1) saying
$$rk(J_p)=n-dim_k(\mathfrak{m}_p/\mathfrak{m}^2_p)$$
Here we assume $k$ is algebraically closed. Assume $Y:=V(I) \subseteq \mathbb{A}^n_k$ (in the notation of chapter I, HH) is an algebraic variety with coordinate ring $A(Y):=k[x_1,..,x_n]/I$ and maximal ideal $\mathfrak{m}_p$ corresponding to the point $p\in Y$. Localize $A:=A(Y)$ at $\mathfrak{m}:=\mathfrak{m}_p$ you get a noetherian local ring $(A_{\mathfrak{m}}, \tilde{\mathfrak{m}})$ with
$$krdim(A)=krdim(A_{\mathfrak{m}}) \leq dim_k(\tilde{\mathfrak{m}}/\tilde{\mathfrak{m}}^2)=dim_k(\mathfrak{m}/\mathfrak{m}^2).$$
Hence
$$dim(V):=krdim(A) \leq dim_k(\mathfrak{m}/\mathfrak{m}^2) =n-rk(J_p(I))=dim_k(T_p(V)).$$
Edit: Let me briefly explain why I think this is true. If $T_p(V)$ is the tangent space of $V$ at $p$, then I think the following two statements are true: $dim(V)≤dim_k T_p(V)$ and $dim_k(T_p(V))=n−rank(J_p(A))$. They immediately imply what I stated. Am I correct?
Answer: By the above argument: This claim is correct.