@R.vanDobbendeBruyn say we fix the polynomial map $(f_1,\dots , f_m):\mathbb{A}^n\rightarrow \mathbb{A}^m$. The cokernel then becomes elements $v\in\mathbb{A}^m$ with $\sum_j v_j\frac{\partial f_j}{x_i}$ for $i=1,\dots,n$. In particular, this sum only varies twith $f$′s differentials in a particular coordinate (rather than the coordinate derivatives in the tangent space case). Can we say anything about vectors of this form relative to $(f_1,\dots , f_m)$ or better yet, the variety $V$ that is generated by these polynomials?

With "the implicit function theorem" I mean the standard theorem for $\mathcal{C}^1$ functions $\mathbb{R}^{n+m}\rightarrow \mathbb{R}^n$. The curve a priori is any smooth curve satisfying $G(x(t), u(t)) = 0$ such that the corresponding Jacobian is invertible along the curve. $V(t)$ are the open neighborhoods the classic implicit function theorem provides at every point of the curve.