# ag.algebraic geometry – Definition of smooth morphism of schemes

In Wedhorn’s and Görtz’s book, a morphism $$f: X to Y$$ is said to be smooth of relative dimension $$d$$ when for every $$x in X$$, there exists affine open neighborhoods $$U$$ of $$x$$ and $$V = operatorname{Spec} R$$ of $$y=f(x)$$ such that $$f(U) subseteq V$$, and an open immersion of $$R$$-schemes $$j: U hookrightarrow operatorname{Spec} R(T_1, dots, T_n)/(g_1, dots, g_{n-d})$$ such that the matrix $$left (frac{partial g_i}{partial T_j}(x)right )$$ has rank $$n-d$$.

In Bosch’s book, $$f$$ is smooth of relative dimension $$d$$ if for every $$x in X$$, there exists an open neighborhood $$U$$ of $$x$$ and an $$Y$$-morphism $$j:U to W subseteq mathbb{A}_Y^n$$ (here, $$mathbb{A}^n_Y = underline{operatorname{Spec}}_Y mathscr{O}_Y(T_1, dots, T_n)$$) giving rise to a closed immersion from $$U$$ over an open subscheme $$W subseteq mathbb{A}^n_Y$$ satisfying:

If $$mathcal{I} subseteq mathscr{O}_W$$ is the ideal correspondent to $$j$$, then there are $$n-d$$ sections $$g_1, dots g_{n-d}$$ generating $$mathcal{I}$$ in a neighborhood of $$z = j(x)$$ and such that the matrix $$left(frac{partial g_i}{partial t_j}(z)right)$$ has rank $$n-d$$, where $$t_i$$ are the coordinate functions of $$mathbb{A}^n_Y$$.

To be honest, I can’t quite make sense of Bosch’s definition: what does $$frac{partial g_i}{partial t_j}$$ even mean, given that the $$t_j$$ are global sections and the $$g_i$$ are sections on a neighborhood of $$z$$, and, thus, not necessarily polynomials. Am I missing something? How can I prove that these two definitions are equivalent? I can sort of see how Bosch’s definition imply Wedhorn’s and Görtz’s, if I squint. Can someone help me?

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