Let $f: X = \operatorname{Spec} A \rightarrow S = \operatorname{Spec} R$ be a morphism of schemes. Assume $A = R[t_1, ... , t_n]/\mathfrak a$, for $\mathfrak a = (g_1, ... , g_t)$ a finitely generated ideal. Let $x = \mathfrak P \in X$ be a prime of $R[\underline{t}] = R[t_1, ... , t_n]$, lying over $s=\mathfrak p \in S$.
We say $f$ is unramified at $x$ if the images of $dg_1, ... , dg_t \in \Omega_{R[\underline{t}]/R}$ generate $\Omega_{R[\underline{t}]/R} \otimes_{R[\underline{t}]} \kappa(\mathfrak P)$. Equivalently, $\mathfrak PA_{\mathfrak P}$ is generated by $\mathfrak p$, and $\kappa(\mathfrak P)$ is a finite separable extension of $\kappa(\mathfrak p)$.
If $f$ is unramified at $x$, then the stalk of the fiber $f^{-1}(s)$ at $x$ is exactly $\kappa(\mathfrak P)$, and in particular, $f$ has relative dimension zero at $x$.
The book I'm reading, Neron Models, makes a more general claim which I'm trying to understand. I know that $\Omega_{R[\underline{t}]/R}$ is a free $R[\underline{t}]$-module of rank $n$.
Condition (e) is the statement "Equivalently, $\mathfrak PA_{\mathfrak P}$ is generated..."
I'm not sure how to begin verifying the claim about relative dimension. I thought I should start with the case where $\mathfrak a = (g_2, ... , g_n)$, and assuming the images of $dg_2, ... , dg_n$ are linearly independent. I know that $\Omega_{R[\underline{t}]/R} \otimes_{R[\underline{t}]} \kappa(\mathfrak P)$ has dimension $n$ as a vector space over $\kappa(\mathfrak P)$. Perhaps I can let $g_1$ be an element of $R[\underline{t}]$ such that the image of $dg_1$ forms a basis with $dg_2, ... , dg_n$, and let $\mathfrak a' = (g_1, ... , g_n)$.