This definition is from the book Neron Models.
I want to understand what this definition is saying in the affine case. Suppose $X = \operatorname{Spec} B, S = \operatorname{Spec A}$, and $f$ comes from a finitely presented ring homomorphism $\phi: A \rightarrow B$.
Then $B = A[T_1, … , T_n]/\mathfrak a$ for some $n$ and some finitely generated ideal $\mathfrak a$, and we have a closed $A$-immersion $j:X \rightarrow \mathbb A_A^n$. So in the definition we can take the open set $U$ to be all of $X$, and the sheaf of ideals $\mathscr I$ in the definition identifies with the ideal $\mathfrak a$.
Let $\mathfrak q$ be a prime of $B$ at which $f$ is unramified, say $\mathfrak q = \mathfrak Q/\mathfrak a$ for a prime $\mathfrak Q$ of $A[\underline{T}] = A[T_1, … , T_n]$, contracting to a prime $\mathfrak p$ of $A$.
We have the universal $A$-linear map $d: A[\underline{T}] \rightarrow \Omega_{A[\underline{T}]/A}$ which localizes nicely:
$$d_{\mathfrak p}: A_{\mathfrak p}[\underline{T}] \rightarrow \Omega_{A_{\mathfrak p}[\underline{T}]/A_{\mathfrak p}}$$
There is also an exact sequence of $B = A[\underline{T}]/\mathfrak a$ modules:
$$\mathfrak a/\mathfrak a^2 \rightarrow \Omega_{A[\underline{T}]/A} \otimes_{A[\underline{T}]} B \rightarrow \Omega_{B/A} \rightarrow 0$$
where the map on the left is induced by $d$.
I cannot quite work out what condition (b) is saying in this case. I get the general idea that the image of some localization of $\mathfrak a$ under $d$ should generate some localization of $\Omega_{A[\underline{T}]/A}$.
Best Answer
(One of) the definition(s) of $X \to Y$ being unramified is that the sheaf of relative differentials $\Omega_{X/Y}$ is trivial. This is a local condition so can be checked affine locally or at a point on stalks (Stacks project 40.4.1). The exact sequence you have says that on affines you can compute the module of relative differentials $\Omega_{B/A}$ as the cokernel of $d: \mathfrak{a}/\mathfrak{a}^2 \to B \otimes \Omega_{A[T]/A} \cong \oplus BdT_i$. If $\mathfrak{a} = (f_1,\ldots,f_n)$, then the image of $d$ is generated by $(df_1,\ldots,df_n)$, so if that generates everything the coker will be trivial. Since forming modules of differentials commutes with localization of $B$, all the stalks die too. The stalks at $x$ are $j^{*} ( \Omega_{{A[T]/A} })_{x}$ = $B_x \otimes \Omega_{A[T]_x/A}$ are generated by $1 \otimes df_i$ and the same holds after passing to the residue field $k(x) \otimes (*)$. By Nakayama you can lift $df_i$ back up to generators for $\Omega_{A[T]_x}$