I'm reading Neron Models and am trying to understand the different equivalent definitions of a morphism of schemes being unramified. I'm currently trying to understand the equivalence of (b) and (c).
The book says the equivalence of (b) and (c) follows directly from the definition of $\Omega_{X/S}$ and Nakayama's lemma.
Let's consider the affine case $S = \operatorname{Spec}R, A = \operatorname{Spec} X$. Let $J$ be the kernel of the codiagonal $A \otimes_R A \rightarrow A$, so that $\Omega_{X/S} = J/J^2$ is naturally a module over $A = A\otimes_R A/J$.
Let $\mathfrak p \in X$ with preimage $\mathfrak P \in \operatorname{Spec} A\otimes_R A$. Then the equivalence of (b) and (c) can be stated as:
$(J/J^2)_{\mathfrak p} = 0$ if and only if $(A \otimes_R A)_{\mathfrak P} \rightarrow A_{\mathfrak p}$ is injective.
Maybe there is nothing special about the surjection $A \otimes_R A \rightarrow A$, and we can work more generally with a surjective ring homomorphism $\pi: B \rightarrow B/J$ with kernel $J$. We need to assume $J$ is finitely generated.
The quotient $J/J^2$ is naturally a module over $B/J$. Let $\mathfrak p = \mathfrak P/J$ be a prime of $B/J$.
To answer my question, it would be enough to show that $B_{\mathfrak P} \rightarrow (B/J)_{\mathfrak p}$ is injective if and only if $(J/J^2)_{\mathfrak p} = 0$.
This seems very strange. Since $(B/J)_{\mathfrak p} = B_{\mathfrak P}/J_{\mathfrak P}$, the injectivity of this last homomorphism is equivalent to $J_{\mathfrak P} = 0$. On the other hand, $$(J/J^2)_{\mathfrak p} = (J/J^2) \otimes_{B/J} ((B/J)/\mathfrak p) = J/J^2 \otimes_{B/J} B/\mathfrak P$$
Apparently this equivalence should fall out of Nakayama's lemma by tensoring with $B_{\mathfrak P}/\mathfrak PB_{\mathfrak P}$.
Best Answer
I do not agree with the first equation in your last equation environment. I guess you simply confused localization at an ideal with modding out by it for a moment. Viewing $(J/J^2)_{\mathfrak{p}}$ as a $B_{\mathfrak{P}}$-module, the equation should read: $$ (J/J^2)_{\mathfrak{p}}=J/J^2 \otimes_{B/J} (B/J)_{\mathfrak{p}}=J/J^2 \otimes_{B/J} (B/J)_{\mathfrak{P}}=J/J^2\otimes_{B/J} (B/J\otimes_B B_{\mathfrak{P}})=(J/J^2)_{\mathfrak{P}}=J_{\mathfrak{P}}/J_{\mathfrak{P}}^2. $$ Hence by Nakayama's Lemma, $(J/J^2)_{\mathfrak{p}}=0$ if and only if $J_{\mathfrak{P}}=0$.
By the way, (in case you were not aware of it) the finiteness condition on the ideal sheaf defining $X$ as a closed subscheme of some open in $X\times_S X$ is implied by the assumption on $f\colon X\to S$ being locally of finite presentation. Check out Tag 0818, Stacks Project.