## 41.5 The functorial characterization of unramified morphisms

In basic algebraic geometry we learn that some classes of morphisms can be characterized functorially, and that such descriptions are quite useful. Unramified morphisms too have such a characterization.

Theorem 41.5.1. Let $f : X \to S$ be a morphism of schemes. Assume $S$ is a locally Noetherian scheme, and $f$ is locally of finite type. Then the following are equivalent:

1. $f$ is unramified,

2. the morphism $f$ is formally unramified: for any affine $S$-scheme $T$ and subscheme $T_0$ of $T$ defined by a square-zero ideal, the natural map

$\mathop{\mathrm{Hom}}\nolimits _ S(T, X) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ S(T_0, X)$

is injective.

Proof. See More on Morphisms, Lemma 37.6.8 for a more general statement and proof. What follows is a sketch of the proof in the current case.

Firstly, one checks both properties are local on the source and the target. This we may assume that $S$ and $X$ are affine. Say $X = \mathop{\mathrm{Spec}}(B)$ and $S = \mathop{\mathrm{Spec}}(R)$. Say $T = \mathop{\mathrm{Spec}}(C)$. Let $J$ be the square-zero ideal of $C$ with $T_0 = \mathop{\mathrm{Spec}}(C/J)$. Assume that we are given the diagram

$\xymatrix{ & B \ar[d]^\phi \ar[rd]^{\bar{\phi }} & \\ R \ar[r] \ar[ur] & C \ar[r] & C/J }$

Secondly, one checks that the association $\phi ' \mapsto \phi ' - \phi$ gives a bijection between the set of liftings of $\bar{\phi }$ and the module $\text{Der}_ R(B, J)$. Thus, we obtain the implication (1) $\Rightarrow$ (2) via the description of unramified morphisms having trivial module of differentials, see Theorem 41.4.1.

To obtain the reverse implication, consider the surjection $q : C = (B \otimes _ R B)/I^2 \to B = C/J$ defined by the square zero ideal $J = I/I^2$ where $I$ is the kernel of the multiplication map $B \otimes _ R B \to B$. We already have a lifting $B \to C$ defined by, say, $b \mapsto b \otimes 1$. Thus, by the same reasoning as above, we obtain a bijective correspondence between liftings of $\text{id} : B \to C/J$ and $\text{Der}_ R(B, J)$. The hypothesis therefore implies that the latter module is trivial. But we know that $J \cong \Omega _{B/R}$. Thus, $B/R$ is unramified. $\square$

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