41.4 Three other characterizations of unramified morphisms

The following theorem gives three equivalent notions of being unramified at a point. See Morphisms, Lemma 29.35.14 for (part of) the statement for general schemes.

Theorem 41.4.1. Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism of schemes which is locally of finite type. Let $x$ be a point of $X$. The following are equivalent

1. $f$ is unramified at $x$,

2. the stalk $\Omega _{X/Y, x}$ of the module of relative differentials at $x$ is trivial,

3. there exist open neighbourhoods $U$ of $x$ and $V$ of $f(x)$, and a commutative diagram

$\xymatrix{ U \ar[rr]_ i \ar[rd] & & \mathbf{A}^ n_ V \ar[ld] \\ & V }$

where $i$ is a closed immersion defined by a quasi-coherent sheaf of ideals $\mathcal{I}$ such that the differentials $\text{d}g$ for $g \in \mathcal{I}_{i(x)}$ generate $\Omega _{\mathbf{A}^ n_ V/V, i(x)}$, and

4. the diagonal $\Delta _{X/Y} : X \to X \times _ Y X$ is a local isomorphism at $x$.

Proof. The equivalence of (1) and (2) is proved in Morphisms, Lemma 29.35.14.

If $f$ is unramified at $x$, then $f$ is unramified in an open neighbourhood of $x$; this does not follow immediately from Definition 41.3.5 of this chapter but it does follow from Morphisms, Definition 29.35.1 which we proved to be equivalent in Lemma 41.3.6. Choose affine opens $V \subset Y$, $U \subset X$ with $f(U) \subset V$ and $x \in U$, such that $f$ is unramified on $U$, i.e., $f|_ U : U \to V$ is unramified. By Morphisms, Lemma 29.35.13 the morphism $U \to U \times _ V U$ is an open immersion. This proves that (1) implies (4).

If $\Delta _{X/Y}$ is a local isomorphism at $x$, then $\Omega _{X/Y, x} = 0$ by Morphisms, Lemma 29.32.7. Hence we see that (4) implies (2). At this point we know that (1), (2) and (4) are all equivalent.

Assume (3). The assumption on the diagram combined with Morphisms, Lemma 29.32.15 show that $\Omega _{U/V, x} = 0$. Since $\Omega _{U/V, x} = \Omega _{X/Y, x}$ we conclude (2) holds.

Finally, assume that (2) holds. To prove (3) we may localize on $X$ and $Y$ and assume that $X$ and $Y$ are affine. Say $X = \mathop{\mathrm{Spec}}(B)$ and $Y = \mathop{\mathrm{Spec}}(A)$. The point $x \in X$ corresponds to a prime $\mathfrak q \subset B$. Our assumption is that $\Omega _{B/A, \mathfrak q} = 0$ (see Morphisms, Lemma 29.32.5 for the relationship between differentials on schemes and modules of differentials in commutative algebra). Since $Y$ is locally Noetherian and $f$ locally of finite type we see that $A$ is Noetherian and $B \cong A[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$, see Properties, Lemma 28.5.2 and Morphisms, Lemma 29.15.2. In particular, $\Omega _{B/A}$ is a finite $B$-module. Hence we can find a single $g \in B$, $g \not\in \mathfrak q$ such that the principal localization $(\Omega _{B/A})_ g$ is zero. Hence after replacing $B$ by $B_ g$ we see that $\Omega _{B/A} = 0$ (formation of modules of differentials commutes with localization, see Algebra, Lemma 10.131.8). This means that $\text{d}(f_ j)$ generate the kernel of the canonical map $\Omega _{A[x_1, \ldots , x_ n]/A} \otimes _ A B \to \Omega _{B/A}$. Thus the surjection $A[x_1, \ldots , x_ n] \to B$ of $A$-algebras gives the commutative diagram of (3), and the theorem is proved. $\square$

How can we use this theorem? Well, here are a few remarks:

1. Suppose that $f : X \to Y$ and $g : Y \to Z$ are two morphisms locally of finite type between locally Noetherian schemes. There is a canonical short exact sequence

$f^*(\Omega _{Y/Z}) \to \Omega _{X/Z} \to \Omega _{X/Y} \to 0$

see Morphisms, Lemma 29.32.9. The theorem therefore implies that if $g \circ f$ is unramified, then so is $f$. This is Morphisms, Lemma 29.35.16.

2. Since $\Omega _{X/Y}$ is isomorphic to the conormal sheaf of the diagonal morphism (Morphisms, Lemma 29.32.7) we see that if $X \to Y$ is a monomorphism of locally Noetherian schemes and locally of finite type, then $X \to Y$ is unramified. In particular, open and closed immersions of locally Noetherian schemes are unramified. See Morphisms, Lemmas 29.35.7 and 29.35.8.

3. The theorem also implies that the set of points where a morphism $f : X \to Y$ (locally of finite type of locally Noetherian schemes) is not unramified is the support of the coherent sheaf $\Omega _{X/Y}$. This allows one to give a scheme theoretic definition to the “ramification locus”.

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