## 75.14 Formally unramified morphisms

In this section we work out what it means that a morphism of algebraic spaces is formally unramified.

Definition 75.14.1. Let $S$ be a scheme. A morphism $f : X \to Y$ of algebraic spaces over $S$ is said to be formally unramified if it is formally unramified as a transformation of functors as in Definition 75.13.1.

We will not restate the results proved in the more general setting of formally unramified transformations of functors in Section 75.13. It turns out we can characterize this property in terms of vanishing of the module of relative differentials, see Lemma 75.14.6.

Lemma 75.14.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

1. $f$ is formally unramified,

2. for every diagram

$\xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^ f & Y }$

where $U$ and $V$ are schemes and the vertical arrows are étale the morphism of schemes $\psi$ is formally unramified (as in More on Morphisms, Definition 37.6.1), and

3. for one such diagram with surjective vertical arrows the morphism $\psi$ is formally unramified.

Proof. Assume $f$ is formally unramified. By Lemma 75.13.5 the morphisms $U \to X$ and $V \to Y$ are formally unramified. Thus by Lemma 75.13.3 the composition $U \to Y$ is formally unramified. Then it follows from Lemma 75.13.8 that $U \to V$ is formally unramified. Thus (1) implies (2). And (2) implies (3) trivially

Assume given a diagram as in (3). By Lemma 75.13.5 the morphism $V \to Y$ is formally unramified. Thus by Lemma 75.13.3 the composition $U \to Y$ is formally unramified. Then it follows from Lemma 75.13.6 that $X \to Y$ is formally unramified, i.e., (1) holds. $\square$

Lemma 75.14.3. Let $S$ be a scheme. If $f : X \to Y$ is a formally unramified morphism of algebraic spaces over $S$, then given any solid commutative diagram

$\xymatrix{ X \ar[d]_ f & T \ar[d]^ i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] }$

where $T \subset T'$ is a first order thickening of algebraic spaces over $S$ there exists at most one dotted arrow making the diagram commute. In other words, in Definition 75.14.1 the condition that $T$ be an affine scheme may be dropped.

Proof. This is true because there exists a surjective étale morphism $U' \to T'$ where $U'$ is a disjoint union of affine schemes (see Properties of Spaces, Lemma 65.6.1) and a morphism $T' \to X$ is determined by its restriction to $U'$. $\square$

Lemma 75.14.4. A composition of formally unramified morphisms is formally unramified.

Proof. This is formal. $\square$

Lemma 75.14.5. A base change of a formally unramified morphism is formally unramified.

Proof. This is formal. $\square$

Lemma 75.14.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

1. $f$ is formally unramified, and

2. $\Omega _{X/Y} = 0$.

Proof. This is a combination of Lemma 75.14.2, More on Morphisms, Lemma 37.6.7, and Lemma 75.7.3. $\square$

Lemma 75.14.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

1. The morphism $f$ is unramified,

2. the morphism $f$ is locally of finite type and $\Omega _{X/Y} = 0$, and

3. the morphism $f$ is locally of finite type and formally unramified.

Proof. Choose a diagram

$\xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^ f & Y }$

where $U$ and $V$ are schemes and the vertical arrows are étale and surjective. Then we see

\begin{align*} f\text{ unramified} & \Leftrightarrow \psi \text{ unramified} \\ & \Leftrightarrow \psi \text{ locally finite type and }\Omega _{U/V} = 0 \\ & \Leftrightarrow f\text{ locally finite type and }\Omega _{X/Y} = 0 \\ & \Leftrightarrow f\text{ locally finite type and formally unramified} \end{align*}

Here we have used Morphisms, Lemma 29.35.2 and Lemma 75.14.6. $\square$

Lemma 75.14.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

1. $f$ is unramified and a monomorphism,

2. $f$ is unramified and universally injective,

3. $f$ is locally of finite type and a monomorphism,

4. $f$ is universally injective, locally of finite type, and formally unramified.

Moreover, in this case $f$ is also representable, separated, and locally quasi-finite.

Proof. We have seen in Lemma 75.14.7 that being formally unramified and locally of finite type is the same thing as being unramified. Hence (4) is equivalent to (2). A monomorphism is certainly formally unramified hence (3) implies (4). It is clear that (1) implies (3). Finally, if (2) holds, then $\Delta : X \to X \times _ Y X$ is both an open immersion (Morphisms of Spaces, Lemma 66.38.9) and surjective (Morphisms of Spaces, Lemma 66.19.2) hence an isomorphism, i.e., $f$ is a monomorphism. In this way we see that (2) implies (1). Finally, we see that $f$ is representable, separated, and locally quasi-finite by Morphisms of Spaces, Lemmas 66.27.10 and 66.51.1. $\square$

Lemma 75.14.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

1. $f$ is a closed immersion,

2. $f$ is universally closed, unramified, and a monomorphism,

3. $f$ is universally closed, unramified, and universally injective,

4. $f$ is universally closed, locally of finite type, and a monomorphism,

5. $f$ is universally closed, universally injective, locally of finite type, and formally unramified.

Proof. The equivalence of (2) – (5) follows immediately from Lemma 75.14.8. Moreover, if (2) – (5) are satisfied then $f$ is representable. Similarly, if (1) is satisfied then $f$ is representable. Hence the result follows from the case of schemes, see Étale Morphisms, Lemma 41.7.2. $\square$

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