37.81 The extensive criterion for closed immersions

In this section, we give a criterion for a morphism of schemes to be a closed immersion.

Lemma 37.81.1. A morphism $f : X \to Y$ of affine schemes is a closed immersion if and only if for every injective ring map $A \to B$ and commutative square

$\xymatrix{ \mathop{\mathrm{Spec}}(B) \ar[d] \ar[r] & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{..>}[ur] & Y }$

there exists a lift $\mathop{\mathrm{Spec}}(A) \to X$ making the two triangles commute.

Proof. Let the morphism $f$ be given by the ring map $\phi : R \to S$. Then $f$ is a closed immersion if and only if $\phi$ is surjective.

First, we assume that $\phi$ is surjective. Let $\psi : A \to B$ be an injective ring map, and suppose we are given a commutative diagram

$\xymatrix{ R \ar[r]^\alpha \ar[d]^\phi & A \ar[d]^\psi \\ S \ar[r]^\beta \ar@{..>}[ur] & B }$

Then we define a lift $S \to A$ by $s \mapsto \alpha (r)$, where $r \in R$ is such that $\phi (r) = s$. This is well-defined because $\psi$ is injective and the square commutes. Since taking the ring spectrum defines an anti-equivalence between commutative rings and affine schemes, the desired lifting property for $f$ holds.

Next, we assume that $\phi$ has lifts against all injective ring maps $\psi : A \to B$. Note that $\phi (R)$ is a subring of $S$, so we obtain a commutative square

$\xymatrix{ R \ar[r] \ar[d]^\phi & \phi (R) \ar[d] \\ S \ar@{=}[r] \ar@{..>}[ur] & S }$

in which a lift $S \to \phi (R)$ exists. Hence, the inclusion $\phi (R) \to S$ must be an isomorphism, which shows that $\phi$ is surjective, and we win. $\square$

Lemma 37.81.2. Let $X$ be a scheme. If the canonical morphism $X \to \mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X))$ of Schemes, Lemma 26.6.4 has a retraction, then $X$ is an affine scheme.

Proof. Write $S = \mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X))$ and $f : X \to S$ the morphism given in the lemma. Let $s : S \to X$ be a retraction; so $\text{id}_ X = sf$. Then $f s f = \text{id}_ S f$. Since $f$ induces an isomorphism $\Gamma (S, \mathcal{O}_ S) \to \Gamma (X, \mathcal{O}_ X)$ this means that $fs$ and $\text{id}_ S$ induce the same map on $\Gamma (S, \mathcal{O}_ S)$. Whence $f s = \text{id}_ S$ as $S$ is affine. Hence $f$ is an isomorphism and $X$ is an affine scheme, as was to be shown. $\square$

Lemma 37.81.3. Let $X$ be a scheme. Let $f : X \to S = \mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X))$ be the canonical morphism of Schemes, Lemma 26.6.4. The largest quasi-coherent $\mathcal{O}_ S$-module contained in the kernel of $f^\sharp : \mathcal{O}_ S \to f_*\mathcal{O}_ X$ is zero. If $X$ is quasi-compact, then $f^\sharp$ is injective. In particular, if $X$ is quasi-compact, then $f$ is a dominant morphism.

Proof. Let $M \subset \Gamma (S, \mathcal{O}_ S)$ be the submodule corresponding to the largest quasi-coherent $\mathcal{O}_ S$-module contained in the kernel of $f^\sharp$. Then any element $a \in M$ is mapped to zero by $f^\sharp$. However, $f^\sharp (a)$ is the element of

$\Gamma (S, f_*\mathcal{O}_ X) = \Gamma (X, \mathcal{O}_ X) = \Gamma (S, \mathcal{O}_ S)$

corresponding to $a$ itself! Thus $a = 0$. Hence $M = 0$ which proves the first assertion. Note that this is equivalent to the morphism $f : X \to S$ being scheme-theoretically surjective.

If $X$ is quasi-compact, then $\mathop{\mathrm{Ker}}(f^\sharp )$ is quasi-coherent by Morphisms, Lemma 29.6.3. Hence $\mathop{\mathrm{Ker}}(f^\sharp ) = 0$ and $f^\sharp$ is injective. In this case, $f$ is a dominant morphism by part (4) of Morphisms, Lemma 29.6.3. $\square$

Lemma 37.81.4. Let $f: X \to Y$ be a quasi-compact morphism of schemes. Then $f$ is a closed immersion if and only if for every injective ring map $A \to B$ and commutative square

$\xymatrix{ \mathop{\mathrm{Spec}}(B) \ar[d] \ar[r] & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{..>}[ur] & Y }$

there exists a lift $\mathop{\mathrm{Spec}}A \to X$ making the diagram commute.

Proof. Assume that $f$ is a closed immersion. Let $A \to B$ be an injective ring map and consider a commutative square

$\xymatrix{ \mathop{\mathrm{Spec}}(B) \ar[d] \ar[r] & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{..>}[ur] & Y }$

Then $\mathop{\mathrm{Spec}}(A) \times _ Y X \to \mathop{\mathrm{Spec}}(A)$ is a closed immersion and hence we get an ideal $I \subset A$ and a commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(B) \ar[d] \ar[r] & \mathop{\mathrm{Spec}}(A/I) \ar[r] \ar[d] & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{..>}[ur] & \mathop{\mathrm{Spec}}(A) \ar[r] & Y }$

We obtain a lift by Lemma 37.81.1.

Assume that $f$ has the lifting property stated in the lemma. To prove that $f$ is a closed immersion is local on $Y$, hence we may and do assume $Y$ is affine. In particular, $Y$ is quasi-compact and therefore $X$ is quasi-compact. Hence there exists a finite affine open covering $X = U_1 \cup \ldots \cup U_ n$. The source of the morphism

$\pi : U = \coprod U_ i \longrightarrow X$

is affine and the induced ring map $\Gamma (X, \mathcal{O}_ X) \to \Gamma (U, \mathcal{O}_ U)$ is injective. By assumption, there exists a lift in the diagram

$\xymatrix{ U \ar[r]^\pi \ar[d] & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X)) \ar[r]^-{f'} \ar@{..>}[ur]^ h & Y }$

where $f'$ is the morphism of affine schemes corresponding to the ring map $\Gamma (Y, \mathcal{O}_ Y) \to \Gamma (X, \mathcal{O}_ X)$. It follows from the fact that $\pi$ is an epimorphism that the morphism $h$ is a retraction of the canonical morphism $X \to \mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X))$; details omitted. Hence $X$ is affine by Lemma 37.81.2. By Lemma 37.81.1 we conclude that $f$ is a closed immersion. $\square$

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