64.20 Affine morphisms

We have already defined in Section 64.3 what it means for a representable morphism of algebraic spaces to be affine.

Lemma 64.20.1. Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is affine (in the sense of Section 64.3) if and only if for all affine schemes $Z$ and morphisms $Z \to Y$ the scheme $X \times _ Y Z$ is affine.

Proof. This follows directly from the definition of an affine morphism of schemes (Morphisms, Definition 29.11.1). $\square$

This clears the way for the following definition.

Definition 64.20.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say $f$ is affine if for every affine scheme $Z$ and morphism $Z \to Y$ the algebraic space $X \times _ Y Z$ is representable by an affine scheme.

Lemma 64.20.3. 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 representable and affine,

2. $f$ is affine,

3. for every affine scheme $V$ and étale morphism $V \to Y$ the scheme $X \times _ Y V$ is affine,

4. there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $V \times _ Y X \to V$ is affine, and

5. there exists a Zariski covering $Y = \bigcup Y_ i$ such that each of the morphisms $f^{-1}(Y_ i) \to Y_ i$ is affine.

Proof. It is clear that (1) implies (2), that (2) implies (3), and that (3) implies (4) by taking $V$ to be a disjoint union of affines étale over $Y$, see Properties of Spaces, Lemma 63.6.1. Assume $V \to Y$ is as in (4). Then for every affine open $W$ of $V$ we see that $W \times _ Y X$ is an affine open of $V \times _ Y X$. Hence by Properties of Spaces, Lemma 63.13.1 we conclude that $V \times _ Y X$ is a scheme. Moreover the morphism $V \times _ Y X \to V$ is affine. This means we can apply Spaces, Lemma 62.11.5 because the class of affine morphisms satisfies all the required properties (see Morphisms, Lemmas 29.11.8 and Descent, Lemmas 35.20.18 and 35.34.1). The conclusion of applying this lemma is that $f$ is representable and affine, i.e., (1) holds.

The equivalence of (1) and (5) follows from the fact that being affine is Zariski local on the target (the reference above shows that being affine is in fact fpqc local on the target). $\square$

Proof. Omitted. Hint: Transitivity of fibre products. $\square$

Proof. Omitted. Hint: Transitivity of fibre products. $\square$

Proof. Follows immediately from the corresponding statement for morphisms of schemes, see Morphisms, Lemma 29.11.9. $\square$

Lemma 64.20.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. There is an anti-equivalence of categories

$\begin{matrix} \text{algebraic spaces} \\ \text{affine over }X \end{matrix} \longleftrightarrow \begin{matrix} \text{quasi-coherent sheaves} \\ \text{of }\mathcal{O}_ X\text{-algebras} \end{matrix}$

which associates to $f : Y \to X$ the sheaf $f_*\mathcal{O}_ Y$. Moreover, this equivalence is compatible with arbitrary base change.

Proof. This lemma is the analogue of Morphisms, Lemma 29.11.5. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras. We will construct an affine morphism of algebraic spaces $\pi : Y = \underline{\mathop{\mathrm{Spec}}}_ X(\mathcal{A}) \to X$ with $\pi _*\mathcal{O}_ Y \cong \mathcal{A}$. To do this, choose a scheme $U$ and a surjective étale morphism $\varphi : U \to X$. As usual denote $R = U \times _ X U$ with projections $s, t : R \to U$. Denote $\psi : R \to X$ the composition $\psi = \varphi \circ s = \varphi \circ t$. By the aforementioned lemma there exists an affine morphisms of schemes $\pi _0 : V \to U$ and $\pi _1 : W \to R$ with $\pi _{0, *}\mathcal{O}_ V \cong \varphi ^*\mathcal{A}$ and $\pi _{1, *}\mathcal{O}_ W \cong \psi ^*\mathcal{A}$. Since the construction is compatible with base change there exist morphisms $s', t' : W \to V$ such that the diagrams

$\vcenter { \xymatrix{ W \ar[r]_{s'} \ar[d] & V \ar[d] \\ R \ar[r]^ s & U } } \quad \text{and}\quad \vcenter { \xymatrix{ W \ar[r]_{t'} \ar[d] & V \ar[d] \\ R \ar[r]^ t & U } }$

are cartesian. It follows that $s', t'$ are étale. It is a formal consequence of the above that $(t', s') : W \to V \times _ S V$ is a monomorphism. We omit the verification that $W \to V \times _ S V$ is an equivalence relation (hint: think about the pullback of $\mathcal{A}$ to $U \times _ X U \times _ X U = R \times _{s, U, t} R$). The quotient sheaf $Y = V/W$ is an algebraic space, see Spaces, Theorem 62.10.5. By Groupoids, Lemma 39.20.7 we see that $Y \times _ X U \cong V$. Hence $Y \to X$ is affine by Lemma 64.20.3. Finally, the isomorphism of

$(Y \times _ X U \to U)_*\mathcal{O}_{Y \times _ X U} = \pi _{0, *}\mathcal{O}_ V \cong \varphi ^*\mathcal{A}$

is compatible with glueing isomorphisms, whence $(Y \to X)_*\mathcal{O}_ Y \cong \mathcal{A}$ by Properties of Spaces, Proposition 63.32.1. We omit the verification that this construction is compatible with base change. $\square$

Definition 64.20.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras. The relative spectrum of $\mathcal{A}$ over $X$, or simply the spectrum of $\mathcal{A}$ over $X$ is the affine morphism $\underline{\mathop{\mathrm{Spec}}}(\mathcal{A}) \to X$ corresponding to $\mathcal{A}$ under the equivalence of categories of Lemma 64.20.7.

Forming the relative spectrum commutes with arbitrary base change.

Remark 64.20.9. Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Then $f$ has a canonical factorization

$Y \longrightarrow \underline{\mathop{\mathrm{Spec}}}_ X(f_*\mathcal{O}_ Y) \longrightarrow X$

This makes sense because $f_*\mathcal{O}_ Y$ is quasi-coherent by Lemma 64.11.2. The morphism $Y \to \underline{\mathop{\mathrm{Spec}}}_ X(f_*\mathcal{O}_ Y)$ comes from the canonical $\mathcal{O}_ Y$-algebra map $f^*f_*\mathcal{O}_ Y \to \mathcal{O}_ Y$ which corresponds to a canonical morphism $Y \to Y \times _ X \underline{\mathop{\mathrm{Spec}}}_ X(f_*\mathcal{O}_ Y)$ over $Y$ (see Lemma 64.20.7) whence a factorization of $f$ as above.

Lemma 64.20.10. Let $S$ be a scheme. Let $f : Y \to X$ be an affine morphism of algebraic spaces over $S$. Let $\mathcal{A} = f_*\mathcal{O}_ Y$. The functor $\mathcal{F} \mapsto f_*\mathcal{F}$ induces an equivalence of categories

$\left\{ \begin{matrix} \text{category of quasi-coherent} \\ \mathcal{O}_ Y\text{-modules} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{category of quasi-coherent} \\ \mathcal{A}\text{-modules} \end{matrix} \right\}$

Moreover, an $\mathcal{A}$-module is quasi-coherent as an $\mathcal{O}_ X$-module if and only if it is quasi-coherent as an $\mathcal{A}$-module.

Proof. Omitted. $\square$

Lemma 64.20.11. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Suppose $g : X \to Y$ is a morphism of algebraic spaces over $B$.

1. If $X$ is affine over $B$ and $\Delta : Y \to Y \times _ B Y$ is affine, then $g$ is affine.

2. If $X$ is affine over $B$ and $Y$ is separated over $B$, then $g$ is affine.

3. A morphism from an affine scheme to an algebraic space with affine diagonal is affine.

4. A morphism from an affine scheme to a separated algebraic space is affine.

Proof. Proof of (1). The base change $X \times _ B Y \to Y$ is affine by Lemma 64.20.5. The morphism $(1, g) : X \to X \times _ B Y$ is the base change of $Y \to Y \times _ B Y$ by the morphism $X \times _ B Y \to Y \times _ B Y$. Hence it is affine by Lemma 64.20.5. The composition of affine morphisms is affine (see Lemma 64.20.4) and (1) follows. Part (2) follows from (1) as a closed immersion is affine (see Lemma 64.20.6) and $Y/B$ separated means $\Delta$ is a closed immersion. Parts (3) and (4) are special cases of (1) and (2). $\square$

Lemma 64.20.12. Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $A$ be an Artinian ring. Any morphism $\mathop{\mathrm{Spec}}(A) \to X$ is affine.

Proof. Let $U \to X$ be an étale morphism with $U$ affine. To prove the lemma we have to show that $\mathop{\mathrm{Spec}}(A) \times _ X U$ is affine, see Lemma 64.20.3. Since $X$ is quasi-separated the scheme $\mathop{\mathrm{Spec}}(A) \times _ X U$ is quasi-compact. Moreover, the projection morphism $\mathop{\mathrm{Spec}}(A) \times _ X U \to \mathop{\mathrm{Spec}}(A)$ is étale. Hence this morphism has finite discrete fibers and moreover the topology on $\mathop{\mathrm{Spec}}(A)$ is discrete. Thus $\mathop{\mathrm{Spec}}(A) \times _ X U$ is a scheme whose underlying topological space is a finite discrete set. We are done by Schemes, Lemma 26.11.8. $\square$

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