The Stacks project

Lemma 81.13.1. Let $S$ be a scheme. Let

\[ g : Y \longrightarrow X \]

be a morphism of algebraic spaces over $S$. Assume $X$ is locally Noetherian, and $g$ is proper. Let $R = Y \times _ X Y$ with projection morphisms $t, s : R \to Y$. There exists a coequalizer $X'$ of $s, t : R \to Y$ in the category of algebraic spaces over $S$. Moreover

  1. The morphism $X' \to X$ is finite.

  2. The morphism $Y \to X'$ is proper.

  3. The morphism $Y \to X'$ is surjective.

  4. The morphism $X' \to X$ is universally injective.

  5. If $g$ is surjective, the morphism $X' \to X$ is a universal homeomorphism.

Proof. Denote $h : R \to X$ denote the composition of either $s$ or $t$ with $g$. Then $h$ is proper by Morphisms of Spaces, Lemmas 67.40.3 and 67.40.4. The sheaves

\[ g_*\mathcal{O}_ Y \quad \text{and}\quad h_*\mathcal{O}_ R \]

are coherent $\mathcal{O}_ X$-algebras by Cohomology of Spaces, Lemma 69.20.2. The $X$-morphisms $s$, $t$ induce $\mathcal{O}_ X$-algebra maps $s^\sharp , t^\sharp $ from the first to the second. Set

\[ \mathcal{A} = \text{Equalizer}\left(s^\sharp , t^\sharp : g_*\mathcal{O}_ Y \longrightarrow h_*\mathcal{O}_ R\right) \]

Then $\mathcal{A}$ is a coherent $\mathcal{O}_ X$-algebra and we can define

\[ X' = \underline{\mathop{\mathrm{Spec}}}_ X(\mathcal{A}) \]

as in Morphisms of Spaces, Definition 67.20.8. By Morphisms of Spaces, Remark 67.20.9 and functoriality of the $\underline{\mathop{\mathrm{Spec}}}$ construction there is a factorization

\[ Y \longrightarrow X' \longrightarrow X \]

and the morphism $g' : Y \to X'$ equalizes $s$ and $t$.

Before we show that $X'$ is the coequalizer of $s$ and $t$, we show that $Y \to X'$ and $X' \to X$ have the desired properties. Since $\mathcal{A}$ is a coherent $\mathcal{O}_ X$-module it is clear that $X' \to X$ is a finite morphism of algebraic spaces. This proves (1). The morphism $Y \to X'$ is proper by Morphisms of Spaces, Lemma 67.40.6. This proves (2). Denote $Y \to Y' \to X$ with $Y' = \underline{\mathop{\mathrm{Spec}}}_ X(g_*\mathcal{O}_ Y)$ the Stein factorization of $g$, see More on Morphisms of Spaces, Theorem 76.36.4. Of course we obtain morphisms $Y \to Y' \to X' \to X$ fitting with the morphisms studied above. Since $\mathcal{O}_{X'} \subset g_*\mathcal{O}_ Y$ is a finite extension we see that $Y' \to X'$ is finite and surjective. Some details omitted; hint: use Algebra, Lemma 10.36.17 and reduce to the affine case by ├ętale localization. Since $Y \to Y'$ is surjective (with geometrically connected fibres) we conclude that $Y \to X'$ is surjective. This proves (3). To show that $X' \to X$ is universally injective, we have to show that $X' \to X' \times _ X X'$ is surjective, see Morphisms of Spaces, Definition 67.19.3 and Lemma 67.19.2. Since $Y \to X'$ is surjective (see above) and since base changes and compositions of surjective morphisms are surjective by Morphisms of Spaces, Lemmas 67.5.5 and 67.5.4 we see that $Y \times _ X Y \to X' \times _ X X'$ is surjective. However, since $Y \to X'$ equalizes $s$ and $t$, we see that $Y \times _ X Y \to X' \times _ X X'$ factors through $X' \to X' \times _ X X'$ and we conclude this latter map is surjective. This proves (4). Finally, if $g$ is surjective, then since $g$ factors through $X' \to X$ we see that $X' \to X$ is surjective. Since a surjective, universally injective, finite morphism is a universal homeomorphism (because it is universally bijective and universally closed), this proves (5).

In the rest of the proof we show that $Y \to X'$ is the coequalizer of $s$ and $t$ in the category of algebraic spaces over $S$. Observe that $X'$ is locally Noetherian (Morphisms of Spaces, Lemma 67.23.5). Moreover, observe that $Y \times _{X'} Y \to Y \times _ X Y$ is an isomorphism as $Y \to X'$ equalizes $s$ and $t$ (this is a categorical statement). Hence in order to prove the statement that $Y \to X'$ is the coequalizer of $s$ and $t$, we may and do assume $X = X'$. In other words, $\mathcal{O}_ X$ is the equalizer of the maps $s^\sharp , t^\sharp : g_*\mathcal{O}_ Y \to h_*\mathcal{O}_ R$.

Let $X_1 \to X$ be a flat morphism of algebraic spaces over $S$ with $X_1$ locally Noetherian. Denote $g_1 : Y_1 \to X_1$, $h_1 : R_1 \to X_1$ and $s_1, t_1 : R_1 \to Y_1$ the base changes of $g, h, s, t$ to $X_1$. Of course $g_1$ is proper and $R_1 = Y_1 \times _{X_1} Y_1$. Since we have flat base change for pushforward of quasi-coherent modules, Cohomology of Spaces, Lemma 69.11.2, we see that $\mathcal{O}_{X_1}$ is the equalizer of the maps $s_1^\sharp , t_1^\sharp : g_{1, *}\mathcal{O}_{Y_1} \to h_{1, *}\mathcal{O}_{R_1}$. Hence all the assumptions we have are preserved by this base change.

At this point we are going to check conditions (1) and (2) of Lemma 81.3.3. Condition (1) follows from Lemma 81.5.1 and the fact that $g$ is proper and surjective (because $X = X'$). To check condition (2), by the remarks on base change above, we reduce to the statement discussed and proved in the next paragraph.

Assume $S = \mathop{\mathrm{Spec}}(A)$ is an affine scheme, $X = X'$ is an affine scheme, and $Z$ is an affine scheme over $S$. We have to show that

\[ \mathop{\mathrm{Mor}}\nolimits _ S(X, Z) \longrightarrow \text{Equalizer}(s, t : \mathop{\mathrm{Mor}}\nolimits _ S(Y, Z) \to \mathop{\mathrm{Mor}}\nolimits _ S(R, Z)) \]

is bijective. However, this is clear from the fact that $X = X'$ which implies $\mathcal{O}_ X$ is the equalizer of the maps $s^\sharp , t^\sharp : g_*\mathcal{O}_ Y \to h_*\mathcal{O}_ R$ which in turn implies

\[ \Gamma (X, \mathcal{O}_ X) = \text{Equalizer}\left( s^\sharp , t^\sharp : \Gamma (Y, \mathcal{O}_ Y) \to \Gamma (R, \mathcal{O}_ R) \right) \]

Namely, we have

\[ \mathop{\mathrm{Mor}}\nolimits _ S(X, Z) = \mathop{\mathrm{Hom}}\nolimits _ A(\Gamma (Z, \mathcal{O}_ Z), \Gamma (X, \mathcal{O}_ X)) \]

and similarly for $Y$ and $R$, see Properties of Spaces, Lemma 66.33.1. $\square$


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