The Stacks project

Lemma 35.13.2. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be a family of morphisms of affine schemes. The following are equivalent

  1. for any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we have

    \[ \Gamma (X, \mathcal{F}) = \text{Equalizer}\left( \xymatrix{ \prod \nolimits _{i \in I} \Gamma (X_ i, f_ i^*\mathcal{F}) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \nolimits _{i, j \in I} \Gamma (X_ i \times _ X X_ j, (f_ i \times f_ j)^*\mathcal{F}) } \right) \]
  2. $\{ f_ i : X_ i \to X\} _{i \in I}$ is a universal effective epimorphism (Sites, Definition 7.12.1) in the category of affine schemes.

Proof. Assume (2) holds and let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Consider the scheme (Constructions, Section 27.4)

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

where $\mathcal{O}_ X \oplus \mathcal{F}$ is an $\mathcal{O}_ X$-algebra with multiplication $(f, s)(f', s') = (ff', fs' + f's)$. If $s_ i \in \Gamma (X_ i, f_ i^*\mathcal{F})$ is a section, then $s_ i$ determines a unique element of

\[ \Gamma (X' \times _ X X_ i, \mathcal{O}_{X' \times _ X X_ i}) = \Gamma (X_ i, \mathcal{O}_{X_ i}) \oplus \Gamma (X_ i, f_ i^*\mathcal{F}) \]

Proof of equality omitted. If $(s_ i)_{i \in I}$ is in the equalizer of (1), then, using the equality

\[ \mathop{\mathrm{Mor}}\nolimits (T, \mathbf{A}^1_\mathbf {Z}) = \Gamma (T, \mathcal{O}_ T) \]

which holds for any scheme $T$, we see that these sections define a family of morphisms $h_ i : X' \times _ X X_ i \to \mathbf{A}^1_\mathbf {Z}$ with $h_ i \circ \text{pr}_1 = h_ j \circ \text{pr}_2$ as morphisms $(X' \times _ X X_ i) \times _{X'} (X' \times _ X X_ j) \to \mathbf{A}^1_\mathbf {Z}$. Since we've assume (2) we obtain a morphism $h : X' \to \mathbf{A}^1_\mathbf {Z}$ compatible with the morphisms $h_ i$ which in turn determines an element $s \in \Gamma (X, \mathcal{F})$. We omit the verification that $s$ maps to $s_ i$ in $\Gamma (X_ i, f_ i^*\mathcal{F})$.

Assume (1). Let $T$ be an affine scheme and let $h_ i : X_ i \to T$ be a family of morphisms such that $h_ i \circ \text{pr}_1 = h_ j \circ \text{pr}_2$ on $X_ i \times _ X X_ j$ for all $i, j \in I$. Then

\[ \prod h_ i^\sharp : \Gamma (T, \mathcal{O}_ T) \to \prod \Gamma (X_ i, \mathcal{O}_{X_ i}) \]

maps into the equalizer and we find that we get a ring map $\Gamma (T, \mathcal{O}_ T) \to \Gamma (X, \mathcal{O}_ X)$ by the assumption of the lemma for $\mathcal{F} = \mathcal{O}_ X$. This ring map corresponds to a morphism $h : X \to T$ such that $h_ i = h \circ f_ i$. Hence our family is an effective epimorphism.

Let $p : Y \to X$ be a morphism of affines. We will show the base changes $g_ i : Y_ i \to Y$ of $f_ i$ form an effective epimorphism by applying the result of the previous paragraph. Namely, if $\mathcal{G}$ is a quasi-coherent $\mathcal{O}_ Y$-module, then

\[ \Gamma (Y, \mathcal{G}) = \Gamma (X, p_*\mathcal{G}),\quad \Gamma (Y_ i, g_ i^*\mathcal{G}) = \Gamma (X, f_ i^*p_*\mathcal{G}), \]

and

\[ \Gamma (Y_ i \times _ Y Y_ j, (g_ i \times g_ j)^*\mathcal{G}) = \Gamma (X, (f_ i \times f_ j)^*p_*\mathcal{G}) \]

by the trivial base change formula (Cohomology of Schemes, Lemma 30.5.1). Thus we see property (1) lemma holds for the family $g_ i$. $\square$


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