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 $g_ i$. $\square$

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0EUA. Beware of the difference between the letter 'O' and the digit '0'.