31.25 Meromorphic functions and sections; reduced case
For a scheme which is reduced and which locally has finitely many irreducible components, the sheaf of meromorphic functions is quasi-coherent.
Lemma 31.25.1. Let $X$ be a reduced scheme such that any quasi-compact open has a finite number of irreducible components. Let $X^0$ be the set of generic points of irreducible components of $X$. Then we have
\[ \mathcal{K}_ X = \bigoplus \nolimits _{\eta \in X^0} j_{\eta , *}\kappa (\eta ) = \prod \nolimits _{\eta \in X^0} j_{\eta , *}\kappa (\eta ) \]
where $j_\eta : \mathop{\mathrm{Spec}}(\kappa (\eta )) \to X$ is the canonical map of Schemes, Section 26.13. Moreover
$\mathcal{K}_ X$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras,
for every quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the sheaf
\[ \mathcal{K}_ X(\mathcal{F}) = \bigoplus \nolimits _{\eta \in X^0} j_{\eta , *}\mathcal{F}_\eta = \prod \nolimits _{\eta \in X^0} j_{\eta , *}\mathcal{F}_\eta \]
of meromorphic sections of $\mathcal{F}$ is quasi-coherent,
$\mathcal{S}_ x \subset \mathcal{O}_{X, x}$ is the set of nonzerodivisors for any $x \in X$,
$\mathcal{K}_{X, x}$ is the total quotient ring of $\mathcal{O}_{X, x}$ for any $x \in X$,
$\mathcal{K}_ X(U)$ equals the total quotient ring of $\mathcal{O}_ X(U)$ for any affine open $U \subset X$,
the ring of rational functions of $X$ is the ring of meromorphic functions on $X$, in a formula: $R(X) = \Gamma (X, \mathcal{K}_ X)$.
Proof.
This lemma is a special case of Lemma 31.23.6 because on a reduced scheme the weakly associated points are the generic points by Lemma 31.5.12.
$\square$
Lemma 31.25.2. Let $X$ be a scheme. Assume $X$ is reduced and any quasi-compact open $U \subset X$ has a finite number of irreducible components. Then the normalization morphism $\nu : X^\nu \to X$ is the morphism
\[ \underline{\mathop{\mathrm{Spec}}}_ X(\mathcal{O}') \longrightarrow X \]
where $\mathcal{O}' \subset \mathcal{K}_ X$ is the integral closure of $\mathcal{O}_ X$ in the sheaf of meromorphic functions.
Proof.
Compare the definition of the normalization morphism $\nu : X^\nu \to X$ (see Morphisms, Definition 29.54.1) with the description of $\mathcal{K}_ X$ in Lemma 31.25.1 above.
$\square$
Lemma 31.25.3. Let $X$ be an integral scheme with generic point $\eta $. We have
the sheaf of meromorphic functions is isomorphic to the constant sheaf with value the function field (see Morphisms, Definition 29.49.6) of $X$.
for any quasi-coherent sheaf $\mathcal{F}$ on $X$ the sheaf $\mathcal{K}_ X(\mathcal{F})$ is isomorphic to the constant sheaf with value $\mathcal{F}_\eta $.
Proof.
Omitted.
$\square$
In some cases we can show regular meromorphic sections exist.
Lemma 31.25.4. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. In each of the following cases $\mathcal{L}$ has a regular meromorphic section:
$X$ is integral,
$X$ is reduced and any quasi-compact open has a finite number of irreducible components,
$X$ is locally Noetherian and has no embedded points.
Proof.
In case (1) let $\eta \in X$ be the generic point. We have seen in Lemma 31.25.3 that $\mathcal{K}_ X$, resp. $\mathcal{K}_ X(\mathcal{L})$ is the constant sheaf with value $\kappa (\eta )$, resp. $\mathcal{L}_\eta $. Since $\dim _{\kappa (\eta )} \mathcal{L}_\eta = 1$ we can pick a nonzero element $s \in \mathcal{L}_\eta $. Clearly $s$ is a regular meromorphic section of $\mathcal{L}$. In case (2) pick $s_\eta \in \mathcal{L}_\eta $ nonzero for all generic points $\eta $ of $X$; this is possible as $\mathcal{L}_\eta $ is a $1$-dimensional vector space over $\kappa (\eta )$. It follows immediately from the description of $\mathcal{K}_ X$ and $\mathcal{K}_ X(\mathcal{L})$ in Lemma 31.25.1 that $s = \prod s_\eta $ is a regular meromorphic section of $\mathcal{L}$. Case (3) is Lemma 31.24.4.
$\square$
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