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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