Lemma 31.24.3. Let $X$ be a locally Noetherian scheme having no embedded points. Let $X^0$ be the set of generic points of irreducible components of $X$. Then we have
\[ \mathcal{K}_ X = \bigoplus \nolimits _{\eta \in X^0} j_{\eta , *}\mathcal{O}_{X, \eta } = \prod \nolimits _{\eta \in X^0} j_{\eta , *}\mathcal{O}_{X, \eta } \]
where $j_\eta : \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \eta }) \to X$ 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, and
the ring of rational functions of $X$ is the ring of meromorphic functions on $X$, in a formula: $R(X) = \Gamma (X, \mathcal{K}_ X)$.
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