The Stacks project

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

  1. $\mathcal{K}_ X$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras,

  2. 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,

  3. $\mathcal{S}_ x \subset \mathcal{O}_{X, x}$ is the set of nonzerodivisors for any $x \in X$,

  4. $\mathcal{K}_{X, x}$ is the total quotient ring of $\mathcal{O}_{X, x}$ for any $x \in X$,

  5. $\mathcal{K}_ X(U)$ equals the total quotient ring of $\mathcal{O}_ X(U)$ for any affine open $U \subset X$,

  6. 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. $\square$

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