The Stacks project

Lemma 70.10.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume

  1. every weakly associated point of $X$ is a point of codimension $0$, and

  2. $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma 66.49.1,

  3. every codimension $0$ point of $X$ can be represented by a monomorphism $\mathop{\mathrm{Spec}}(k) \to X$.

Let $X^0 \subset |X|$ be the set of codimension $0$ points 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; this makes sense because $X^0$ is contained in the schematic locus of $X$. Similarly, for every quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we obtain the formula

\[ \mathcal{K}_ X(\mathcal{F}) = \bigoplus \nolimits _{\eta \in X^0} j_{\eta , *}\mathcal{F}_\eta = \prod \nolimits _{\eta \in X^0} j_{\eta , *}\mathcal{F}_\eta \]

for the sheaf of meromorphic sections of $\mathcal{F}$. Finally, 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. By Decent Spaces, Lemma 67.20.3 and Section 67.6 we see that $X$ is decent1. Thus $X^0 \subset |X|$ is the set of generic points of irreducible components (Decent Spaces, Lemma 67.20.1) and $X^0$ is locally finite in $|X|$ by (b). It follows that $X^0$ is contained in every dense open subset of $|X|$. In particular, $X^0$ is contained in the schematic locus (Decent Spaces, Theorem 67.10.2). Thus the local rings $\mathcal{O}_{X, \eta }$ and the morphisms $j_\eta $ are defined.

Observe that a locally finite direct sum of sheaves of modules is equal to the product. This and the fact that $X^0$ is locally finite in $|X|$ explains the equalities between direct sums and products in the statement. Then since $\mathcal{K}_ X(\mathcal{F}) = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{K}_ X$ we see that the second equality follows from the first.

Let $j : Y = \coprod \nolimits _{\eta \in X^0} \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \eta }) \to X$ be the product of the morphisms $j_\eta $. We have to show that $\mathcal{K}_ X = j_*\mathcal{O}_ Y$. Observe that $\mathcal{K}_ Y = \mathcal{O}_ Y$ as $Y$ is a disjoint union of spectra of local rings of dimension $0$: in a local ring of dimension zero any nonzerodivisor is a unit. Next, note that pullbacks of meromorphic functions are defined for $j$ by Lemma 70.10.7. This gives a map

\[ \mathcal{K}_ X \longrightarrow j_*\mathcal{O}_ Y. \]

Let $U \in X_{\acute{e}tale}$ be affine. By Lemma 70.10.4 the left hand side evaluates to total ring of fractions of $\mathcal{O}_ X(U)$. On the other hand, the right hand side is equal to the product of the local rings of $U$ at the codimension $0$ points, i.e., the generic points of $U$. These two rings are equal (as we already saw in the proof of Lemma 70.10.4) by Algebra, Lemmas 10.25.4 and 10.66.7. Thus our map is an isomorphism.

Finally, we have to show that $R(X) = \Gamma (X, \mathcal{K}_ X)$. This follows from the case of schemes (Divisors, Lemma 31.23.6) applied to the schematic locus $X' \subset X$. Namely, the ring of rational functions of $X$ is by definition the same as the ring of rational functions on $X'$ as it is a dense open subspace of $X$ (see above). Certainly, $R(X')$ agrees with the ring of rational functions when $X'$ is viewed as a scheme. On the other hand, by our description of $\mathcal{K}_ X$ above, and the fact, seen above, that $X^0 \subset |X'|$ is contained in any dense open, we see that $\Gamma (X, \mathcal{K}_ X) = \Gamma (X', \mathcal{K}_{X'})$. Finally, use the compatibility recorded in Lemma 70.10.5. $\square$

[1] Conversely, if $X$ is decent, then condition (c) holds automatically.

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