Definition 71.10.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The sheaf of meromorphic functions on $X$ is the sheaf $\mathcal{K}_ X$ on $X_{\acute{e}tale}$ associated to the presheaf displayed above. A meromorphic function on $X$ is a global section of $\mathcal{K}_ X$.
71.10 Meromorphic functions and sections
This section is the analogue of Divisors, Section 31.23. Beware: it is even easier to make mistakes with this material in the case of algebraic space, than it is in the case of schemes!
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For any scheme $U$ étale over $X$ we have defined the set $\mathcal{S}(U) \subset \mathcal{O}_ X(U)$ of regular sections of $\mathcal{O}_ X$ over $U$, see Definition 71.7.4. The restriction of a regular section to $V/U$ étale is regular. Hence $\mathcal{S} : U \mapsto \mathcal{S}(U)$ is a subsheaf (of sets) of $\mathcal{O}_ X$. We sometimes denote $\mathcal{S} = \mathcal{S}_ X$ if we want to indicate the dependence on $X$. Moreover, $\mathcal{S}(U)$ is a multiplicative subset of the ring $\mathcal{O}_ X(U)$ for each $U$. Hence we may consider the presheaf of rings
\[ U \longmapsto \mathcal{S}(U)^{-1} \mathcal{O}_ X(U), \]
on $X_{\acute{e}tale}$ and its sheafification, see Modules on Sites, Section 18.44.
Since each element of each $\mathcal{S}(U)$ is a nonzerodivisor on $\mathcal{O}_ X(U)$ we see that the natural map of sheaves of rings $\mathcal{O}_ X \to \mathcal{K}_ X$ is injective. Moreover, by the compatibility of sheafification and taking stalks we see that
\[ \mathcal{K}_{X, \overline{x}} = \mathcal{S}_{\overline{x}}^{-1}\mathcal{O}_{X, \overline{x}} \]
for any geometric point $\overline{x}$ of $X$. The set $\mathcal{S}_{\overline{x}}$ is a subset of the set of nonzerodivisors of $\mathcal{O}_{X, \overline{x}}$, but in general not equal to this.
Lemma 71.10.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $U$ affine and étale over $X$ the set $\mathcal{S}_ X(U)$ is the set of nonzerodivisors in $\mathcal{O}_ X(U)$.
Proof. Follows from Lemma 71.7.5. $\square$
Next, let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules on $X_{\acute{e}tale}$. Consider the presheaf $U \mapsto \mathcal{S}(U)^{-1}\mathcal{F}(U)$. Its sheafification is the sheaf $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{K}_ X$, see Modules on Sites, Lemma 18.44.2.
Definition 71.10.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules on $X_{\acute{e}tale}$.
We denote $\mathcal{K}_ X(\mathcal{F})$ the sheaf of $\mathcal{K}_ X$-modules which is the sheafification of the presheaf $U \mapsto \mathcal{S}(U)^{-1}\mathcal{F}(U)$. Equivalently $\mathcal{K}_ X(\mathcal{F}) = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{K}_ X$ (see above).
A meromorphic section of $\mathcal{F}$ is a global section of $\mathcal{K}_ X(\mathcal{F})$.
In particular we have
\[ \mathcal{K}_ X(\mathcal{F})_{\overline{x}} = \mathcal{F}_{\overline{x}} \otimes _{\mathcal{O}_{X, \overline{x}}} \mathcal{K}_{X, \overline{x}} = \mathcal{S}_{\overline{x}}^{-1}\mathcal{F}_{\overline{x}} \]
for any geometric point $\overline{x}$ of $X$. However, one has to be careful since it may not be the case that $\mathcal{S}_{\overline{x}}$ is the set of nonzerodivisors in the étale local ring $\mathcal{O}_{X, \overline{x}}$ as we pointed out above. The sheaves of meromorphic sections aren't quasi-coherent modules in general, but they do have some properties in common with quasi-coherent modules.
Lemma 71.10.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume
every weakly associated point of $X$ is a point of codimension $0$, and
$X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma 67.49.1.
Then
$\mathcal{K}_ X$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras,
for $U \in X_{\acute{e}tale}$ affine $\mathcal{K}_ X(U)$ is the total ring of fractions of $\mathcal{O}_ X(U)$,
for a geometric point $\overline{x}$ the set $\mathcal{S}_{\overline{x}}$ the set of nonzerodivisors of $\mathcal{O}_{X, \overline{x}}$, and
for a geometric point $\overline{x}$ the ring $\mathcal{K}_{X, \overline{x}}$ is the total ring of fractions of $\mathcal{O}_{X, \overline{x}}$.
Proof. By Lemma 71.7.5 we see that $U \in X_{\acute{e}tale}$ affine $\mathcal{S}_ X(U) \subset \mathcal{O}_ X(U)$ is the set of nonzerodivisors in $\mathcal{O}_ X(U)$. Thus the presheaf $\mathcal{S}^{-1}\mathcal{O}_ X$ is equal to
\[ U \longmapsto Q(\mathcal{O}_ X(U)) \]
on $X_{affine, {\acute{e}tale}}$, with notation as in Algebra, Example 10.9.8. Observe that the codimension $0$ points of $X$ correspond to the generic points of $U$, see Properties of Spaces, Lemma 66.11.1. Hence if $U = \mathop{\mathrm{Spec}}(A)$, then $A$ is a ring with finitely many minimal primes such that any weakly associated prime of $A$ is minimal. The same is true for any étale extension of $A$ (because the spectrum of such is an affine scheme étale over $X$ hence can play the role of $A$ in the previous sentence). In order to show that our presheaf is a sheaf and quasi-coherent it suffices to show that
\[ Q(A) \otimes _ A B \longrightarrow Q(B) \]
is an isomorphism when $A \to B$ is an étale ring map, see Properties of Spaces, Lemma 66.29.3. (To define the displayed arrow, observe that since $A \to B$ is flat it maps nonzerodivisors to nonzerodivisors.) By Algebra, Lemmas 10.25.4 and 10.66.7. we have
\[ Q(A) = \prod \nolimits _{\mathfrak p \subset A\text{ minimal}} A_\mathfrak p \quad \text{and}\quad Q(B) = \prod \nolimits _{\mathfrak q \subset B\text{ minimal}} B_\mathfrak q \]
Since $A \to B$ is étale, the minimal primes of $B$ are exactly the primes of $B$ lying over the minimal primes of $A$ (for example by More on Algebra, Lemma 15.44.2). By Algebra, Lemmas 10.153.10, 10.153.3 (13), and 10.153.5 we see that $A_\mathfrak p \otimes _ A B$ is a finite product of local rings finite étale over $A_\mathfrak p$. This cleary implies that $A_\mathfrak p \otimes _ A B = \prod _{\mathfrak q\text{ lies over }\mathfrak p} B_\mathfrak q$ as desired.
At this point we know that (1) and (2) hold. Proof of (3). Let $s \in \mathcal{O}_{X, \overline{x}}$ be a nonzerodivisor. Then we can find an étale neighbourhood $(U, \overline{u}) \to (X, \overline{x})$ and $f \in \mathcal{O}_ X(U)$ mapping to $s$. Let $u \in U$ be the point determined by $\overline{u}$. Since $\mathcal{O}_{U, u} \to \mathcal{O}_{X, \overline{x}}$ is faithfully flat (as a strict henselization), we see that $f$ maps to a nonzerodivisor in $\mathcal{O}_{U, u}$. By Divisors, Lemma 31.23.6 after shrinking $U$ we find that $f$ is a nonzerodivisor and hence a section of $\mathcal{S}_ X(U)$. Part (4) follows from (3) by computing stalks. $\square$
Lemma 71.10.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume
every weakly associated point of $X$ is a point of codimension $0$, and
$X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma 67.49.1.
$X$ is representable by a scheme $X_0$ (awkward but temporary notation).
Then the sheaf of meromorphic functions $\mathcal{K}_ X$ is the quasi-coherent sheaf of $\mathcal{O}_ X$-algebras associated to the quasi-coherent sheaf of meromorphic functions $\mathcal{K}_{X_0}$.
Proof. For the equivalence between $\mathit{QCoh}(\mathcal{O}_ X)$ and $\mathit{QCoh}(\mathcal{O}_{X_0})$, please see Properties of Spaces, Section 66.29. The lemma is true because $\mathcal{K}_ X$ and $\mathcal{K}_{X_0}$ are quasi-coherent and have the same value on corresponding affine opens of $X$ and $X_0$ by Lemma 71.10.4 and Divisors, Lemma 31.23.6. $\square$
Definition 71.10.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We say that pullbacks of meromorphic functions are defined for $f$ if for every commutative diagram with $U \in X_{\acute{e}tale}$ and $V \in Y_{\acute{e}tale}$ and any section $s \in \mathcal{S}_ Y(V)$ the pullback $f^\sharp (s) \in \mathcal{O}_ X(U)$ is an element of $\mathcal{S}_ X(U)$.
\[ \xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & Y } \]
In this case there is an induced map $f^\sharp : f_{small}^{-1}\mathcal{K}_ Y \to \mathcal{K}_ X$, in other words we obtain a commutative diagram of morphisms of ringed topoi
\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{K}_ X) \ar[r] \ar[d]^{f_{small}} & (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \ar[d]^{f_{small}} \\ (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{K}_ Y) \ar[r] & (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y) } \]
We sometimes denote $f^*(s) = f^\sharp (s)$ for a section $s \in \Gamma (Y, \mathcal{K}_ Y)$.
Lemma 71.10.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Pullbacks of meromorphic sections are defined in each of the following cases
weakly associated points of $X$ are mapped to points of codimension $0$ on $Y$,
$f$ is flat,
add more here as needed.
Proof. Working étale locally, this translates into the case of schemes, see Divisors, Lemma 31.23.5. To do the translation use Lemma 71.7.5 (description of regular sections), Definition 71.2.2 (definition of weakly associated points), and Properties of Spaces, Lemma 66.11.1 (description of codimension $0$ points). $\square$
Lemma 71.10.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume
every weakly associated point of $X$ is a point of codimension $0$, and
$X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma 67.49.1,
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 68.20.3 and Section 68.6 we see that $X$ is decent1. Thus $X^0 \subset |X|$ is the set of generic points of irreducible components (Decent Spaces, Lemma 68.20.1) 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 68.10.2). 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 71.10.7. This gives a map
\[ \mathcal{K}_ X \longrightarrow j_*\mathcal{O}_ Y. \]
Let $U \in X_{\acute{e}tale}$ be affine. By Lemma 71.10.4 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 71.10.4) 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 71.10.5. $\square$
Definition 71.10.9. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. A meromorphic section $s$ of $\mathcal{L}$ is said to be regular if the induced map $\mathcal{K}_ X \to \mathcal{K}_ X(\mathcal{L})$ is injective.
Let us spell out when (regular) meromorphic sections can be pulled back.
Lemma 71.10.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that pullbacks of meromorphic functions are defined for $f$ (see Definition 71.10.6).
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ Y$-modules. There is a canonical pullback map $f^* : \Gamma (Y, \mathcal{K}_ Y(\mathcal{F})) \to \Gamma (X, \mathcal{K}_ X(f^*\mathcal{F}))$ for meromorphic sections of $\mathcal{F}$.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. A regular meromorphic section $s$ of $\mathcal{L}$ pulls back to a regular meromorphic section $f^*s$ of $f^*\mathcal{L}$.
Proof. Omitted. $\square$
Lemma 71.10.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying (a), (b), and (c) of Lemma 71.10.8. Then every invertible $\mathcal{O}_ X$-module $\mathcal{L}$ has a regular meromorphic section.
Proof. With notation as in Lemma 71.10.8 the stalk $\mathcal{L}_\eta $ of $\mathcal{L}$ at is defined for all $\eta \in X^0$ and it is a rank $1$ free $\mathcal{O}_{X, \eta }$-module. Pick a generator $s_\eta \in \mathcal{L}_\eta $ for all $\eta \in X^0$. It follows immediately from the description of $\mathcal{K}_ X$ and $\mathcal{K}_ X(\mathcal{L})$ in Lemma 71.10.8 that $s = \prod s_\eta $ is a regular meromorphic section of $\mathcal{L}$. $\square$
[1] Conversely, if $X$ is decent, then condition (c) holds automatically.
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