Definition 31.23.1. Let $(X, \mathcal{O}_ X)$ be a locally ringed space. The sheaf of meromorphic functions on $X$ is the sheaf $\mathcal{K}_ X$ associated to the presheaf displayed above. A meromorphic function on $X$ is a global section of $\mathcal{K}_ X$.
31.23 Meromorphic functions and sections
This section contains only the general definitions and some elementary results. See [misconceptions] for some possible pitfalls1.
Let $(X, \mathcal{O}_ X)$ be a locally ringed space. For any open $U \subset X$ we have defined the set $\mathcal{S}(U) \subset \mathcal{O}_ X(U)$ of regular sections of $\mathcal{O}_ X$ over $U$, see Definition 31.14.6. The restriction of a regular section to a smaller open 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
\[ U \longmapsto \mathcal{S}(U)^{-1} \mathcal{O}_ X(U), \]
see Modules, Lemma 17.27.1.
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.
Example 31.23.2. Let $A = \mathbf{C}[x, \{ y_\alpha \} _{\alpha \in \mathbf{C}}]/ ((x - \alpha )y_\alpha , y_\alpha y_\beta )$. Any element of $A$ can be written uniquely as $f(x) + \sum \lambda _\alpha y_\alpha $ with $f(x) \in \mathbf{C}[x]$ and $\lambda _\alpha \in \mathbf{C}$. Let $X = \mathop{\mathrm{Spec}}(A)$. In this case $\mathcal{O}_ X = \mathcal{K}_ X$, since on any affine open $D(f)$ the ring $A_ f$ any nonzerodivisor is a unit (proof omitted).
Let $(X, \mathcal{O}_ X)$ be a locally ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. 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, Lemma 17.27.2.
Definition 31.23.3. Let $X$ be a locally ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules.
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
\[ \mathcal{K}_ X(\mathcal{F})_ x = \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \mathcal{K}_{X, x} = \mathcal{S}_ x^{-1}\mathcal{F}_ x \]
for any point $x \in X$. However, one has to be careful since it may not be the case that $\mathcal{S}_ x$ is the set of nonzerodivisors in the local ring $\mathcal{O}_{X, x}$. Namely, there is always an injective map
\[ \mathcal{K}_{X, x} \longrightarrow Q(\mathcal{O}_{X, x}) \]
to the total quotient ring. It is also surjective if and only if $\mathcal{S}_ x$ is the set of nonzerodivisors in $\mathcal{O}_{X, x}$. The sheaves of meromorphic sections aren't quasi-coherent modules in general, but they do have some properties in common with quasi-coherent modules.
Definition 31.23.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of locally ringed spaces. We say that pullbacks of meromorphic functions are defined for $f$ if for every pair of open $U \subset X$, $V \subset Y$ such that $f(U) \subset V$, and any section $s \in \Gamma (V, \mathcal{S}_ Y)$ the pullback $f^\sharp (s) \in \Gamma (U, \mathcal{O}_ X)$ is an element of $\Gamma (U, \mathcal{S}_ X)$.
In this case there is an induced map $f^\sharp : f^{-1}\mathcal{K}_ Y \to \mathcal{K}_ X$, in other words we obtain a commutative diagram of morphisms of ringed spaces
\[ \xymatrix{ (X, \mathcal{K}_ X) \ar[r] \ar[d]^ f & (X, \mathcal{O}_ X) \ar[d]^ f \\ (Y, \mathcal{K}_ Y) \ar[r] & (Y, \mathcal{O}_ Y) } \]
We sometimes denote $f^*(s) = f^\sharp (s)$ for a section $s \in \Gamma (Y, \mathcal{K}_ Y)$.
Lemma 31.23.5. Let $f : X \to Y$ be a morphism of schemes. In each of the following cases pullbacks of meromorphic functions are defined.
every weakly associated point of $X$ maps to a generic point of an irreducible component of $Y$,
$X$, $Y$ are integral and $f$ is dominant,
$X$ is integral and the generic point of $X$ maps to a generic point of an irreducible component of $Y$,
$X$ is reduced and every generic point of every irreducible component of $X$ maps to the generic point of an irreducible component of $Y$,
$X$ is locally Noetherian, and any associated point of $X$ maps to a generic point of an irreducible component of $Y$,
$X$ is locally Noetherian, has no embedded points and any generic point of an irreducible component of $X$ maps to the generic point of an irreducible component of $Y$, and
$f$ is flat.
Proof. The question is local on $X$ and $Y$. Hence we reduce to the case where $X = \mathop{\mathrm{Spec}}(A)$, $Y = \mathop{\mathrm{Spec}}(R)$ and $f$ is given by a ring map $\varphi : R \to A$. By the characterization of regular sections of the structure sheaf in Lemma 31.14.7 we have to show that $R \to A$ maps nonzerodivisors to nonzerodivisors. Let $t \in R$ be a nonzerodivisor.
If $R \to A$ is flat, then $t : R \to R$ being injective shows that $t : A \to A$ is injective. This proves (7).
In the other cases we note that $t$ is not contained in any of the minimal primes of $R$ (because every element of a minimal prime in a ring is a zerodivisor). Hence in case (1) we see that $\varphi (t)$ is not contained in any weakly associated prime of $A$. Thus this case follows from Algebra, Lemma 10.66.7. Case (5) is a special case of (1) by Lemma 31.5.8. Case (6) follows from (5) and the definitions. Case (4) is a special case of (1) by Lemma 31.5.12. Cases (2) and (3) are special cases of (4). $\square$
Lemma 31.23.6. Let $X$ be a scheme such that
every weakly associated point of $X$ is a generic point of an irreducible component of $X$, and
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
\[ \mathcal{K}_ X = \bigoplus \nolimits _{\eta \in X^0} j_{\eta , *}\mathcal{O}_{X, \eta } = \prod \nolimits _{\eta \in X^0} j_{\eta , *}\mathcal{O}_{X, \eta } \]
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,
$\mathcal{S}_ x \subset \mathcal{O}_{X, x}$ 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 $X$ (Morphisms, Definition 29.49.3) is the ring of meromorphic functions on $X$, in a formula: $R(X) = \Gamma (X, \mathcal{K}_ X)$.
Proof. Observe that a locally finite direct sum of sheaves of modules is equal to the product since you can check this on stalks for example. Then since $\mathcal{K}_ X(\mathcal{F}) = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{K}_ X$ we see that (2) follows from the other statements. Also, observe that part (6) follows from the initial statement of the lemma and Morphisms, Lemma 29.49.5 when $X^0$ is finite; the general case of (6) follows from this by glueing (argument omitted).
Let $j : Y = \coprod \nolimits _{\eta \in X^0} \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \eta }) \to X$ be the coproduct of the morphisms $j_\eta $. We have to show that $\mathcal{K}_ X = j_*\mathcal{O}_ Y$. First note 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 31.23.5. This gives a map
\[ \mathcal{K}_ X \longrightarrow j_*\mathcal{O}_ Y. \]
Let $\mathop{\mathrm{Spec}}(A) = U \subset X$ be an affine open. Then $A$ is a ring with finitely many minimal primes $\mathfrak q_1, \ldots , \mathfrak q_ t$ and every weakly associated prime of $A$ is one of the $\mathfrak q_ i$. We obtain $Q(A) = \prod A_{\mathfrak q_ i}$ by Algebra, Lemmas 10.25.4 and 10.66.7. In other words, already the value of the presheaf $U \mapsto \mathcal{S}(U)^{-1}\mathcal{O}_ X(U)$ agrees with $j_*\mathcal{O}_ Y(U)$ on our affine open $U$. Hence the displayed map is an isomorphism which proves the first displayed equality in the statement of the lemma.
Finally, we prove (1), (3), (4), and (5). Part (5) we saw during the course of the proof that $\mathcal{K}_ X = j_*\mathcal{O}_ Y$. The morphism $j$ is quasi-compact by our assumption that the set of irreducible components of $X$ is locally finite. Hence $j$ is quasi-compact and quasi-separated (as $Y$ is separated). By Schemes, Lemma 26.24.1 $j_*\mathcal{O}_ Y$ is quasi-coherent. This proves (1). Let $x \in X$. We may choose an affine open neighbourhood $U = \mathop{\mathrm{Spec}}(A)$ of $x$ all of whose irreducible components pass through $x$. Then $A \subset A_\mathfrak p$ because every weakly associated prime of $A$ is contained in $\mathfrak p$ hence elements of $A \setminus \mathfrak p$ are nonzerodivisors by Algebra, Lemma 10.66.7. It follows easily that any nonzerodivisor of $A_\mathfrak p$ is the image of a nonzerodivisor on a (possibly smaller) affine open neighbourhood of $x$. This proves (3). Part (4) follows from part (3) by computing stalks. $\square$
Definition 31.23.7. Let $X$ be a locally ringed space. 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. In other words, $s$ is a regular section of the invertible $\mathcal{K}_ X$-module $\mathcal{K}_ X(\mathcal{L})$, see Definition 31.14.6.
Let us spell out when (regular) meromorphic sections can be pulled back.
Lemma 31.23.8. Let $f : X \to Y$ be a morphism of locally ringed spaces. Assume that pullbacks of meromorphic functions are defined for $f$ (see Definition 31.23.4).
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 $\mathcal{O}_ Y$-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 31.23.9. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s$ be a regular meromorphic section of $\mathcal{L}$. Let us denote $\mathcal{I} \subset \mathcal{O}_ X$ the sheaf of ideals defined by the rule The formula makes sense since $\mathcal{L}(V) \subset \mathcal{K}_ X(\mathcal{L})(V)$. Then $\mathcal{I}$ is a quasi-coherent sheaf of ideals and we have injective maps whose cokernels are supported on closed nowhere dense subsets of $X$.
\[ \mathcal{I}(V) = \{ f \in \mathcal{O}_ X(V) \mid fs \in \mathcal{L}(V)\} . \]
\[ 1 : \mathcal{I} \longrightarrow \mathcal{O}_ X, \quad s : \mathcal{I} \longrightarrow \mathcal{L} \]
Proof. The question is local on $X$. Hence we may assume that $X = \mathop{\mathrm{Spec}}(A)$, and $\mathcal{L} = \mathcal{O}_ X$. After shrinking further we may assume that $s = a/b$ with $a, b \in A$ both nonzerodivisors in $A$. Set $I = \{ x \in A \mid x(a/b) \in A\} $.
To show that $\mathcal{I}$ is quasi-coherent we have to show that $I_ f = \{ x \in A_ f \mid x(a/b) \in A_ f\} $ for every $f \in A$. If $c/f^ n \in A_ f$, $(c/f^ n)(a/b) \in A_ f$, then we see that $f^ mc(a/b) \in A$ for some $m$, hence $c/f^ n \in I_ f$. Conversely it is easy to see that $I_ f$ is contained in $\{ x \in A_ f \mid x(a/b) \in A_ f\} $. This proves quasi-coherence.
Let us prove the final statement. It is clear that $(b) \subset I$. Hence $V(I) \subset V(b)$ is a nowhere dense subset as $b$ is a nonzerodivisor. Thus the cokernel of $1$ is supported in a nowhere dense closed set. The same argument works for the cokernel of $s$ since $s(b) = (a) \subset sI \subset A$. $\square$
Definition 31.23.10. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s$ be a regular meromorphic section of $\mathcal{L}$. The sheaf of ideals $\mathcal{I}$ constructed in Lemma 31.23.9 is called the ideal sheaf of denominators of $s$.
[1] Danger, Will Robinson!
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