Éléments de Géométrie Algébrique — English

§20. Meromorphic functions; pseudo-morphisms

20.0. Introduction

Most of the notions and results of §§20 and 21 attach directly to chap. I, and depend hardly at all on chaps. II to IV, except for the occasional use of the notion of depth and of regular local ring (in (20.6), (21.11), (21.13) and (21.15)), of Zariski's "Main theorem" in (20.4) and (21.12), and of properties of transversely regular immersions in (20.6) and (21.15).

In §20, we introduce several variants of the notion of rational map, already studied in (I, 7) from a point of view still rather close to the classical viewpoint, and for this reason rather poorly adapted to the case of preschemes that are not necessarily reduced. The notions and results of §20 are used in §21 (nos. 21.1 to 21.7) to develop the general notion of divisor and its most elementary properties. This notion is especially convenient when the local rings of the preschemes under consideration are Noetherian and integrally closed, and especially when they are moreover factorial (21.6 and 21.7), because of its identification in this latter case with the notion of 1-codimensional cycle (linear combination of irreducible subpreschemes of codimension 1). In (21.9), one determines the divisors on a Noetherian prescheme of dimension 1 but not necessarily normal, which is useful for various applications. Nos. (21.11) and (21.12) give two important theorems, due respectively to Auslander-Buchsbaum and Van der Waerden, and related to the notion of factorial ring (nos. (21.9), (21.11) and (21.12) are independent of one another). In nos. (21.13) and (21.14), also independent of the previous three, we study a useful variant of the notion of factorial local ring, that of parafactorial local ring, which is introduced notably [41] in the development of comparison theorems between the Picard group of a projective prescheme $X$ over a field $k$ and that of a "hyperplane section". One will see in (21.14.1) (Ramanujam-Samuel theorem) that parafactorial local rings are much more numerous than one might a priori have expected.

In (20.5), (20.6) and (21.15), we take up the preceding notions again but from a viewpoint "relative" to a fixed base prescheme. For the moment these notions are used only rather rarely; in particular, the notion of relative divisor is scarcely used except when one is dealing with positive divisors, and in this case it is explained advantageously without recourse to the notion of relative meromorphic function, by means of the notion of transversely regular immersion of codimension 1. The reader will therefore find it advantageous to omit these sections on a first reading.

20.1. Meromorphic functions

(20.1.1). Let $(X,{\mathcal{O}}_X)$ be a ringed space, and let $\mathcal{S}$ be a subsheaf of sets of ${\mathcal{O}}_X$. For every open $U$ of $X$, consider the ring of fractions $\Gamma (U,{\mathcal{O}}_X)[\Gamma (U,\mathcal{S}{)}^{-1}]$ (Bourbaki, Alg. comm., chap. II, §2, n° 1). It is immediate that the map $U\mapsto \Gamma (U,{\mathcal{O}}_X)[\Gamma (U,\mathcal{S}{)}^{-1}]$ is a presheaf of rings $(0_I,\mathrm{1.5.1}and\mathrm{1.5.7})$. We denote by ${\mathcal{O}}_X[{\mathcal{S}}^{-1}]$ the sheaf of rings associated to this presheaf and we say that this is the sheaf of rings of fractions of ${\mathcal{O}}_X$ with denominators in $\mathcal{S}$; it is a flat ${\mathcal{O}}_X$-module. It is immediate that for every $x\in X$, one has a canonical isomorphism

$$(\mathrm{20.1.1.1})({\mathcal{O}}_X[{\mathcal{S}}^{-1}]{)}_x\stackrel{\sim }{\to }{\mathcal{O}}_x[{\mathcal{S}}_x^{-1}],$$

since the reasoning of $(0_I,\mathrm{1.4.5})$ generalizes immediately to the case where one has an inductive system $(A_{\alpha },{\varphi }_{\beta \alpha })$ of rings, and for each index $\alpha$ a subset $S_{\alpha }$ of $A_{\alpha }$ such that

${\varphi }_{\beta \alpha }(S_{\alpha })\subset S_{\beta }$ for $\alpha \le \beta$; one then takes for $S$ the inductive limit in $A=\lim A_{\alpha }$ of the inductive system of subsets $(S_{\alpha })$.

(20.1.2). Let now $\mathcal{F}$ be an ${\mathcal{O}}_X$-module. One then sets

  (20.1.2.1)             ℱ[𝒮⁻¹] = ℱ ⊗_{𝒪_X} 𝒪_X[𝒮⁻¹]

and one says that this is the sheaf of modules of fractions of $\mathcal{F}$ with denominators in $\mathcal{S}$; it is immediate that it is associated to the presheaf of modules $U\mapsto \Gamma (U,\mathcal{F})[\Gamma (U,\mathcal{S}{)}^{-1}]$, and that for every $x\in X$ one has a canonical isomorphism

$$(\mathrm{20.1.2.2})(\mathcal{F}[{\mathcal{S}}^{-1}]{)}_x\stackrel{\sim }{\to }{\mathcal{F}}_x[{\mathcal{S}}_x^{-1}].$$

(20.1.3). We shall be interested here in the case where $\mathcal{S}$ is the subsheaf $\mathcal{S}({\mathcal{O}}_X)$ of ${\mathcal{O}}_X$ such that for every open $U$, $\Gamma (U,\mathcal{S})$ is the set of regular elements of the ring $\Gamma (U,{\mathcal{O}}_X)$; it is immediate that this is a sheaf (and not only a presheaf), the regularity of a section of ${\mathcal{O}}_X$ over $U$ being verified "fibre by fibre" (i.e. meaning that the germ of the section at $x$ is regular in ${\mathcal{O}}_{X,x}$ for every $x\in U$); in other words $\mathcal{S}({\mathcal{O}}_X{)}_x$ is none other than the set of regular elements of ${\mathcal{O}}_{X,x}$. The corresponding sheaf of rings

$${\mathcal{M}}_X={\mathcal{O}}_X[{\mathcal{S}}^{-1}]$$

is called the sheaf of germs of meromorphic functions on $X$, and the sections of ${\mathcal{M}}_X$ over $X$ are called the meromorphic functions on $X$; they form a ring which one denotes $M(X)$. For every ${\mathcal{O}}_X$-Module $\mathcal{F}$,

                         ℱ ⊗_{𝒪_X} 𝓜_X = ℱ[𝒮⁻¹]

is also denoted ${\mathcal{M}}_X(\mathcal{F})$ and called the sheaf of germs of meromorphic sections of $\mathcal{F}$; its sections over $X$ form an $M(X)$-module denoted $M(X,\mathcal{F})$, whose elements are called meromorphic sections of $\mathcal{F}$ over $X$. These definitions imply that for every open $U$ of $X$, one has a canonical isomorphism ${\mathcal{M}}_X(\mathcal{F})|U\stackrel{\sim }{\to }{\mathcal{M}}_U(\mathcal{F}|U)$, in particular ${\mathcal{M}}_X|U\stackrel{\sim }{\to }{\mathcal{M}}_U$.

(20.1.3.1). If $X$ is a reduced prescheme, one will note that if an element $s\in \Gamma (U,{\mathcal{O}}_X)$ is such that $s_{\xi }\ne 0$ for every maximal point $\xi$ of $U$, then $s$ is regular. Indeed, if $st=0$ for a $t\in \Gamma (U,{\mathcal{O}}_X)$, one has $s_{\xi }{t}_{\xi }=0$, hence $t_{\xi }=0$ since ${\mathcal{O}}_{X,\xi }$ is a field, and to say that $t_{\xi }=0$ for every maximal point $\xi$ of $X$ means that $t=0$: one is at once reduced to the case where $U$ is affine, and an element of a reduced ring belonging to every minimal prime ideal is zero by definition. The converse is true if the set of irreducible components of $X$ is locally finite. One is at once reduced to the case where $X=\mathrm{Spec}(A)$ is affine; if ${\mathfrak{p}}_i$ ($1\le i\le n$) are the minimal prime ideals of $A$ and $s\in {\mathfrak{p}}_i$ for some index $i$, then there exists $t\in A$ such that $t\in {\mathfrak{p}}_j$ for $j\ne i$ and $t\notin {\mathfrak{p}}_i$ (Bourbaki, Alg. comm., chap. II, §1, n° 1, prop. 1); one therefore has $st\in {\mathfrak{p}}_i$ for every $i$, hence $st=0$ since $A$ is reduced; so $s$ is not regular.

(20.1.4). For every open $U$ of $X$, the homomorphism $t\mapsto t/1$ from $\Gamma (U,{\mathcal{O}}_X)$ to $\Gamma (U,{\mathcal{O}}_X)[\Gamma (U,\mathcal{S}{)}^{-1}]$ (which is none other than the total ring of fractions of

$\Gamma (U,{\mathcal{O}}_X)$) is injective; these homomorphisms therefore define a canonical injective homomorphism

$$(\mathrm{20.1.4.1})i:{\mathcal{O}}_X\to {\mathcal{M}}_X$$

which allows one to identify ${\mathcal{O}}_X$ with a subsheaf of ${\mathcal{M}}_X$. Given a meromorphic function $\varphi \in M(X)$, one says that $\varphi$ is defined on an open $U$ of $X$ if $\varphi |U$ is a section of ${\mathcal{O}}_X$ over $U$; the sheaf axioms show that, for a given section $\varphi$, there is a largest open on which $\varphi$ is defined; one calls this the domain of definition of $\varphi$ and denotes it $dom(\varphi )$.

(20.1.5). For every ${\mathcal{O}}_X$-Module $\mathcal{F}$, one deduces from (20.1.4.1) a di-homomorphism formed of $i$ and the homomorphism of sheaves of additive groups

  (20.1.5.1)             1_ℱ ⊗ i : ℱ → 𝓜_X(ℱ) = ℱ ⊗_{𝒪_X} 𝓜_X.

One will note that the latter is no longer injective in general; when it is injective, one says that $\mathcal{F}$ is strictly torsion-free: this means that for every open $U$ of $X$ and every section $s\in \Gamma (U,{\mathcal{O}}_X)$ which is a regular element of that ring, the homothety $z\mapsto sz$ of $\Gamma (U,\mathcal{F})$ is injective; this condition is evidently satisfied if $\mathcal{F}$ is locally free.

Proposition (20.1.6).

Let $X$ be a locally Noetherian prescheme, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module. For $\mathcal{F}$ to be strictly torsion-free, it is necessary and sufficient that $Ass(\mathcal{F})\subset Ass({\mathcal{O}}_X)$.

One is at once reduced to the case where $X=\mathrm{Spec}(A)$ is affine, $\mathcal{F}=\tilde{M}$, and one knows that the elements $s$ of $A$ belonging to an ideal of $Ass(M)$ are exactly those for which the homothety $z\mapsto sz$ is not injective (Bourbaki, Alg. comm., chap. IV, §1, n° 1, cor. 2 of prop. 2).

(20.1.7). If $u$ is a section of ${\mathcal{M}}_X(\mathcal{F})$ over $X$, one says that $u$ is defined at a point $x\in X$ if there exists an open neighbourhood $V$ of $x$ in $X$ such that $u|V$ is the image of a section of $\mathcal{F}$ over $V$ under the di-homomorphism (20.1.5.1). One says that $u$ is defined on an open $U$ of $X$ if it is defined at every point of $U$; there is again a largest open on which $u$ is defined, called the domain of definition of $u$ and denoted $dom(u)$. When $\mathcal{F}$ is strictly torsion-free, so that $\mathcal{F}$ is identified by (20.1.5.1) with a subsheaf of ${\mathcal{M}}_X(\mathcal{F})$, saying that $u$ is defined on $U$ means that $u|U$ is a section of $\mathcal{F}$ over $U$.

(20.1.8). In accordance with the general notation $(0_I,\mathrm{5.4.7})$, one denotes by ${\mathcal{M}}_X^{\times }$ the sheaf of multiplicative groups such that $\Gamma (U,{\mathcal{M}}_X^{\times })$ is (for every open $U$ of $X$) the group of invertible elements of $\Gamma (U,{\mathcal{M}}_X)$. This sheaf is none other than the sheaf $\mathcal{S}({\mathcal{M}}_X)$ defined in (20.1.3): indeed, if $s\in \Gamma (U,\mathcal{S}({\mathcal{M}}_X))$, then for every $x\in U$ there exists an open neighbourhood $V\subset U$ of $x$ such that $s|V$ is a regular element in the total ring of fractions of $\Gamma (V,{\mathcal{O}}_X)$, and one knows that such an element is necessarily invertible in this ring of fractions. We shall say that the sections of ${\mathcal{M}}_X^{\times }$ over $X$ are the regular meromorphic functions (one will note that we are departing here from the terminology followed by certain authors, who call "regular" meromorphic functions those which are sections of ${\mathcal{O}}_X$, identified with a subsheaf of ${\mathcal{M}}_X$).

Let $\mathcal{L}$ be an invertible ${\mathcal{O}}_X$-Module $(0_I,\mathrm{5.4.1})$; then it is clear that ${\mathcal{M}}_X(\mathcal{L})=\mathcal{L}{\otimes }_{{\mathcal{O}}_X}{\mathcal{M}}_X$

is an invertible ${\mathcal{M}}_X$-Module. Let $U$ be an open such that $\mathcal{L}|U$ is isomorphic to ${\mathcal{O}}_U$; since every automorphism of ${\mathcal{M}}_U$ is multiplication by an invertible element of $\Gamma (U,{\mathcal{M}}_X)$ $(0_I,\mathrm{5.4.7})$, it amounts to the same thing to say that a section $s\in \Gamma (U,{\mathcal{M}}_X(\mathcal{L}))$ has invertible image in $\Gamma (U,{\mathcal{M}}_X)$ under an isomorphism or under every isomorphism onto $\Gamma (U,{\mathcal{M}}_X)$; one will say in this case that $s$ is a regular meromorphic section of $\mathcal{L}$ over $U$; a section $s$ of $\mathcal{L}$ over $X$ will be called a regular meromorphic section of $\mathcal{L}$ if, for every open $U$ such that $\mathcal{L}|U$ is isomorphic to ${\mathcal{O}}_U$, $s|U$ is a regular meromorphic section of $\mathcal{L}$ over $U$. One denotes by $({\mathcal{M}}_X(\mathcal{L}){)}^{\times }$ the subsheaf of ${\mathcal{M}}_X(\mathcal{L})$ such that for every open $U$, $\Gamma (U,({\mathcal{M}}_X(\mathcal{L}){)}^{\times })$ is the set of regular meromorphic sections of $\mathcal{L}$ over $U$. Let $s$ be a meromorphic section of $\mathcal{L}$ over $X$ (i.e. a section of ${\mathcal{M}}_X(\mathcal{L})$); it defines a homomorphism $h_s:{\mathcal{M}}_X\to {\mathcal{M}}_X(\mathcal{L})$ which to every section $t$ of ${\mathcal{M}}_X$ over an open $U$ associates $(s|U)t$. It follows at once from the foregoing that, for $s$ to be regular, it is necessary and sufficient that $h_s$ be injective, and in fact $h_s$ is then a bijective homomorphism from ${\mathcal{M}}_X$ to ${\mathcal{M}}_X(\mathcal{L})$, and its restriction to ${\mathcal{M}}_X^{\times }$ is a bijection onto $({\mathcal{M}}_X(\mathcal{L}){)}^{\times }$. One concludes that the homothety $t\mapsto ts$ is an isomorphism from $M(X)$ onto $M(X,\mathcal{L})$.

(20.1.9). Let $s$ be a regular meromorphic section of the invertible ${\mathcal{O}}_X$-Module $\mathcal{L}$ over $X$; then, for every ${\mathcal{O}}_X$-Module $\mathcal{F}$, $s$ likewise defines a homomorphism $h_s\otimes 1_{\mathcal{F}}:{\mathcal{M}}_X(\mathcal{F})\to {\mathcal{M}}_X(\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{L})$, which is again bijective.

(20.1.10). Let $s$ be a meromorphic section of an invertible ${\mathcal{O}}_X$-Module $\mathcal{L}$ over $X$; for $s$ to be regular, it is necessary and sufficient that there exist a meromorphic section $s^{\prime }$ of ${\mathcal{L}}^{-1}$ over $X$ such that the canonical image of $s\otimes s^{\prime }$ in ${\mathcal{M}}_X$ $(0_I,\mathrm{5.4.3})$ is the unit section, and this section $s^{\prime }$ is then unique: indeed, the necessity of the local existence of such a section is evident, and its local uniqueness entails its global existence (and uniqueness); moreover, the existence of $s^{\prime }$ is trivially sufficient for $s$ to be regular. One will set $s^{\prime }=s^{-1}$.

Finally, if ${\mathcal{L}}^{\prime }$ is a second invertible ${\mathcal{O}}_X$-Module, $s$ (resp. $s^{\prime }$) a regular meromorphic section of $\mathcal{L}$ (resp. ${\mathcal{L}}^{\prime }$) over $X$, then $s\otimes s^{\prime }$ is evidently a regular meromorphic section of $\mathcal{L}\otimes {\mathcal{L}}^{\prime }$ over $X$.

(20.1.11). If $f:X^{\prime }\to X$ is a morphism of ringed spaces, there is in general no natural map sending a meromorphic function on $X$ to a meromorphic function on $X^{\prime }$. For example, if $X$ is the spectrum of an integral local ring $A$, $X^{\prime }$ that of its residue field $k$, there is no natural homomorphism from the field of fractions $K$ of $A$ to $k$, and one can send an element of $K$ to an element of $k$ only if it is already in $A$.

More generally, if $f=(\psi ,\theta )$, denote, for every open $U$ of $X$, by ${\mathcal{S}}_f(U)$ the set of regular sections $s\in \Gamma (U,{\mathcal{O}}_X)$ such that the image of $s$ under

                         Γ(θ♯) : Γ(U, 𝒪_X) → Γ(f⁻¹(U), 𝒪_{X'})

is a regular section. It is immediate that $U\mapsto {\mathcal{S}}_f(U)$ is a subsheaf of the sheaf of sets $\mathcal{S}({\mathcal{O}}_X)$, which one denotes ${\mathcal{S}}_f$. One sets ${\mathcal{M}}_f={\mathcal{O}}_X[{\mathcal{S}}_f^{-1}]$; this is a subsheaf

of rings of ${\mathcal{M}}_X$, and one canonically deduces from $\theta ♯:\psi \ast ({\mathcal{O}}_X)\to {\mathcal{O}}_{X^{\prime }}$ a homomorphism of sheaves of rings ${\theta }^{\prime }♯:\psi \ast ({\mathcal{M}}_f)\to {\mathcal{M}}_{X^{\prime }}$ extending $\theta ♯$ (Bourbaki, Alg. comm., chap. II, §2, n° 1, prop. 2); whence, recalling that $f\ast ({\mathcal{M}}_f)=\psi \ast ({\mathcal{M}}_f){\otimes }_{\psi \ast ({\mathcal{O}}_X)}{\mathcal{O}}_{X^{\prime }}$, a canonical homomorphism of ${\mathcal{O}}_{X^{\prime }}$-Algebras

$$(\mathrm{20.1.11.1})f\ast ({\mathcal{M}}_f)\to {\mathcal{M}}_{X^{\prime }}.$$

For every meromorphic function $\varphi$ on $X$ which is a section of ${\mathcal{M}}_f$, $\Gamma ({\theta }^{\prime }♯)(\varphi )$ is a meromorphic function on $X^{\prime }$, called the inverse image of $\varphi$ under $f$, and denoted $\varphi \circ f$ if this entails no confusion.

Similarly, if $\mathcal{F}$ is an ${\mathcal{O}}_X$-Module, one sets ${\mathcal{M}}_f(\mathcal{F})=\mathcal{F}{\otimes }_{{\mathcal{O}}_X}{\mathcal{M}}_f$, and one immediately deduces from ${\theta }^{\prime }♯$ a canonical homomorphism (also written $u\mapsto u\circ f$)

                         Γ(X, 𝓜_f(ℱ)) → Γ(X', 𝓜_{X'}(f*(ℱ))).

Moreover, if $u\in \Gamma (X,{\mathcal{M}}_f(\mathcal{F}))$ is defined (20.1.7) at a point $x$, $u$ coincides, on a neighbourhood $U$ of $x$, with a section of the form ${\sum }_i{h}_i\otimes (t_i/s_i)$, where the $h_i$ belong to $\Gamma (U,\mathcal{F})$, the $t_i$ to $\Gamma (U,{\mathcal{O}}_X)$, and the $s_i$ to $\Gamma (U,{\mathcal{S}}_f)$. As by hypothesis the images of the $s_i$ in $\Gamma (f^{-1}(U),{\mathcal{O}}_{X^{\prime }})$ are regular, one sees that $u\circ f$ is defined at every point of $f^{-1}(U)$; in other words, one has

$$(\mathrm{20.1.11.2})f^{-1}(dom(u))\subset dom(u\circ f).$$

We shall see later (20.6.5, (i)) examples (with $\mathcal{F}={\mathcal{O}}_X$) where the two sides of (20.1.11.2) may be distinct.

Consider in particular the case where ${\mathcal{M}}_f={\mathcal{M}}_X$; then, if $\mathcal{L}$ is an invertible ${\mathcal{O}}_X$-Module, the image in ${\mathcal{M}}_{X^{\prime }}(f\ast (\mathcal{L}))$, under $\Gamma ({\theta }^{\prime }♯)$, of a regular meromorphic section of $\mathcal{L}$ over $X$ (20.1.8) is a regular meromorphic section of $f\ast (\mathcal{L})$ over $X^{\prime }$, as follows at once from the definition of these sections and from the fact that a homomorphism of rings sends an invertible element to an invertible element.

Let $f^{\prime }:X^{\prime \prime }\to X^{\prime }$ be a second morphism of ringed spaces, and suppose that ${\mathcal{M}}_f={\mathcal{M}}_X$ and ${\mathcal{M}}_{f^{\prime }}={\mathcal{M}}_{X^{\prime }}$; then, if one sets $f^{\prime \prime }=f\circ f^{\prime }$, one also has ${\mathcal{M}}_{f^{\prime \prime }}={\mathcal{M}}_X$, and one sees at once that for every meromorphic section $u$ of $\mathcal{F}$ over $X$, one has $u\circ f^{\prime \prime }=(u\circ f)\circ f^{\prime }$.

Proposition (20.1.12).

If the morphism $f:X^{\prime }\to X$ is flat $(0_I,\mathrm{6.7.1})$, one has ${\mathcal{M}}_f={\mathcal{M}}_X$, and the homomorphism $\varphi \mapsto \varphi \circ f$ is defined on all of $M(X)$. Moreover, if $f$ is a (flat) morphism of ringed spaces in local rings, one has $dom(\varphi \circ f)=f^{-1}(dom(\varphi ))$; if in addition $f$ is surjective (hence faithfully flat), the homomorphism $\varphi \mapsto \varphi \circ f$ is injective.

The first assertion follows from the fact that, if $B$ is an $A$-algebra which is a flat $A$-module, every element of $A$ which is not a zero-divisor in $A$ is not a zero-divisor in $B$ $(0_I,\mathrm{6.3.4})$. To prove the other assertions, note that, for every $x^{\prime }\in X^{\prime }$, if $x=f(x^{\prime })$, ${\mathcal{O}}_{X^{\prime },x^{\prime }}$ is a flat ${\mathcal{O}}_{X,x}$-module, and since the homomorphism ${\mathcal{O}}_{X,x}\to {\mathcal{O}}_{X^{\prime },x^{\prime }}$ is local by hypothesis, it is injective $(0_I,\mathrm{6.5.1}and\mathrm{6.6.2})$; if one sets $A={\mathcal{O}}_{X,x}$, $B={\mathcal{O}}_{X^{\prime },x^{\prime }}$, so that $A$ identifies with a subring of $B$, $(f\ast ({\mathcal{M}}_X){)}_{x^{\prime }}$ is equal to $S^{-1}A{\otimes }_{A}B=S^{-1}B$, where $S$ is the set of regular elements of $A$, $({\mathcal{M}}_{X^{\prime }}{)}_{x^{\prime }}$ is equal to $T^{-1}B$, where $T$ is the set

of regular elements of $B$, and as we have seen that $S\subset T$, the homomorphism $S^{-1}B\to T^{-1}B$ is injective; in other words, this proves that the homomorphism (20.1.11.1) $f\ast ({\mathcal{M}}_X)\to {\mathcal{M}}_{X^{\prime }}$ is injective (whence the last assertion of the statement). The quotient $f\ast ({\mathcal{M}}_X)/{\mathcal{O}}_{X^{\prime }}$ identifies with an ${\mathcal{O}}_{X^{\prime }}$-submodule of ${\mathcal{M}}_{X^{\prime }}/{\mathcal{O}}_{X^{\prime }}$, and $(f\ast ({\mathcal{M}}_X)/{\mathcal{O}}_{X^{\prime }}{)}_{x^{\prime }}$ identifies with $({\mathcal{M}}_X/{\mathcal{O}}_X{)}_x{\otimes }_{{\mathcal{O}}_{X,x}}{\mathcal{O}}_{X^{\prime },x^{\prime }}$. Now suppose that $x\notin dom(\varphi )$; the image of ${\varphi }_x$ in $({\mathcal{M}}_X/{\mathcal{O}}_X{)}_x$ is therefore $\ne 0$; by faithful flatness, one deduces that the same holds for the image of $(\varphi \circ f{)}_{x^{\prime }}$ in $({\mathcal{M}}_{X^{\prime }}/{\mathcal{O}}_{X^{\prime }}{)}_{x^{\prime }}$, hence $x^{\prime }\notin dom(\varphi \circ f)$, which finishes the proof.

Remark (20.1.13).

Let $X$ be a reduced complex analytic space; then the notion of meromorphic function on $X$ defined above coincides with the usual notion. Consider on the other hand a prescheme $Y$, locally of finite type over the field $\mathbb{C}$; one then knows that one can associate to $Y$ an analytic space $Y^{an}$ having the same underlying topological space, and that the canonical morphism $f:Y^{an}\to Y$ is flat [37]; by virtue of (20.1.12), the canonical homomorphism $u\mapsto u\circ f$ from $M(Y)$ to $M(Y^{an})$ is therefore everywhere defined and injective; but it is not surjective in general. For example, when $Y={\mathbb{V}}_0^r$ ($Er{r}_{III},14$) is the affine space of dimension $r$ over $\mathbb{C}$, $M(Y)$ identifies canonically with the field $R(Y)$ of rational functions on $Y$ (20.2.13, (i)), while $M(Y^{an})$ is the field of usual meromorphic functions on ${\mathbb{C}}^r$. By reason of this fact, it is often preferable, in algebraic geometry, to refrain from the terminology introduced in this section, and to use the equivalent terminology of "pseudo-function" which will be defined below.

20.2. Pseudo-morphisms and pseudo-functions

The only ringed spaces considered in this section are preschemes.

(20.2.1). Recall (11.10.2) that in a prescheme $X$ one says that an open $U$ is schematically dense if, for every open $V$ of $X$, the canonical homomorphism $\Gamma (V,{\mathcal{O}}_X)\to \Gamma (V\cap U,{\mathcal{O}}_X)$ is injective.

Consider two preschemes $X$, $Y$, and two schematically dense opens $U$, $U^{\prime }$ of $X$; one says that two morphisms $u:U\to Y$, $u^{\prime }:U^{\prime }\to Y$ are equivalent if there exists an open $U^{\prime \prime }\subset U\cap U^{\prime }$, schematically dense in $X$, such that $u|U^{\prime \prime }=u^{\prime }|U^{\prime \prime }$. As it follows at once from the definition of schematically dense opens that the intersection of two such opens is again one, it is immediate that the previous relation is indeed an equivalence relation. An equivalence class under this relation is called a pseudo-morphism of $X$ into $Y$, or a strict rational map of $X$ into $Y$.

If $S$ is a prescheme and $X$, $Y$ are two $S$-preschemes, one calls pseudo-$S$-morphism the equivalence class (for the foregoing relation) of an $S$-morphism from a schematically dense open of $X$ to $Y$. A pseudo-morphism is therefore nothing other than a pseudo-$S$-morphism for $S=\mathrm{Spec}(\mathbb{Z})$.

One denotes by Ps.hom(X, Y) (resp. $Ps.ho{m}_S(X,Y)$) the set of pseudo-morphisms (resp. pseudo-$S$-morphisms) of $X$ into $Y$.

(20.2.2). It follows from the definition recalled above that if $U$ is a schematically dense open in $X$, then for every open $V$ of $X$, $U\cap V$ is schematically dense in $V$. If two morphisms $u:U\to Y$, $u^{\prime }:U^{\prime }\to Y$ of schematically dense opens of $X$ into $Y$ are equivalent, it follows that their restrictions $u|(V\cap U)$ and $u^{\prime }|(V\cap U^{\prime })$ are also equivalent morphisms (for the prescheme induced on $V$); their class is called the restriction to $V$ of the pseudo-morphism $\omega$, the class of $u$, and this pseudo-morphism is denoted $\omega |V$. If $W\subset V$ is an open of $X$, it is clear that $(\omega |V)|W=\omega |W$. This shows that the restriction maps define a presheaf of sets $U\mapsto Ps.hom(U,Y)$; one will note that this presheaf is not in general a sheaf (cf. (20.2.16)); the associated sheaf is denoted $\mathcal{P}s.hom(X,Y)$. For pseudo-$S$-morphisms, one sees likewise that $U\mapsto Ps.ho{m}_S(U,Y)$ is a presheaf of sets, whose associated sheaf is denoted $\mathcal{P}s.ho{m}_S(X,Y)$.

If $V$ is schematically dense in $X$, then for every open $U$ schematically dense in $X$, $U\cap V$ is also schematically dense in $X$, so the map $\omega \mapsto \omega |V$ is a bijection from Ps.hom(X, Y) (resp. $Ps.ho{m}_S(X,Y)$) onto Ps.hom(V, Y) (resp. $Ps.ho{m}_S(V,Y)$).

(20.2.3). Given a pseudo-$S$-morphism $\omega$ of $X$ into $Y$, one says that it is defined at a point $x\in X$ if there exists an open neighbourhood $V$ of $x$ in $X$, an open $U\subset V$ containing $x$ and schematically dense in $V$, and an $S$-morphism $u:U\to Y$ whose class in $Ps.ho{m}_S(V,Y)$ equals $\omega |V$ (20.2.2); one calls domain of definition of $\omega$, and one denotes by $do{m}_S(\omega )$ (or simply $dom(\omega )$ if $S=\mathrm{Spec}(\mathbb{Z})$), the set of $x\in X$ where $\omega$ is defined; it is evidently an open of $X$. Moreover, for every open $W$ of $X$, one has

  (20.2.3.1)             dom_S(ω | W) = (dom_S(ω)) ∩ W

by virtue of the property of schematically dense opens recalled in (20.2.2).

Proposition (20.2.4).

Suppose that $X$, $Y$ are $S$-preschemes such that the structure morphism $Y\to S$ is separated; then, if $\omega$ is a pseudo-$S$-morphism of $X$ into $Y$, $do{m}_S(\omega )$ is the largest of the schematically dense opens $U$ of $X$ such that there exists an $S$-morphism $u:U\to Y$ belonging to the class $\omega$.

It evidently suffices to prove that if $U$, $U^{\prime }$ are two schematically dense opens in $X$ such that two $S$-morphisms $u:U\to Y$ and $u^{\prime }:U^{\prime }\to Y$ are equivalent, then the restrictions of $u$ and $u^{\prime }$ to $U\cap U^{\prime }$ are equal. Now by hypothesis there exists an open $U^{\prime \prime }\subset U\cap U^{\prime }$, schematically dense in $X$ and on which $u$ and $u^{\prime }$ coincide; as U'' is also schematically dense in $U\cap U^{\prime }$, our assertion follows from (11.10.1, d).

Corollary (20.2.5).

Let $S$ be an S_0-scheme, $X$, $Y$ two $S$-preschemes such that the composite $Y\to S\to S_0$ is separated (which implies (I, 5.5.1) that $Y\to S$ is also separated). For every pseudo-$S$-morphism $\omega$ of $X$ into $Y$, one has then $do{m}_S(\omega )=do{m}_{S_0}(\omega )$. In particular, if $Y$ is a scheme, one has $do{m}_S(\omega )=dom(\omega )$.

Indeed, it is clear that $do{m}_S(\omega )\subset do{m}_{S_0}(\omega )$ without any separation hypothesis; by virtue of (20.2.4), it suffices to prove that if an S_0-morphism $u_0:U_0\to Y$ from a schematically dense

open U_0 of $X$ into $Y$ is such that the composite $v_0:U_0\to Y\to S$ coincides with the restriction of the structure morphism $w_0:U_0\to S$ on an open $U\subset U_0$ schematically dense in $X$, then $v_0=w_0$. But by virtue of the hypothesis that the morphism $S\to S_0$ is separated, this again follows from (11.10.1, d).

Corollary (20.2.6).

Under the hypotheses of (20.2.4), the presheaf $U\mapsto Ps.ho{m}_S(U,Y)$ is a sheaf.

Indeed, let $U$ be an open of $X$, $(U_{\alpha })$ a cover of $U$ by opens of $U$; suppose given for each $\alpha$ a pseudo-$S$-morphism ${\omega }_{\alpha }$ of $U_{\alpha }$ into $Y$, so that for every pair of indices $\alpha$, $\beta$, the restrictions (20.2.2) of the pseudo-$S$-morphisms ${\omega }_{\alpha }$ and ${\omega }_{\beta }$ to $U_{\alpha }\cap U_{\beta }$ are equal; by virtue of (20.2.3.1), this entails $do{m}_S({\omega }_{\alpha })\cap U_{\beta }=do{m}_S({\omega }_{\beta })\cap U_{\alpha }$. Let $W$ be the union of the opens $do{m}_S({\omega }_{\alpha })$, and, for each $\alpha$, let $u_{\alpha }$ be the unique $S$-morphism $do{m}_S({\omega }_{\alpha })\to Y$ belonging to the class ${\omega }_{\alpha }$ (20.2.4); by reason of the hypothesis and of (20.2.4), the restrictions of $u_{\alpha }$ and $u_{\beta }$ to $do{m}_S({\omega }_{\alpha })\cap do{m}_S({\omega }_{\beta })$ are equal, so there exists a morphism $u:W\to Y$ whose restriction to each open $do{m}_S({\omega }_{\alpha })$ equals $u_{\alpha }$. It is clear that $W$ is schematically dense in $U$, hence defines a pseudo-$S$-morphism $\omega$ of $U$ into $Y$ whose restrictions to the $U_{\alpha }$ are the ${\omega }_{\alpha }$.

Remark (20.2.7).

One knows (11.10.4) that when $X$ is reduced, it amounts to the same to say that an open of $X$ is dense in $X$ or that it is schematically dense in $X$; the notion of pseudo-morphism (resp. pseudo-$S$-morphism) of $X$ into $Y$ then coincides with that of rational map (resp. $S$-rational map) of $X$ into $Y$ (I, 7.1.2). In the general case, the notion of pseudo-morphism seems more useful than that of rational map.

(20.2.8). One calls pseudo-function on $X$ a pseudo-morphism of $X$ into $\mathrm{Spec}(\mathbb{Z}[T])$ ($T$ indeterminate), or, what amounts to the same, an $X$-pseudo-morphism of $X$ into $X{\otimes }_{\mathbb{Z}}\mathbb{Z}[T]$; it amounts also to the same (I, 3.3.15) to say that a pseudo-function on $X$ is an equivalence class of sections of ${\mathcal{O}}_X$ over schematically dense opens of $X$, two sections $g\in \Gamma (U,{\mathcal{O}}_X)$, $g^{\prime }\in \Gamma (U^{\prime },{\mathcal{O}}_X)$ over such opens being equivalent if there exists an open $U^{\prime \prime }\subset U\cap U^{\prime }$, schematically dense in $X$, on which $g$ and $g^{\prime }$ coincide. One may here apply (20.2.4) with $S=X$ and $Y=X{\otimes }_{\mathbb{Z}}\mathbb{Z}[T]$; hence, for every pseudo-function $\varphi$ on $X$, there exists a largest schematically dense open $dom(\varphi )$ in $X$ on which there is a section of ${\mathcal{O}}_X$ over $dom(\varphi )$ belonging to the class $\varphi$. It is clear that the sheaf $\mathcal{P}s.hom(X,X{\otimes }_{\mathbb{Z}}\mathbb{Z}[T])$ is here a sheaf of rings, even an ${\mathcal{O}}_X$-Algebra, which we shall denote ${\mathcal{M}}_X^{\prime }$; it follows from (20.2.6) that, for every open $U$ of $X$, $\Gamma (U,{\mathcal{M}}_X^{\prime })$ equals the set of pseudo-morphisms of $U$ into $\mathrm{Spec}(\mathbb{Z}[T])$; one sets $M^{\prime }(X)=\Gamma (X,{\mathcal{M}}_X^{\prime })$. To say that a section $\varphi$ of ${\mathcal{M}}_X^{\prime }$ over $U$ is invertible in the ring $\Gamma (U,{\mathcal{M}}_X^{\prime })$ means that there exists an open $U^{\prime }$ schematically dense in $dom(\varphi )$, hence in $U$, such that, if $g$ is the unique section of ${\mathcal{O}}_X$ over $dom(\varphi )$ belonging to $\varphi$, $g|U^{\prime }$ is invertible in $\Gamma (U^{\prime },{\mathcal{O}}_X)$. It follows from (I, 3.3.15) that, in the canonical correspondence between $\Gamma (V,{\mathcal{O}}_X)$ and $\mathrm{Hom}(V,\mathbb{Z}[T])$ ($V$ open

of $X$), the invertible elements of $\Gamma (V,{\mathcal{O}}_X)$ correspond to morphisms which factor as $V\to \mathrm{Spec}(\mathbb{Z}[T,T^{-1}])\to \mathrm{Spec}(\mathbb{Z}[T])$. One concludes that the sheaf ${\mathcal{M}}_X^{\prime \times }$ of germs of invertible sections of ${\mathcal{M}}_X^{\prime }$ identifies canonically with the sheaf $\mathcal{P}s.hom(X,X{\otimes }_{\mathbb{Z}}\mathbb{Z}[T,T^{-1}])$.

Lemma (20.2.9).

Let $U$ be an open of $X$, $s$ a regular element of the ring $\Gamma (U,{\mathcal{O}}_X)$; then the open set $U_s$ of $x\in U$ such that $s(x)\ne 0$ is schematically dense in $U$.

Indeed, let $V$ be an open of $U$, $t$ a section of ${\mathcal{O}}_X$ over $V$ such that $t|(V\cap U_s)=0$. For every affine open $W\subset V$, there then exists an integer $n$ such that $s^{n}t|W=0$ (I, 1.4.1); but since $s$ is a regular element of $\Gamma (U,{\mathcal{O}}_X)$, this entails $t|W=0$, whence $t=0$.

(20.2.10). Consider a meromorphic function $f\in M(X)$ (20.1.4); then $dom(f)$ is schematically dense in $X$: indeed, every point of $X$ admits an open neighbourhood $U$ such that there exists a section $s\in \Gamma (U,\mathcal{S})$ which is a regular element of this ring, and such that $s(f|U)\in \Gamma (U,{\mathcal{O}}_X)$; since $s|U_s$ is invertible, one concludes that $f|U_s\in \Gamma (U_s,{\mathcal{O}}_X)$, hence $U_s\subset dom(f)$ by definition (20.1.4), and our assertion follows from the lemma (20.2.9). One may therefore associate to $f$ the pseudo-function equal to the class of the section of ${\mathcal{O}}_X$ over $dom(f)$, the restriction of $f$; operating similarly with $X$ replaced by an arbitrary open of $X$, one obtains in this way a canonical homomorphism of ${\mathcal{O}}_X$-Algebras

$$(\mathrm{20.2.10.1}){\mathcal{M}}_X\to {\mathcal{M}}_X^{\prime }$$

which, by restriction, evidently gives a homomorphism of sheaves of abelian groups

$$(\mathrm{20.2.10.2}){\mathcal{M}}_X^{\times }\to {\mathcal{M}}_X^{\prime \times }$$

for the sheaves of germs of invertible sections of ${\mathcal{M}}_X$ and ${\mathcal{M}}_X^{\prime }$.

Proposition (20.2.11).

(i) The canonical homomorphism (20.2.10.1) (and consequently also (20.2.10.2)) is injective.

(ii) Suppose either that $X$ is locally Noetherian, or that $X$ is reduced and the set of its irreducible components is locally finite. Then the canonical homomorphism (20.2.10.1) (and consequently also (20.2.10.2)) is bijective.

The questions being local on $X$, one may restrict to the case $X=\mathrm{Spec}(A)$ affine, and then show that the canonical homomorphism $M(X)\to M^{\prime }(X)$ is always injective, and that it is bijective in the cases considered in (ii). Taking into account the definition of the sheaf ${\mathcal{M}}_X$ (20.1.3), one may moreover note that (20.2.10.1) actually comes from a homomorphism of presheaves

                         Γ(U, 𝒪_X)[Γ(U, 𝒮)⁻¹] → Γ(U, 𝓜'_X)

and it suffices to show that, for $U$ affine, this latter is injective (resp. bijective under the hypotheses of (ii)). Denoting by $S$ the set of regular elements of $A$ (so that $S^{-1}A$ is the total ring of fractions of $A$), one must therefore consider the canonical homomorphism

$$(\mathrm{20.2.11.1})S^{-1}A\to M^{\prime }(X)$$

and show that it is injective (resp. bijective under the conditions of (ii)). Now, one may write $S^{-1}A=\lim A_t$, where $t$ runs over the set of regular elements of $A$, ordered by the relation "$t$ is a divisor of $t^{\prime }$" $(0_I,\mathrm{1.4.5})$; one may also write $A_t=\Gamma (D(t),{\mathcal{O}}_X)$. On the other hand, one has by definition $M^{\prime }(X)=\lim \Gamma (U,{\mathcal{O}}_X)$, where $U$ runs over the set of schematically dense opens of $X$ (ordered by $\supset$), and the map (20.2.11.1) is none other than the canonical map coming from the fact that the $D(t)$ constitute a part of the set of schematically dense opens in $X$ (20.2.9). Note that, by definition of a schematically dense open, the homomorphism $\Gamma (U,{\mathcal{O}}_X)\to \Gamma (U^{\prime },{\mathcal{O}}_X)$ for two such opens $U^{\prime }\subset U$ is always injective, hence so are the homomorphisms $\Gamma (U,{\mathcal{O}}_X)\to M^{\prime }(X)$, and one concludes that (20.2.11.1) is injective. To prove that (20.2.11.1) is bijective, it suffices to show that the set of $D(t)$ is cofinal in the set of schematically dense opens, in other words that for such an open $U$, there exists $t$ regular in $A$ such that $D(t)\subset U$. Now, when $X$ is reduced and the irreducible components of $X$ form a locally finite set, this set is finite since $X$ was supposed affine, in other words $A$ has only a finite number of minimal prime ideals ${\mathfrak{p}}_i$; as $A$ is reduced, the intersection of the ${\mathfrak{p}}_i$ reduces to 0, and to say that $t$ is regular is therefore equivalent to saying that $t$ does not belong to any of the ${\mathfrak{p}}_i$; one concludes by the reasoning of (I, 7.1.9.1). When $A$ is Noetherian, saying that $U=X-Y$ (where $Y=V(\mathfrak{i})$ is closed in $X$) is schematically dense means (5.10.2) that $Y$ does not meet $Ass({\mathcal{O}}_X)$, and by virtue of Bourbaki, Alg. comm., chap. IV, §1, n° 4, prop. 8, this entails the existence of a $t\in \mathfrak{i}$ such that $t$ is $A$-regular, hence $U\supset D(t)$.

One has moreover proved in the course of this proof the

Corollary (20.2.12).

If $X=\mathrm{Spec}(A)$, where $A$ is Noetherian, or reduced and having only a finite number of minimal prime ideals, the schematically dense opens in $X$ are those which contain an open of the form $D(t)$, where $t$ is regular in $A$, and $M(X)=M^{\prime }(X)$ is the total ring of fractions $S^{-1}A$, where $S$ is the set of regular elements of $X$.

Remarks (20.2.13).

(i) Let $\varphi$ be an element of $M(X)$, ${\varphi }^{\prime }$ its image in M'(X); one has evidently, by definition ((20.1.4) and (20.2.8)), $dom(\varphi )\subset dom({\varphi }^{\prime })$; but in fact one has equality $dom(\varphi )=dom({\varphi }^{\prime })$, since there exists a section of ${\mathcal{O}}_X$ over $dom({\varphi }^{\prime })$ belonging to the class ${\varphi }^{\prime }$, and the corresponding meromorphic function equals $\varphi$ (20.2.11, (i)), so $dom({\varphi }^{\prime })\subset dom(\varphi )$.

(ii) One has already noted that when $X$ is reduced, one has ${\mathcal{M}}_X^{\prime }=\mathcal{R}(X)$ (sheaf of rational functions on $X$ (I, 7.3.2)); if moreover the irreducible components of $X$ form a locally finite set, one has ${\mathcal{M}}_X={\mathcal{M}}_X^{\prime }=\mathcal{R}(X)$. In general, since every schematically dense open set is dense, one has a canonical homomorphism ${\mathcal{M}}_X^{\prime }\to \mathcal{R}(X)$, but even when $X$ is locally Noetherian, this homomorphism is not necessarily injective. For example, if $X=\mathrm{Spec}(A)$, where $A$ is a Noetherian ring such that $Ass(A)$ contains immersed prime ideals (which entails that $A$ is not reduced), one has seen that $M(X)$ and M'(X) identify with the total ring of fractions $S^{-1}A$,

where $S$ is the set of regular elements of $A$, the complement of the union of the ideals $\mathfrak{p}\in Ass(A)$; on the other hand, $R(X)$ identifies with $Q^{-1}A$, where $Q$ is the complement of the union of the minimal prime ideals of $A$ (I, 7.1.9), and the canonical homomorphism $A\to Q^{-1}A$ (and a fortiori $S^{-1}A\to Q^{-1}A$) is therefore not injective, since there exist in $A-Q$ elements $\ne 0$ of $A$ annihilated by elements of $Q$ (Bourbaki, Alg. comm., chap. IV, §1, n° 1, cor. 2 of prop. 1).

(iii) One will note that even when $X$ is locally Noetherian, the ${\mathcal{O}}_X$-Module ${\mathcal{M}}_X$ is not necessarily quasi-coherent. Consider for example a Noetherian local ring $A$ of dimension $\ge 2$, whose maximal ideal $\mathfrak{m}$ is such that $\mathfrak{m}\in Ass(A)$ and such that, on setting $X=\mathrm{Spec}(A)$, the scheme induced on the open $U$ of $X$, complement of $\mathfrak{m}$, is integral. The only regular elements of $A$ are then the invertible elements, so $\Gamma (X,{\mathcal{M}}_X)=M(X)=A$; if ${\mathcal{M}}_X$ were quasi-coherent, it would therefore equal ${\mathcal{O}}_X$, but as $U$ is an integral scheme, ${\mathcal{M}}_X|U=R(U)$ is a simple sheaf (I, 7.3.5), whereas ${\mathcal{O}}_X$ is not a simple sheaf since $\dim (A)\ge 2$.

It remains to give an example of a ring $A$ having the preceding properties. It suffices to consider an integral local ring $B$ of dimension $\ge 2$ and residue field $k$, and to take $A=B\oplus k$ with the multiplication law $(b,x)(b^{\prime },x^{\prime })=(b{b}^{\prime },b{x}^{\prime }+b^{\prime }x)$.

(iv) If $X$ is locally Noetherian, it follows from (5.10.2) that the schematically dense opens in $X$ are those which contain the set $Ass({\mathcal{O}}_X)$.

(20.2.14). Let $X$ be a prescheme, $\mathcal{F}$ a quasi-coherent and strictly torsion-free ${\mathcal{O}}_X$-Module (20.1.5), so that $\mathcal{F}$ identifies with an ${\mathcal{O}}_X$-submodule of ${\mathcal{M}}_X(\mathcal{F})$. For every meromorphic section $u$ of $\mathcal{F}$ over $X$, one calls Ideal of denominators of $u$ the annihilator $\mathcal{J}$ of the section ū image of $u$ in ${\mathcal{M}}_X(\mathcal{F})/\mathcal{F}$. The Ideal $\mathcal{J}$ is quasi-coherent: indeed, the question being local on $X$, one may restrict to the case where $X$ is affine, and there exists a section $s\in \Gamma (X,\mathcal{S}({\mathcal{O}}_X))$ such that $v=su\in \Gamma (X,\mathcal{F})$. To say that, for an open $U\subset X$, a section $f\in \Gamma (U,{\mathcal{O}}_X)$ belongs to $\Gamma (U,\mathcal{J})$ means that $f(u|U)\in \Gamma (U,\mathcal{F})$, and since $s|U$ is a regular element of $\Gamma (U,{\mathcal{O}}_X)$ and $\mathcal{F}$ is strictly torsion-free, the preceding relation is again equivalent to $f((sv)|U)\in \Gamma (U,s\mathcal{F})$; if $\bar{v}$ is the section of $\mathcal{F}/s\mathcal{F}$ which is the canonical image of $v$, one sees therefore that $\mathcal{J}$ is the kernel of the homomorphism ${\mathcal{O}}_X\to \mathcal{F}/s\mathcal{F}$ obtained by multiplication by the section $\bar{v}$. As $\mathcal{F}/s\mathcal{F}$ is quasi-coherent, so is $\mathcal{J}$.

It follows at once from the foregoing definition that $dom(u)$ is the open complement of the closed subprescheme of $X$ defined by the Ideal of denominators of $u$.

Proposition (20.2.15).

Let $f:X^{\prime }\to X$ be a morphism, $\mathcal{F}$ a quasi-coherent ${\mathcal{O}}_X$-Module, $\varphi$ a section of ${\mathcal{M}}_X(\mathcal{F})$ over $X$ (20.1.11). Then $f^{-1}(dom(\varphi ))$ is schematically dense in $X^{\prime }$.

The question being evidently local on $X$ and $X^{\prime }$, one may suppose $X=\mathrm{Spec}(A)$, $X^{\prime }=\mathrm{Spec}(A^{\prime })$ affine, $\mathcal{F}=\tilde{M}$, and $\varphi =h\otimes (1/s)$, where $h\in M$ and $s$ is a regular element of $A$ whose image $s^{\prime }$ in $A^{\prime }$ is a regular element. One knows then (20.2.9) that $D(s^{\prime })$ is a schematically dense open in $X^{\prime }$, and it is the inverse image under $f$ of $D(s)$, which is contained in $dom(\varphi )$.

Remark (20.2.16).

When $Y$ is not separated, the presheaf $U\mapsto Ps.ho{m}_S(U,Y)$ on $X$ is not necessarily a sheaf, even when $X$ is Noetherian, as the following example shows. We shall take for $X$ a spectrum of a semi-local Noetherian ring $A$ of dimension $\ge 2$, having exactly two maximal ideals

${\mathfrak{m}}^{\prime }$, ${\mathfrak{m}}^{\prime \prime }$ (so that $X$ has exactly two closed points $x^{\prime }$, x''), such that ${\mathfrak{m}}^{\prime }$ and ${\mathfrak{m}}^{\prime \prime }$ belong to $Ass(A)$ and such that the open $U=X-x^{\prime },x^{\prime \prime }$ is integral. Let $U^{\prime }=X-x^{\prime }$, $U^{\prime \prime }=X-x^{\prime \prime }$, so that $U=U^{\prime }\cap U^{\prime \prime }$. Note that the only schematically dense opens in $X$ are those that contain $x^{\prime }$ and x'' (20.2.13, (iv)), so they necessarily equal $X$. To have a counter-example, it will therefore suffice to define two $S$-morphisms $u^{\prime }:U^{\prime }\to Y$, $u^{\prime \prime }:U^{\prime \prime }\to Y$ (with $S=X$) whose restrictions to $U$ belong to the same pseudo-morphism of $U$ into $Y$, but which are such that, for every neighbourhood $V^{\prime }$ of x'' in $U^{\prime }$ and every neighbourhood V'' of $x^{\prime }$ in U'', the restrictions of $u^{\prime }$ and u'' to $V^{\prime }\cap V^{\prime \prime }$ are distinct. For this, consider an irreducible closed subset $Z$ of $X$ containing $x^{\prime }$ and x'', distinct from $X$; let $Y$ be the $X$-prescheme obtained by gluing two preschemes $Y^{\prime }$, Y'' isomorphic to $X$ along the everywhere dense open $X-Z$ (I, 2.3.1). One will take for $u^{\prime }$ and u'' respectively the restrictions to $U^{\prime }$ and U'' of the inverse isomorphisms of the structural isomorphisms $Y^{\prime }\stackrel{\sim }{\to }X$, $Y^{\prime \prime }\stackrel{\sim }{\to }X$. Since $V^{\prime }$ and V'' contain the generic point of $Z$, the restrictions of $u^{\prime }$ and u'' to $V^{\prime }\cap V^{\prime \prime }$ are distinct, but the restrictions of $u^{\prime }$ and u'' to $U-(U\cap Z)$ are equal, and $U-(U\cap Z)$ is a dense open in $U$, hence schematically dense since $U$ is reduced.

It remains then only to define $A$ and $Z$ so as to have the preceding properties. Let $X_0=\mathrm{Spec}(B)$ be an integral affine scheme (for example the affine plane over a field $k$), $Y$ an irreducible closed subset of X_0 (for example an affine line) containing two distinct closed points $x^{\prime }$ and x'', corresponding to maximal ideals ${\mathfrak{n}}^{\prime }$, ${\mathfrak{n}}^{\prime \prime }$ of $B$. Consider the ring $C=B\oplus (B/{\mathfrak{n}}^{\prime }\oplus B/{\mathfrak{n}}^{\prime \prime })$ with the multiplication $(b,z)(b^{\prime },z^{\prime })=(b{b}^{\prime },b{z}^{\prime }+b^{\prime }z)$. If $X_1=\mathrm{Spec}(C)$, one has $X_0=(X_1{)}_{red}$ and X_1 is reduced except at the points $x^{\prime }$, x''. It then suffices to take $A=R^{-1}C$, where $R$ is the complement of the union of the maximal ideals of $C$ at the points $x^{\prime }$, x'', and for $Z$ the trace of $Y$ on $X=\mathrm{Spec}(A)$.

20.3. Composition of pseudo-morphisms

(20.3.1). Let $X$, $Y$, $Z$ be three preschemes, $\omega$ a pseudo-morphism of $X$ into $Y$, $f:Y\to Z$ a morphism. It is clear that if $U^{\prime }$, U'' are two schematically dense opens in $X$, $u^{\prime }:U^{\prime }\to Y$, $u^{\prime \prime }:U^{\prime \prime }\to Y$ two morphisms of the class $\omega$, the morphisms $f\circ u^{\prime }$ and $f\circ u^{\prime \prime }$ are equivalent (for the relation defined in (20.2.1)); hence, for all morphisms $u$ of the class $\omega$, the morphisms $f\circ u$ belong to one and the same equivalence class, which one denotes $f\circ \omega$ and which one calls the pseudo-morphism of $X$ into $Z$ composed of $f$ and $\omega$. One has $dom(f\circ \omega )=dom(\omega )$. If $g:Z\to T$ is a morphism, it is clear that $g\circ (f\circ \omega )=(g\circ f)\circ \omega$.

(20.3.2). Suppose now given a pseudo-$S$-morphism $\omega$ of $X$ into $Y$, where $Y$ is separated over $S$, so that there exists an $S$-morphism $u:do{m}_S(\omega )\to Y$ of the class $\omega$ (20.2.4). Let $f:X^{\prime }\to X$ be an $S$-morphism such that the open $V=f^{-1}(do{m}_S(\omega ))$ is schematically dense in $X^{\prime }$; one then says that the class (for the equivalence relation of (20.2.1)) of the $S$-morphism $u\circ (f|V)$ (a class defined by virtue of the foregoing hypothesis) is the pseudo-$S$-morphism composed of $\omega$ and $f$, and one denotes it $\omega \circ f$; it is clear that one has

$$(\mathrm{20.3.2.1})f^{-1}(do{m}_S(\omega ))\subset do{m}_S(\omega \circ f).$$

For $f:X^{\prime }\to X$ given, one denotes by $Ps.ho{m}_S(X,Y{)}^f$ the set of pseudo-$S$-morphisms $\omega$ of $X$ into $Y$ satisfying the foregoing condition. If $\omega$ is such a pseudo-$S$-morphism, it is clear that for every open $V$ of $X$,

                         f⁻¹(dom_S(ω | V)) = f⁻¹(V ∩ dom_S(ω)) = f⁻¹(V) ∩ f⁻¹(dom_S(ω))

is schematically dense in $f^{-1}(V)$, so, if $f^V:f^{-1}(V)\to V$ is the restriction of $f$, the composite $(\omega |V)\circ f^V$ is defined and equal to $(\omega \circ f)|f^{-1}(V)$. One thus defines a

canonical restriction map $Ps.ho{m}_S(X,Y{)}^f\to Ps.ho{m}_S(V,Y{)}^{f^V}$, which permits one to define a presheaf of sets $V\mapsto Ps.ho{m}_S(V,Y{)}^{f^V}$ on $X$, a sub-presheaf of $V\mapsto Ps.ho{m}_S(V,Y)$, whence an associated sheaf of sets which one denotes $\mathcal{P}s.ho{m}_S(X,Y{)}^f$. Moreover, for every open $V$ of $X$, one has a map $\omega \mapsto \omega \circ f^V$ from $Ps.ho{m}_S(V,Y{)}^{f^V}$ to $Ps.ho{m}_S(f^{-1}(V),Y)$, which therefore defines an $f$-morphism of sheaves of sets

                         𝒫𝓈.hom_S(X, Y)^f → 𝒫𝓈.hom_S(X', Y).

(20.3.3). Let now $f^{\prime }:X^{\prime \prime }\to X^{\prime }$ be an $S$-morphism such that the open $f^{\prime -1}(f^{-1}(do{m}_S(\omega )))$ is schematically dense in X''; then $\omega \circ (f\circ f^{\prime })$ is defined and $u\circ f\circ f^{\prime }$ belongs to this pseudo-$S$-morphism; moreover, by virtue of (20.3.2.1), $f^{\prime -1}(do{m}_S(\omega \circ f))$ is a fortiori schematically dense in X'', so $(\omega \circ f)\circ f^{\prime }$ is also defined and $u\circ f\circ f^{\prime }$ belongs to this pseudo-$S$-morphism, so one has $(\omega \circ f)\circ f^{\prime }=\omega \circ (f\circ f^{\prime })$.

On the other hand, for every $S$-morphism $g:Y\to Z$, one has $do{m}_S(g\circ \omega )=do{m}_S(\omega )$ (20.3.1), so $(g\circ \omega )\circ f$ is defined, and $g\circ u\circ f$ belongs to this pseudo-$S$-morphism, which shows that $(g\circ \omega )\circ f=g\circ (\omega \circ f)$.

(20.3.4). The most important case in the definition (20.3.2) is the one where $\mathcal{P}s.ho{m}_S(X,Y{)}^f=\mathcal{P}s.ho{m}_S(X,Y)$: for this it suffices that for every open $U$ of $X$ and every open $V$ schematically dense in $U$, $f^{-1}(V)$ be schematically dense in $f^{-1}(U)$; when this is the case, then, for every open $U$ of $X$ and every pseudo-$S$-morphism $\omega :U\to Y$, one may define the composite $\omega \circ f^U$, even if $Y$ is not separated over $S$. Indeed, if $u^{\prime }:U^{\prime }\to Y$ and $u^{\prime \prime }:U^{\prime \prime }\to Y$ are two morphisms of the class $\omega$, they coincide on an open U_0 schematically dense in $U$, hence the composite morphisms $f^{-1}(U^{\prime })\to U^{\prime }\to Y$ and $f^{-1}(U^{\prime \prime })\to U^{\prime \prime }\to Y$ coincide on $f^{-1}(U_0)$, which is by hypothesis schematically dense in $f^{-1}(U)$; one may therefore define $\omega \circ f^U$ as the class of any of the morphisms $f^{-1}(U^{\prime })\to U^{\prime }\to Y$, where $u^{\prime }$ belongs to $\omega$.

Proposition (20.3.5).

Let $X$, $X^{\prime }$ be two preschemes, $f:X^{\prime }\to X$ a morphism. Then, in each of the following three cases, for every open $U$ of $X$ and every open $V$ schematically dense in $U$, $f^{-1}(V)$ is schematically dense in $f^{-1}(U)$:

(i) $f$ is flat and locally of finite presentation.

(ii) $f$ is flat and the underlying space of $X$ is locally Noetherian.

(iii) $X^{\prime }$ is reduced, the set of irreducible components of $X$ is locally finite, and every irreducible component of $X^{\prime }$ dominates an irreducible component of $X$.

Assertion (i) follows from (11.10.5, (ii), b)); assertion (ii) follows from (11.10.5, (ii), a)), since then every open $V$ in $U$ is retrocompact, in other words the canonical injection $j:V\to U$ is a quasi-compact morphism. Finally, to prove (iii), note that since $X^{\prime }$ is reduced, it suffices to show that for every open $U$ of $X$ and every open $V$ dense in $U$, $f^{-1}(V)$ is dense in $f^{-1}(U)$. Now, for $f^{-1}(V)$ to be dense in $f^{-1}(U)$, it suffices that $f^{-1}(V)$ contain all the maximal points of $X^{\prime }$ belonging to $f^{-1}(U)$; the conclusion therefore follows from the hypothesis that the image under $f$ of every

maximal point of $X^{\prime }$ belonging to $f^{-1}(U)$ is a maximal point of $X$ belonging to $U$, hence to $V$ since the set of irreducible components of $X$ is locally finite.

(20.3.6). Let $X$, $Y$ be two $S$-preschemes; suppose that $X$ satisfies one of the two following hypotheses:

a) $X$ is locally Noetherian;

b) $X$ is reduced and the set of its irreducible components is locally finite.

Then, for every $x\in X$, the canonical $S$-morphism $j:\mathrm{Spec}({\mathcal{O}}_{X,x})\to X$ is flat and satisfies condition (ii) of (20.3.5) in case a), condition (iii) of (20.3.5) in case b). For every pseudo-$S$-morphism $\omega$ of $X$ into $Y$, the composite $\omega \circ j$ is therefore defined and is a pseudo-$S$-morphism of $\mathrm{Spec}({\mathcal{O}}_{X,x})$ into $Y$, called the restriction of $\omega$ to $\mathrm{Spec}({\mathcal{O}}_{X,x})$. Note now that if $X$ satisfies condition a) (resp. b)) of (20.3.6), so does every prescheme induced on an open $U$ of $X$ containing $x$. By passage to the inductive limit, one therefore deduces, from the canonical maps Ps.hom_S(U, Y) → Ps.hom_S(Spec(𝒪_{X,x}), Y) thus obtained, a canonical map

  (20.3.6.1)             (𝒫𝓈.hom_S(X, Y))_x → Ps.hom_S(Spec(𝒪_{X,x}), Y)

where the first member is the fibre at the point $x$ of the sheaf $\mathcal{P}s.ho{m}_S(X,Y)$, the set of germs at $x$ of pseudo-$S$-morphisms from open neighbourhoods of $x$ into $Y$.