Éléments de Géométrie Algébrique — English

§21. Divisors

On the content of the present section, see the comments of the introduction to §20. For the global properties of divisors, the reader is referred to the section devoted to them in chap. V.

21.1. Divisors on a ringed space

(21.1.1). Let $(X,{\mathcal{O}}_X)$ be a ringed space, ${\mathcal{M}}_X$ the sheaf of germs of meromorphic functions on $X$ (20.1.3), ${\mathcal{M}}_X^{\times }$ the sheaf (of multiplicative groups) of germs of regular meromorphic functions on $X$ (20.1.8). It is clear that the sheaf (of multiplicative groups) ${\mathcal{O}}_X^{\times }$ of germs of invertible sections of ${\mathcal{O}}_X$ is canonically identified with a subsheaf (of multiplicative groups) of ${\mathcal{M}}_X^{\times }$.

Definition (21.1.2).

One calls sheaf of divisors on $X$ and denotes by $\mathcal{D}i{v}_X$ the quotient sheaf (of commutative groups) ${\mathcal{M}}_X^{\times }/{\mathcal{O}}_X^{\times }$; the sections of this sheaf over $X$ are called divisors on $X$; they form a commutative group denoted $\mathrm{Div}(X)$. For every section $f$ of ${\mathcal{M}}_X^{\times }$ over $X$ (in other words, every regular meromorphic function on $X$ (20.1.8)), one calls divisor of $f$ and denotes by $div(f)$ (or $di{v}_X(f)$) the divisor on $X$ image of $f$ by the canonical homomorphism $\Gamma (X,{\mathcal{M}}_X^{\times })\to \Gamma (X,\mathcal{D}i{v}_X)$.

The support of a divisor $D$ is the closed set of $x\in X$ such that $D_x\ne 0$. One denotes it $Supp(D)$.

For every open $U$ of $X$, one obviously has ${\mathcal{M}}_X^{\times }|U={\mathcal{M}}_U^{\times }$, ${\mathcal{O}}_X^{\times }|U={\mathcal{O}}_U^{\times }$, hence $\mathcal{D}i{v}_X|U=\mathcal{D}i{v}_U$, and consequently the sheaf $\mathcal{D}i{v}_X$ is equal to the presheaf $U\mapsto \mathrm{Div}(U)$.

When $X=\mathrm{Spec}(A)$ is affine, one writes $\mathrm{Div}(A)$ instead of $\mathrm{Div}(\mathrm{Spec}(A))$.

(21.1.3). We shall always denote the group $\mathrm{Div}(X)$ of divisors on $X$ additively. For two regular meromorphic functions $f$, $g$ on $X$, one therefore has

  (21.1.3.1)             div(fg) = div(f) + div(g),

  (21.1.3.2)             div(f⁻¹) = −div(f).

By definition, for every regular meromorphic function $f$ on $X$, one has the equivalence

  (21.1.3.3)             div(f) = 0  ⟺  f ∈ Γ(X, 𝒪_X^×),

whence, for two regular meromorphic functions $f$, $g$ on $X$,

  (21.1.3.4)             div(f) = div(g)  ⟺  f = ug, u ∈ Γ(X, 𝒪_X^×).

(21.1.4). Let now $\mathcal{L}$ be an invertible ${\mathcal{O}}_X$-Module, and let $s$ be a regular meromorphic section (20.1.8) of $\mathcal{L}$ over $X$. Every $x\in X$ possesses an open neighbourhood $U$ such that $\mathcal{L}|U$ is isomorphic to ${\mathcal{O}}_U$, hence ${\mathcal{M}}_X(\mathcal{L})|U$ isomorphic to ${\mathcal{M}}_U$; by one of these isomorphisms, $s|U$ corresponds to a section $f\in \Gamma (U,{\mathcal{M}}_X^{\times })$, and since two isomorphisms of $\mathcal{L}|U$ onto ${\mathcal{O}}_U$ differ only by multiplication by an element of $\Gamma (U,{\mathcal{O}}_X^{\times })$ $(0_I,\mathrm{5.4.3})$, the element $di{v}_U(f)$ of $\Gamma (U,\mathcal{D}i{v}_X)$ is independent of the chosen isomorphism; it is clear that these elements (for variable $U$) are the restrictions of a section of $\mathcal{D}i{v}_X$ over $X$, which one calls the divisor of $s$ and denotes $div(s)$ (such a divisor is not necessarily of the form $div(g)$ for a regular meromorphic function $g$ on $X$; see (21.2.9)). For $\mathcal{L}={\mathcal{O}}_X$ the definition of $div(s)$ coincides with that of (21.1.2). If $\mathcal{L}$, ${\mathcal{L}}^{\prime }$ are two invertible ${\mathcal{O}}_X$-Modules, $s$ (resp. $s^{\prime }$) a regular meromorphic section of $\mathcal{L}$ (resp. ${\mathcal{L}}^{\prime }$) over $X$, it is immediate that one has

  (21.1.4.1)             div(s ⊗ s') = div(s) + div(s')

  (21.1.4.2)             div(s^⊗ n) = n · div(s)        for every n ∈ ℤ

($s^{-1}$ being the regular meromorphic section of ${\mathcal{L}}^{-1}$ over $X$ defined in (20.1.10)), and, for two regular meromorphic sections $s$, $s^{\prime }$ of $\mathcal{L}$ over $X$, one has the relation

  (21.1.4.3)             div(s) = div(s')  ⟺  s' = ts, with t ∈ Γ(X, 𝒪_X^×).

(21.1.5). The sheaf $\mathcal{S}({\mathcal{O}}_X)$ (20.1.3) whose sections over an open $U$ of $X$ are the regular elements of $\Gamma (U,{\mathcal{O}}_X)$ is a subsheaf of monoids of ${\mathcal{M}}_X^{\times }$; one can write

$$(\mathrm{21.1.5.1})\mathcal{S}({\mathcal{O}}_X)={\mathcal{O}}_X\cap {\mathcal{M}}_X^{\times }.$$

If one denotes by $\mathcal{S}({\mathcal{O}}_X{)}^{-1}$ the sheaf whose sections over $U$ are the inverses in $\Gamma (U,{\mathcal{M}}_X^{\times })$ of the elements of $\Gamma (U,\mathcal{S}({\mathcal{O}}_X))$, it is clear that one has $\Gamma (U,\mathcal{S}({\mathcal{O}}_X))\cap \Gamma (U,\mathcal{S}({\mathcal{O}}_X{)}^{-1})=\Gamma (U,{\mathcal{O}}_X^{\times })$, hence

$$(\mathrm{21.1.5.2})\mathcal{S}({\mathcal{O}}_X)\cap \mathcal{S}({\mathcal{O}}_X{)}^{-1}={\mathcal{O}}_X^{\times }.$$

Definition (21.1.6).

The subsheaf of sets of $\mathcal{D}i{v}_X$ that is the canonical image of the subsheaf $\mathcal{S}({\mathcal{O}}_X)$ of ${\mathcal{M}}_X^{\times }$ is denoted $\mathcal{D}i{v}_X^{+}$; its sections over $X$ are called positive divisors on $X$, and their set is denoted ${\mathrm{Div}}^{+}(X)$.

Since $\mathcal{S}({\mathcal{O}}_X)$ is a sheaf of (multiplicative) monoids, one has

  (21.1.6.1)             Div^+(X) + Div^+(X) ⊂ Div^+(X)

and on the other hand, by virtue of (21.1.5.2) and (21.1.3.3),

$$(\mathrm{21.1.6.2}){\mathrm{Div}}^{+}(X)\cap (-{\mathrm{Div}}^{+}(X))=0.$$

These two relations show that ${\mathrm{Div}}^{+}(X)$ is the set of positive elements for an order structure on the group $\mathrm{Div}(X)$, compatible with this group structure; one denotes this order relation $D\ge D^{\prime }$; in other words, one has

$$(\mathrm{21.1.6.3})D\ge 0⟺D\in {\mathrm{Div}}^{+}(X).$$

We shall always assume in what follows that $\mathrm{Div}(X)$ is endowed with this order structure; it is clear that $\mathcal{D}i{v}_X^{+}|U=\mathcal{D}i{v}_U^{+}$, hence $\Gamma (U,\mathcal{D}i{v}_X^{+})={\mathrm{Div}}^{+}(U)$, and one can therefore say that $\mathcal{D}i{v}_X^{+}$ defines on $\mathcal{D}i{v}_X$ a structure of sheaf of ordered groups. The stalk $(\mathcal{D}i{v}_X{)}_x$ at a point $x$ of the sheaf $\mathcal{D}i{v}_X$ is a submonoid of the group $(\mathcal{D}i{v}_X{)}_x$, the set of elements $\ge 0$ for an order structure compatible with the group structure; for a divisor $D$ on $X$, it amounts to the same to say that $D\ge 0$ or that $D_x\ge 0$ for every $x\in X$.

By definition, for every regular meromorphic function $f$ on $X$, one has the relation

  (21.1.6.4)             div(f) ≥ 0  ⟺  f ∈ Γ(X, 𝒮(𝒪_X));

in other words, $div(f)\ge 0$ signifies that $f$ is a regular section of ${\mathcal{O}}_X$, or also a section of ${\mathcal{O}}_X$ invertible in $M(X)$.

More generally, given a divisor $D$ on $X$, the relation $div(f)\ge D$ is equivalent to the following: for every open $U$ of $X$ such that $D|U=di{v}_U(g)$, where $g\in \Gamma (U,{\mathcal{M}}_X^{\times })$, there exists a regular element $t$ of $\Gamma (U,{\mathcal{O}}_X)$ such that $f|U=tg$.

(21.1.7). Let $\mathcal{L}$ be an invertible ${\mathcal{O}}_X$-Module, $s$ a regular meromorphic section of $\mathcal{L}$ over $X$; one has the relation

  (21.1.7.1)             div(s) ≥ 0  ⟺  s ∈ Γ(X, ℒ) ∩ Γ(X, (𝓜_X(ℒ))^×)

as follows at once from the definitions (21.1.4) and (21.1.6).

Proposition (21.1.8).

Let $X$ be a locally Noetherian prescheme, $D$ a divisor on $X$. Suppose that for every $x\in X$ such that $prof({\mathcal{O}}_{X,x})=1$ one has $D_x\ge 0$ (resp. $D_x=0$). Then one has $D\ge 0$ (resp. $D=0$).

The question being local on $X$, one may assume that $D=div(f)$, $f$ being a

regular meromorphic function on $X$; the relation $D_x\ge 0$ is equivalent to $x\in dom(f)$, hence the hypothesis means that if $T=X-dom(f)$, one has $prof({\mathcal{O}}_{X,x})\ge 2$ for every $x\in T$ (since $dom(f)$ contains the maximal points of $X$). Consequently (5.10.5) the restriction homomorphism $\Gamma (X,{\mathcal{O}}_X)\to \Gamma (X-T,{\mathcal{O}}_X)$ is bijective, which shows that there exists a section $s$ of ${\mathcal{O}}_X$ over $X$ such that $f=s|(X-T)$. But by definition of $T$, this implies $T=\varnothing$, hence $f=s$ and $D\ge 0$. The assertion relative to the relation $D_x=0$ follows at once by applying what precedes to $-D$, by virtue of (21.1.6.2).

Corollary (21.1.9).

Let $X$ be a locally Noetherian prescheme, $D$ a divisor on $X$. Let $S$ be the support of $D$. Then, for every maximal point $x$ of $S$, one has $prof({\mathcal{O}}_{X,x})\le 1$.

Indeed set $X_1=\mathrm{Spec}({\mathcal{O}}_{X,x})$; in view of (20.2.11) and (20.3.6), the sheaf ${\mathcal{O}}_X$ is induced on X_1 by ${\mathcal{O}}_X$, hence one may restrict to the case where $X=X_1$, in which case, since $x\in S$ and $x$ is a maximal point of $S$, one necessarily has $S=x$. If one had $prof({\mathcal{O}}_{X,x})\ne 1$, one would conclude, by virtue of (21.1.8), that $D=0$, which contradicts the definition of $S$.

Proposition (21.1.10).

Let $A$ be a Noetherian local ring; for $\mathrm{Div}(A)=0$, it is necessary and sufficient that $prof(A)=0$ (in other words, that the maximal ideal $\mathfrak{m}$ of $A$ be associated to $A$ (0, 16.4.6)).

Indeed, to say that $\mathrm{Div}(A)=0$ means that in $A$ every regular element is invertible, or also that all the elements of $\mathfrak{m}$ are zero-divisors, which means that $\mathfrak{m}\in Ass(A)$ (Bourbaki, Alg. comm., chap. IV, §1, n° 1, cor. 3 of prop. 2).

21.2. Divisors and invertible fractional Ideals

(21.2.1). Let $(X,{\mathcal{O}}_X)$ be a ringed space. One calls fractional Ideal on $X$ a sub-${\mathcal{O}}_X$-Module of the ${\mathcal{O}}_X$-Module ${\mathcal{M}}_X$ of germs of meromorphic functions on $X$. A fractional Ideal $\mathcal{I}$ on $X$ which is an invertible ${\mathcal{O}}_X$-Module is called an invertible fractional Ideal.

Proposition (21.2.2).

For a fractional Ideal $\mathcal{I}$ on $X$ to be invertible, it is necessary and sufficient that for every $x\in X$, there exist an open neighbourhood $U$ of $x$ and a section $f\in \Gamma (U,{\mathcal{M}}_X^{\times })$ such that $\mathcal{I}|U={\mathcal{O}}_U\cdot f$.

The condition is obviously sufficient, the map $s\mapsto s\cdot (f|V)$ from $\Gamma (V,{\mathcal{O}}_X)$ to $\Gamma (V,\mathcal{I})$ being obviously bijective for every open $V\subset U$. To see that it is necessary, note that by hypothesis there exist an open neighbourhood $U$ of $x$ and an isomorphism of ${\mathcal{O}}_X$-Modules ${\mathcal{O}}_U\stackrel{\sim }{\to }\mathcal{I}|U$. If $f$ is the image of the section $1\in \Gamma (U,{\mathcal{O}}_X)$ by this isomorphism, one may assume, restricting $U$, that $f=u^{-1}s$, where $u\in \Gamma (U,{\mathcal{O}}_X)$ and $s\in \Gamma (U,\mathcal{S}({\mathcal{O}}_X))$, and the considered isomorphism then makes correspond, to every section $v\in \Gamma (V,{\mathcal{O}}_X)$ (where $V$ is an open contained in $U$) the section $v(u|V{)}^{-1}f(s|V{)}^{-1}$; to say that the map thus defined is bijective means that $u|V$ is a regular element of $\Gamma (V,{\mathcal{O}}_X)$, hence $f\in \Gamma (U,{\mathcal{M}}_X^{\times })$.

One will note that for every open $U$ of $X$ such that $\mathcal{I}|U={\mathcal{O}}_U\cdot f$ with $f\in \Gamma (U,{\mathcal{M}}_X^{\times })$, the section $f$ is determined up to multiplication by an element of $\Gamma (U,{\mathcal{O}}_X^{\times })$, since the

multiplication by these elements provides all the automorphisms of the ${\mathcal{O}}_U$-Module ${\mathcal{O}}_U$ $(0_I,\mathrm{5.4.3})$.

Corollary (21.2.3).

(i) Let $\mathcal{I}$ be an invertible fractional Ideal; then the invertible ${\mathcal{O}}_X$-Module ${\mathcal{I}}^{-1}$ is canonically identified with the fractional Ideal ${\mathcal{I}}^{\prime }$ (transporter of $\mathcal{I}$ into ${\mathcal{O}}_X$) defined in the following way: for every open $U$ of $X$ such that $\mathcal{I}|U={\mathcal{O}}_U\cdot f$, where $f\in \Gamma (U,{\mathcal{M}}_X^{\times })$, one has ${\mathcal{I}}^{\prime }|U={\mathcal{O}}_U\cdot f^{-1}$.

(ii) If ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$ are two invertible fractional Ideals, the canonical map ${\mathcal{I}}_1\otimes {\mathcal{I}}_2\to {\mathcal{I}}_1{\mathcal{I}}_2$ is an isomorphism of ${\mathcal{O}}_X$-Modules.

Assertion (ii) follows at once from (21.2.2). On the other hand, the remark made at the end of (21.2.2) shows that there exists indeed one and only one fractional Ideal ${\mathcal{I}}^{\prime }$ defined by the condition of the statement; the canonical isomorphism of ${\mathcal{I}}^{\prime }$ onto ${\mathcal{I}}^{-1}=\mathcal{H}{om}_{{\mathcal{O}}_X}(\mathcal{I},{\mathcal{O}}_X)$ is obtained by making correspond to every section $s\in \Gamma (V,{\mathcal{I}}^{-1})$ (where $V$ is an open contained in $U$ and $f\in \Gamma (V,{\mathcal{M}}_X^{\times })$) the homomorphism $t(f|V{)}^{-1}\mapsto st$ from $\Gamma (V,\mathcal{I})$ to $\Gamma (V,{\mathcal{O}}_X)$.

By virtue of (21.2.3, (i)), one will generally identify the ${\mathcal{O}}_X$-Modules ${\mathcal{I}}^{\prime }$ and ${\mathcal{I}}^{-1}$, considering therefore ${\mathcal{I}}^{-1}$ as a sub-${\mathcal{O}}_X$-Module of ${\mathcal{M}}_X$.

(21.2.4). It follows from (21.2.3) that the set Id.inv(X) of invertible fractional Ideals on $X$ is endowed with a structure of commutative group for the composition law $({\mathcal{I}}_1,{\mathcal{I}}_2)\mapsto {\mathcal{I}}_1{\mathcal{I}}_2$, the neutral element of this group being ${\mathcal{O}}_X$. It is clear that for every open $U$ of $X$, one has $({\mathcal{I}}_1{\mathcal{I}}_2)|U=({\mathcal{I}}_1|U)({\mathcal{I}}_2|U)$, hence the restriction map $\mathcal{I}\mapsto \mathcal{I}|U$ is a homomorphism of groups $Id.inv(X)\to Id.inv(U)$; one thus defines a presheaf of commutative groups $U\mapsto Id.inv(U)$; it is immediate that in fact this presheaf is a sheaf of commutative groups, which one denotes $Id.{inv}_X$.

(21.2.5). For every regular meromorphic function $f\in \Gamma (X,{\mathcal{M}}_X^{\times })$, it follows from (21.2.2) that $\mathcal{J}(f)={\mathcal{O}}_X\cdot f$ is an invertible fractional Ideal, and one obviously has $\mathcal{J}(f_1{f}_2)=\mathcal{J}(f_1)\mathcal{J}(f_2)$; in other words the map $f\mapsto \mathcal{J}(f)$ is a homomorphism from the commutative group $\Gamma (X,{\mathcal{M}}_X^{\times })$ into the commutative group Id.inv(X). Replacing $X$ by any open $U$ of $X$ and noting that the homomorphisms obtained are compatible with the operations of restriction, one obtains a canonical homomorphism of sheaves of commutative groups:

$$(\mathrm{21.2.5.1})\mathcal{J}:{\mathcal{M}}_X^{\times }\to Id.{inv}_X.$$

Note that if $f\in \Gamma (X,{\mathcal{O}}_X^{\times })$, one has $\mathcal{J}(f)={\mathcal{O}}_X$; one deduces at once that the homomorphism $\mathcal{J}$ factors as

  (21.2.5.2)             𝓜_X^× → 𝓜_X^× / 𝒪_X^× = 𝒟iv_X → 𝐼𝑑.𝑖𝑛𝑣_X

where $I$ is a homomorphism from the sheaf of additive groups $\mathcal{D}i{v}_X$ into the sheaf of multiplicative groups $Id.{inv}_X$; one therefore has for every open $U$ of $X$ a homomorphism $I_U:\mathrm{Div}(U)\to Id.inv(U)$ of commutative groups, such that, for every section $f\in \Gamma (U,{\mathcal{M}}_X^{\times })$, one has

$$(\mathrm{21.2.5.3})I_U(di{v}_U(f))={\mathcal{O}}_U\cdot f.$$

One concludes that $I(D)$, for every divisor $D\in \mathrm{Div}(X)$, is the invertible fractional Ideal defined in the following way: for every open $U$ of $X$ such that $D|U=di{v}_U(f)$, where $f\in \Gamma (U,{\mathcal{M}}_X^{\times })$, $I(D)|U$ is the invertible fractional Ideal ${\mathcal{O}}_U\cdot f$. One therefore has, by virtue of (21.1.6), for every regular meromorphic function $f\in \Gamma (X,{\mathcal{M}}_X^{\times })$, the relation

  (21.2.5.4)             f ∈ Γ(X, 𝐼(D))  ⟺  div(f) ≥ D.

Proposition (21.2.6).

The homomorphism $I:\mathcal{D}i{v}_X\to Id.{inv}_X$ is bijective.

One defines a homomorphism $I_X^{\prime }$ from Id.inv(X) into $\mathrm{Div}(X)$ by making correspond to every invertible fractional Ideal $\mathcal{I}$ on $X$ the divisor $I_X^{\prime }(\mathcal{I})$ defined in the following way: for every open $U$ of $X$ such that $\mathcal{I}|U={\mathcal{O}}_U\cdot f$, where $f\in \Gamma (U,{\mathcal{M}}_X^{\times })$ (21.2.2), one takes $I_X^{\prime }(\mathcal{I})|U=di{v}_U(f)$; by virtue of the remark following (21.2.2), this definition is independent of the generator $f$ chosen in $\mathcal{I}|U$, and determines indeed a divisor on $X$. Moreover, this definition shows at once that the homomorphisms $I_X$ and $I_X^{\prime }$ are reciprocal to one another. Replacing $X$ by any open $U$, one deduces the definition of the isomorphism $I^{\prime }:Id.{inv}_X\to \mathcal{D}i{v}_X$ reciprocal to $I$, whence the proposition. One will set $I_X^{\prime }(\mathcal{I})=div(\mathcal{I})$, so that one has, for every regular meromorphic function $f$ on $X$,

$$(\mathrm{21.2.6.1})div({\mathcal{O}}_X\cdot f)=div(f).$$

(21.2.7). One will often identify the sheaves $\mathcal{D}i{v}_X$ and $Id.{inv}_X$ (resp. the groups $\mathrm{Div}(X)$ and Id.inv(X)) by means of the preceding isomorphisms $I$ and $I^{\prime }$ (resp. $I_X$ and $I_X^{\prime }$). One will note that one has the relation

  (21.2.7.1)             D ≥ 0  ⟺  𝐼_X(D) ⊂ 𝒪_X       for D ∈ Div(X)

as follows at once from the definitions (21.1.6) and (21.1.5.1); in other words, the image $I_X({\mathrm{Div}}^{+}(X))$ is the set of Ideals of ${\mathcal{O}}_X$ (also sometimes called integral Ideals) that are invertible ${\mathcal{O}}_X$-Modules: such an Ideal $\mathcal{I}$ is again characterized by the fact that for every $x\in X$, there is an open neighbourhood $U$ of $x$ in $X$ such that $\mathcal{I}|U={\mathcal{O}}_U\cdot f$, where $f$ is a regular element of $\Gamma (U,{\mathcal{O}}_X)$. The set $I_X({\mathrm{Div}}^{+}(X))$ of these Ideals is therefore a submonoid of Id.inv(X), equal to the set of positive elements for an order relation compatible with the group structure of Id.inv(X), and it is immediate that this relation is none other than the relation opposite to inclusion; in other words, one has

  (21.2.7.2)             ℐ_1 ⊂ ℐ_2  ⟺  div(ℐ_1) ≥ div(ℐ_2)

for ${\mathcal{I}}_1$, ${\mathcal{I}}_2$ in Id.inv(X).

(21.2.8). For every divisor $D$ on $X$, one sets

$$(\mathrm{21.2.8.1}){\mathcal{O}}_X(D)=(I_X(D){)}^{-1};$$

${\mathcal{O}}_X(D)$ is therefore an invertible fractional Ideal, defined in the following way: for every open $U$ of $X$

such that $D|U=di{v}_U(f)$, where $f\in \Gamma (U,{\mathcal{M}}_X^{\times })$, ${\mathcal{O}}_X(D)|U$ is the invertible fractional Ideal ${\mathcal{O}}_U\cdot f^{-1}$; by virtue of (21.1.6), for every regular meromorphic function $f$, one has the relation

  (21.2.8.2)             f ∈ Γ(X, 𝒪_X(D))  ⟺  div(f) ≥ −D.

Moreover, it is clear that one has canonical isomorphisms (21.2.3)

  (21.2.8.3)             𝒪_X(0) = 𝒪_X,   𝒪_X(D + D') = 𝒪_X(D) ⊗ 𝒪_X(D'),
                         𝒪_X(nD) = (𝒪_X(D))^⊗ n,   𝒪_X(−D) = (𝒪_X(D))⁻¹

for every integer $n\in \mathbb{Z}$, and for any two divisors $D$, $D^{\prime }$ on $X$.

(21.2.9). Let $\mathcal{I}$ be an invertible fractional Ideal on $X$. The canonical injection $\mathcal{I}\to {\mathcal{M}}_X$ defines by tensorization a homomorphism of ${\mathcal{O}}_X$-Modules

  (21.2.9.1)             𝓜_X(ℐ) = ℐ ⊗_{𝒪_X} 𝓜_X → 𝓜_X ⊗_{𝒪_X} 𝓜_X = 𝓜_X

which is an isomorphism: indeed, if $U$ is an open of $X$ such that $\mathcal{I}|U={\mathcal{O}}_U\cdot f$, where $f\in \Gamma (U,{\mathcal{M}}_X^{\times })$, the homomorphism (21.2.9.1) restricted to $U$ is none other than the isomorphism that to every section $t$ of ${\mathcal{M}}_X(\mathcal{I})|U={\mathcal{M}}_U$ over $V\subset U$ makes correspond the section $t(f|V{)}^{-1}$ of the same sheaf. In the isomorphism (21.2.9.1), to the regular meromorphic sections of $\mathcal{I}$ over $X$ correspond the regular meromorphic functions on $X$.

Consider in particular the case where $\mathcal{I}={\mathcal{O}}_X(D)$, where $D$ is a divisor on $X$; one then has a canonical isomorphism

$$(\mathrm{21.2.9.2}){\mathcal{M}}_X({\mathcal{O}}_X(D))\stackrel{\sim }{\to }{\mathcal{M}}_X$$

and one denotes by $s_D$ the regular meromorphic section of ${\mathcal{O}}_X(D)$ over $X$ which corresponds by this isomorphism to the section 1 of ${\mathcal{M}}_X$. If $U$ is an open of $X$ such that ${\mathcal{O}}_X(D)|U={\mathcal{O}}_U\cdot f^{-1}$, where $f\in \Gamma (U,{\mathcal{M}}_X^{\times })$, one has $s_D|U=f$ in $\Gamma (U,{\mathcal{M}}_X)$; since one then has $D|U=di{v}_U(f)$, one deduces (21.1.4) that one has

$$(\mathrm{21.2.9.3})div(s_D)=D.$$

On the other hand, one deduces at once from the canonical isomorphisms (21.2.8.3) the formulas

  (21.2.9.4)             s_0 = 1,   s_{D + D'} = s_D ⊗ s_{D'},   s_{nD} = s_D^{⊗ n}   (n ∈ ℤ).

(21.2.10). Let us consider, between two pairs $(\mathcal{L},s)$, $({\mathcal{L}}^{\prime },s^{\prime })$, where $\mathcal{L}$ and ${\mathcal{L}}^{\prime }$ are two invertible ${\mathcal{O}}_X$-Modules, $s$ (resp. $s^{\prime }$) a regular meromorphic section of $\mathcal{L}$ (resp. ${\mathcal{L}}^{\prime }$) over $X$, the relation: "there exists an isomorphism $u:\mathcal{L}\stackrel{\sim }{\to }{\mathcal{L}}^{\prime }$ such that $\bar{u}(s)=s^{\prime }$", where $\bar{u}:\Gamma (X,{\mathcal{M}}_X(\mathcal{L}))\stackrel{\sim }{\to }\Gamma (X,{\mathcal{M}}_X({\mathcal{L}}^{\prime }))$ is the isomorphism deduced from $u$ (one will note that the isomorphism $u$ verifying this condition is then uniquely determined). It is clear that this is an equivalence relation, and since there exists a set of invertible ${\mathcal{O}}_X$-Modules such that every invertible ${\mathcal{O}}_X$-Module is isomorphic to an element of this set $(0_I,\mathrm{5.4.7})$, one can speak of the set $D(X)$ of equivalence classes of pairs $(\mathcal{L},s)$ for the preceding relation. For every invertible ${\mathcal{O}}_X$-Module $\mathcal{L}$ and every

regular meromorphic section $s$ of $\mathcal{L}$ over $X$, one will denote by $cl(\mathcal{L},s)$ the element of $D(X)$ corresponding to the pair $(\mathcal{L},s)$. It follows from $(0_I,\mathrm{5.4.3})$ that if $s$, $s^{\prime }$ are two regular meromorphic sections of $\mathcal{L}$ over $X$, the relation $cl(\mathcal{L},s)=cl(\mathcal{L},s^{\prime })$ is equivalent to the existence of a section $t\in \Gamma (X,{\mathcal{O}}_X^{\times })$ such that $s^{\prime }=ts$.

It is immediate that if $(\mathcal{L},s)$ is equivalent to $({\mathcal{L}}_1,s_1)$ and $({\mathcal{L}}^{\prime },s^{\prime })$ to $({\mathcal{L}}_1^{\prime },s_1^{\prime })$, the pairs $(\mathcal{L}\otimes {\mathcal{L}}^{\prime },s\otimes s^{\prime })$ and $({\mathcal{L}}_1\otimes {\mathcal{L}}_1^{\prime },s_1\otimes s_1^{\prime })$ are equivalent; one therefore defines in $D(X)$ a composition law by setting

  (21.2.10.1)            cl(ℒ, s) · cl(ℒ', s') = cl(ℒ ⊗ ℒ', s ⊗ s');

it is immediate that this is a commutative group law, whose neutral element is $cl({\mathcal{O}}_X,1)$ and where the inverse of $cl(\mathcal{L},s)$ is $cl({\mathcal{L}}^{-1},s^{\otimes (-1)})$.

Proposition (21.2.11).

The maps

  (21.2.11.1)            D ↦ cl(𝒪_X(D), s_D),       cl(ℒ, s) ↦ div(s)

are reciprocal isomorphisms of $\mathrm{Div}(X)$ onto $D(X)$ and of $D(X)$ onto $\mathrm{Div}(X)$ respectively.

In view of (21.2.8.3), (21.2.9.4) and (21.2.10.1), it suffices to see that the composites of these two maps are the identity in $\mathrm{Div}(X)$ and $D(X)$ respectively. The first assertion is none other than (21.2.9.3). On the other hand, let $D=div(s)$, where $s$ is a regular meromorphic section of $\mathcal{L}$ over $X$, and let $U$ be an open of $X$ such that there exists an isomorphism of $\mathcal{L}|U$ onto ${\mathcal{O}}_U$ transforming $s|U$ into $f\in \Gamma (U,{\mathcal{M}}_X^{\times })$, so that $D|U=di{v}_U(f)$, ${\mathcal{O}}_X(D)|U={\mathcal{O}}_U\cdot f^{-1}$, and $s_D|U$ is the unit element of $\Gamma (U,{\mathcal{O}}_X^{\times })$. There is therefore an isomorphism $v_U:\mathcal{L}|U\to {\mathcal{O}}_U\cdot f^{-1}={\mathcal{O}}_X(D)|U$ such that ${\bar{v}}_U$ (notation of (21.2.10)) transforms $s|U$ into $f\cdot f^{-1}=1$; one sees at once that these isomorphisms are compatible with the operations of restriction, hence define an isomorphism $v:\mathcal{L}\to {\mathcal{O}}_X(D)$ such that $\bar{v}(s)=s_D$. Q.E.D.

One can transport, by the first of the isomorphisms (21.2.11.1), the ordered group structure of $\mathrm{Div}(X)$ to $D(X)$; the elements $\ge 0$ of $D(X)$ are therefore the classes $cl(\mathcal{L},s)$ such that $div(s)\ge 0$, that is to say (21.1.7.1) such that

                         s ∈ Γ(X, ℒ) ∩ Γ(X, (𝓜_X(ℒ))^×).

(21.2.12). Let $D$ be a positive divisor on a prescheme $X$; the fractional Ideal ${\mathcal{O}}_X(D)$ is therefore an Ideal of ${\mathcal{O}}_X$ which is an invertible ${\mathcal{O}}_X$-Module; let $Y(D)$ be the closed sub-prescheme of $X$ it defines. For every $x\in Y(D)$, there is by hypothesis an open neighbourhood $U$ of $x$ in $X$ and a regular section $t\in \Gamma (U,{\mathcal{O}}_X)$ such that ${\mathcal{O}}_X(D)|U={\mathcal{O}}_U\cdot t\cdot ({\mathcal{O}}_X|U)$; in other words, the canonical immersion $Y(D)\to X$ is regular and of codimension 1 (19.1.4) at every point of $Y(D)$. Conversely, if $Y$ is a closed sub-prescheme of $X$, regularly immersed in $X$ and of codimension 1 at every point of $Y$, there exists one and only one positive divisor $D$ such that $Y(D)=Y$, for every $x\in Y$, there is a neighbourhood $U$ of $x$ in $X$ such that $Y\cap U$ is defined by an Ideal of ${\mathcal{O}}_X$ of the form ${\mathcal{O}}_U\cdot t$, where $t$ is regular in $\Gamma (U,{\mathcal{O}}_X)$.

One will note that one then has $Supp(D)=Y(D)$, for to say that $D_x\ne 0$ means that $t$ (with the notations above) is not invertible, that is to say that $x\in Y(D)$.

21.3. Linear equivalence of divisors

(21.3.1). One says that a divisor $D$ on $X$ is principal if it is of the form $div(f)$, where $f$ is a regular meromorphic function on $X$; the regular meromorphic functions $f^{\prime }$ such that $div(f^{\prime })=D$ are then all those of the form tf, where $t\in \Gamma (X,{\mathcal{O}}_X^{\times })$ (21.1.3.4). The set of principal divisors is a subgroup of $\mathrm{Div}(X)$, denoted $\mathrm{Div}.princ(X)$, isomorphic to $\Gamma (X,{\mathcal{M}}_X^{\times })/\Gamma (X,{\mathcal{O}}_X^{\times })$. Two divisors $D$, $D^{\prime }$ are said to be linearly equivalent if $D-D^{\prime }$ is a principal divisor; the principal divisors are therefore the divisors linearly equivalent to 0.

(21.3.2). Recall $(0_I,\mathrm{5.4.7})$ that one can speak of the set of equivalence classes of invertible ${\mathcal{O}}_X$-Modules for the relation of isomorphy; one denotes this set by $\mathrm{Pic}(X)$, and for every invertible ${\mathcal{O}}_X$-Module $\mathcal{L}$, one denotes by $cl(\mathcal{L})$ the equivalence class of ${\mathcal{O}}_X$-Modules isomorphic to $\mathcal{L}$; moreover, $\mathrm{Pic}(X)$ is a commutative group for the multiplication defined by $cl(\mathcal{L})cl({\mathcal{L}}^{\prime })=cl(\mathcal{L}\otimes {\mathcal{L}}^{\prime })$. It is clear that the map

  (21.3.2.1)             𝓁' : cl(ℒ, s) ↦ cl(ℒ)

is a homomorphism from the group $D(X)$ (21.2.10) into the group $\mathrm{Pic}(X)$. By composition, one therefore deduces a homomorphism

  (21.3.2.2)             𝓁 : Div(X) ⥲ D(X) → Pic(X)

(also denoted $l_X$) such that, for every divisor $D$, one has

$$(\mathrm{21.3.2.3})l(D)=cl({\mathcal{O}}_X(D)).$$

Note finally that, if $u:X^{\prime }\to X$ is a morphism of ringed spaces, ${\mathcal{L}}_1$, ${\mathcal{L}}_2$ two isomorphic invertible ${\mathcal{O}}_X$-Modules, the invertible ${\mathcal{O}}_{X^{\prime }}$-Modules $u^{\ast }({\mathcal{L}}_1)$ and $u^{\ast }({\mathcal{L}}_2)$ $(0_I,\mathrm{5.4.5})$ are isomorphic; since moreover, for any two invertible ${\mathcal{O}}_X$-Modules ${\mathcal{L}}_1$, ${\mathcal{L}}_2$, one has $u^{\ast }({\mathcal{L}}_1\otimes {\mathcal{L}}_2)=u^{\ast }({\mathcal{L}}_1)\otimes u^{\ast }({\mathcal{L}}_2)$ up to canonical isomorphism $(0_I,\mathrm{4.3.3})$, one sees that the morphism $u$ canonically defines a homomorphism of commutative groups

  (21.3.2.4)             Pic(u) : Pic(X) → Pic(X').

Proposition (21.3.3).

(i) The kernel of the canonical homomorphism $l:\mathrm{Div}(X)\to \mathrm{Pic}(X)$ is the subgroup $\mathrm{Div}.princ(X)$; in other words, for ${\mathcal{O}}_X(D)$ and ${\mathcal{O}}_X(D^{\prime })$ to be isomorphic, it is necessary and sufficient that $D$ and $D^{\prime }$ be linearly equivalent. One therefore has a canonical injective homomorphism

  (21.3.3.1)             Div(X) / Div.princ(X) → Pic(X)

deduced from $l$.

(ii) For an invertible ${\mathcal{O}}_X$-Module $\mathcal{L}$ to be such that $cl(\mathcal{L})$ is of the form $l(D)$, or also for $\mathcal{L}$ to be isomorphic to an ${\mathcal{O}}_X$-Module of the form ${\mathcal{O}}_X(D)$, it is necessary and sufficient that there exist a regular meromorphic section of $\mathcal{L}$.

The proposition follows at once from the definitions and from (21.2.10).

Proposition (21.3.4).

Let $X$ be a prescheme satisfying one of the two following hypotheses:

a) $X$ is locally Noetherian and $Ass({\mathcal{O}}_X)$ is contained in an affine open of $X$.

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

Then the canonical homomorphism $l:\mathrm{Div}(X)\to \mathrm{Pic}(X)$ is surjective, and gives, by passage to the quotient, an isomorphism

                         Div(X) / Div.princ(X) ⥲ Pic(X).

It suffices to show that every invertible ${\mathcal{O}}_X$-Module $\mathcal{L}$ admits a regular meromorphic section over $X$ (21.3.3). In the two cases, it suffices, thanks to (20.2.11, (ii)), to define a section $s$ of $({\mathcal{M}}_X(\mathcal{L}){)}^{\times }$ over a schematically dense open $U$ of $X$, or also in case a), over an open $U$ containing $Ass({\mathcal{O}}_X)$ (20.2.13, (iv)). Indeed, let $V$ be any open of $X$ such that $\mathcal{L}|V$ is isomorphic to ${\mathcal{O}}_V$, so that there exists an isomorphism of ${\mathcal{M}}_X(\mathcal{L})|V$ onto ${\mathcal{M}}_X|V$, transforming $s|(U\cap V)$ into a section $f_V$ of ${\mathcal{M}}_X$ over $U\cap V$. Since $U\cap V$ is schematically dense in $V$, it follows from (20.2.11) that there exists one and only one regular meromorphic function $g_V$ on $V$ such that $g_V|(U\cap V)=f_V$, and this section therefore corresponds, by the isomorphism considered, to a regular meromorphic section $u_V$ of $\mathcal{L}$ over $V$ such that $u_V|(U\cap V)=s|(U\cap V)$. Moreover, if $V^{\prime }$ is a second open of $X$ such that $\mathcal{L}|V^{\prime }$ is isomorphic to ${\mathcal{O}}_{V^{\prime }}$, the restrictions of $u_V$ and $u_{V^{\prime }}$ to $V\cap V^{\prime }$ are equal, for they correspond by isomorphism to two meromorphic functions which coincide in a schematically dense open $U\cap V\cap V^{\prime }$, and one concludes again by (20.2.11); the $u_V$ are therefore the restrictions of a section of $({\mathcal{M}}_X(\mathcal{L}){)}^{\times }$ over $X$.

This being so, in case b), one takes for each of the maximal points $x_{\lambda }$ of $X$ an open $U_{\lambda }$ containing $x_{\lambda }$, not meeting any irreducible component of $X$ distinct from $x_{\lambda }$ and such that $\mathcal{L}|U_{\lambda }$ is isomorphic to ${\mathcal{O}}_{U_{\lambda }}$; one will take for $s$ the section such that $s|U_{\lambda }$ is the section of $\mathcal{L}|U_{\lambda }$ corresponding by the preceding isomorphism to the unit section of ${\mathcal{O}}_{U_{\lambda }}$.

In case a), one can take by hypothesis for $U$ an affine open (hence Noetherian); in other words, one may assume that $X=\mathrm{Spec}(A)$, where $A$ is Noetherian, and $\mathcal{L}=\tilde{P}$, where $P$ is a projective $A$-module of rank 1. If $S$ is the set of regular elements of $A$, one has $\Gamma (X,{\mathcal{M}}_X)=S^{-1}A$ (20.2.12) and $\Gamma (X,{\mathcal{M}}_X(\mathcal{L}))=S^{-1}P$. But $S$ is the set of elements not belonging to any of the ideals associated to $A$, hence $S^{-1}A$ is a semi-local ring whose maximal ideals come from the maximal elements of $Ass(A)$, and $S^{-1}P$ is a projective $S^{-1}A$-module of rank 1, hence here free of rank 1 (Bourbaki, Alg. comm., chap. II, §5, n° 3, prop. 5); an element forming a basis of this $S^{-1}A$-module is therefore (20.1.8) a regular meromorphic section of $\mathcal{L}$ over $X$.

Corollary (21.3.5).

If $X$ is a Noetherian prescheme such that there exists an ample invertible ${\mathcal{O}}_X$-Module (II, 4.5.3) (for example a quasi-projective prescheme over the spectrum of a Noetherian ring (II, 5.3.1 and 4.6.6)), the canonical homomorphism $l:\mathrm{Div}(X)\to \mathrm{Pic}(X)$ is surjective.

Indeed (II, 4.5.4), there then exists an affine open neighbourhood of the finite set $Ass({\mathcal{O}}_X)$.

Remark (21.3.6).

Recall $(0_I,\mathrm{5.4.7})$ that one has a canonical isomorphism $\pi :H^1(X,{\mathcal{O}}_X^{\times })\stackrel{\sim }{\to }\mathrm{Pic}(X)$ defined in the following way. One starts from a 1-cocycle $(c_{\alpha \beta })$ with values in ${\mathcal{O}}_X^{\times }$ corresponding to an open cover $(U_{\alpha })$ of $X$, $c_{\alpha \beta }$ being an element of $\Gamma (U_{\alpha }\cap U_{\beta },{\mathcal{O}}_X^{\times })$, and one associates with it the class of the invertible ${\mathcal{O}}_X$-Module obtained by gluing the ${\mathcal{O}}_{U_{\alpha }}$ along the isomorphisms ${\mathcal{O}}_X|(U_{\alpha }\cap U_{\beta })\stackrel{\sim }{\to }{\mathcal{O}}_X|(U_{\alpha }\cap U_{\beta })$ defined by multiplication by $c_{\alpha \beta }$. On the other hand, one deduces from the exact sequence of sheaves of commutative groups

$$1\to {\mathcal{O}}_X^{\times }\to {\mathcal{M}}_X^{\times }\to \mathcal{D}i{v}_X\to 0$$

the connecting homomorphism of the exact cohomology sequence

$$(\mathrm{21.3.6.1})\partial :\mathrm{Div}(X)\to H^1(X,{\mathcal{O}}_X^{\times }).$$

Let us show that the composite homomorphism

                                                ∂              π
                         Div(X) → H^1(X, 𝒪_X^×) ⥲ Pic(X)

is none other than the homomorphism $l$ defined in (21.3.2.2). Indeed, one must start from a divisor $D$ and an open cover $(U_{\alpha })$ of $X$ such that $D|U_{\alpha }=di{v}_{U_{\alpha }}(g_{\alpha })$, where $g_{\alpha }$ is a regular meromorphic function over $U_{\alpha }$; $\partial (D)$ is the cohomology class of the cocycle $(c_{\alpha \beta })$, where $c_{\alpha \beta }=g_{\alpha |\alpha \beta }{g}_{\beta |\alpha \beta }^{-1}$, $g_{\alpha |\alpha \beta }$ denoting the restriction of $g_{\alpha }$ to $U_{\alpha }\cap U_{\beta }$. It is clear that the image by $\pi$ of this cohomology class is the class of the invertible fractional Ideal $\mathcal{L}$ such that for every $\alpha$, $\mathcal{L}|U_{\alpha }={\mathcal{O}}_{U_{\alpha }}\cdot g_{\alpha }^{-1}$, which is none other by definition than ${\mathcal{O}}_X(D)$ (21.2.8).

21.4. Inverse images of divisors

(21.4.1). Let $f:X^{\prime }\to X$ be a morphism of ringed spaces; we propose to give conditions allowing us to associate with a divisor $D$ on $X$ a divisor $D^{\prime }$ on $X^{\prime }$, inverse image of $D$ by $f$. Note first for this that for every section $t\in \Gamma (X,{\mathcal{O}}_X^{\times })$, the image of $t$ by the canonical homomorphism $\Gamma (X,{\mathcal{O}}_X)\to \Gamma (X^{\prime },{\mathcal{O}}_{X^{\prime }})$ is again invertible, in other words belongs to $\Gamma (X^{\prime },{\mathcal{O}}_{X^{\prime }}^{\times })$. Consider then $D$ as given by the equivalence class of a pair $(\mathcal{L},s)$, where $\mathcal{L}$ is an invertible ${\mathcal{O}}_X$-Module and $s$ a regular meromorphic section of $\mathcal{L}$ over $X$ (21.2.11). Form the invertible ${\mathcal{O}}_{X^{\prime }}$-Module $f^{\ast }(\mathcal{L})={\mathcal{L}}^{\prime }$; to say that the inverse image $s\circ f$ of $s$ by $f$ exists (20.1.11) and is a regular meromorphic section of ${\mathcal{L}}^{\prime }$ over $X^{\prime }$ amounts to saying that the inverse images by $f$ of $s$ and of $s^{\otimes (-1)}$ exist, in other words that $s\in \Gamma (X,{\mathcal{M}}_f(\mathcal{L}))$ and $s^{\otimes (-1)}\in \Gamma (X,{\mathcal{M}}_f({\mathcal{L}}^{-1}))$; the remark made above then shows that if $({\mathcal{L}}_1,s_1)$ is a pair equivalent to $(\mathcal{L},s)$ in the sense of (21.2.10), the inverse image $s_1\circ f$ exists and is a regular meromorphic section of ${\mathcal{L}}_1^{\prime }=f^{\ast }({\mathcal{L}}_1)$ over $X^{\prime }$, and the pairs $({\mathcal{L}}^{\prime },s\circ f)$ and $({\mathcal{L}}_1^{\prime },s_1\circ f)$ are equivalent. One can therefore lay down the following definition:

Definition (21.4.2).

Given a morphism $f:X^{\prime }\to X$ of ringed spaces, one says that the inverse image by $f$ of a divisor $D$ on $X$ exists if one has $s_D\in \Gamma (X,{\mathcal{M}}_f({\mathcal{O}}_X(D)))$ and $s_{-D}\in \Gamma (X,{\mathcal{M}}_f({\mathcal{O}}_X(-D)))$ (cf. (20.1.11)). One then calls inverse image of $D$ by $f$, and denotes by $f^{\ast }(D)$, the divisor on $X^{\prime }$ which corresponds canonically (21.2.11) to the class of the pair $(f^{\ast }({\mathcal{O}}_X(D)),s_D\circ f)$.

It follows at once from this definition that if $D$ and $D^{\prime }$ have inverse images under $f$, so do $-D$ and $D+D^{\prime }$ (taking account of (21.2.9.4)) and that one has

$f^{\ast }(D+D^{\prime })=f^{\ast }(D)+f^{\ast }(D^{\prime })$. In other words, the set ${\mathrm{Div}}_f(X)$ of divisors on $X$ whose inverse image by $f$ exists is a subgroup of $\mathrm{Div}(X)$, and the map $D\mapsto f^{\ast }(D)$ is an increasing homomorphism from the ordered subgroup ${\mathrm{Div}}_f(X)$ into the ordered group $\mathrm{Div}(X^{\prime })$, making commutative the diagram

                         Div_f(X) ─𝓁_X─→ Pic(X)
                            │              │
  (21.4.2.1)              f^*│              │ Pic(f)
                            ↓              ↓
                         Div(X') ─𝓁_{X'}→ Pic(X')

(21.4.3). Definition (21.4.2) shows at once that, for $f^{\ast }(D)$ to exist, it is necessary and sufficient that for every open $U$ of $X$, the inverse image by $f_U:f^{-1}(U)\to U$ (restriction of $f$) of $D|U$ exist. Now, if $D=div(g)$, where $g$ is a regular meromorphic function on $X$, to say that the inverse image of $s_D$ by $f$ exists and is a regular meromorphic section of $f^{\ast }({\mathcal{O}}_X(D))$ signifies (21.2.9) that the inverse image of $g$ by $f$ exists and is a regular meromorphic function on $X^{\prime }$. One deduces at once a second description of ${\mathrm{Div}}_f(X)$ and of $f^{\ast }(D)$: consider the subsheaf of groups of ${\mathcal{M}}_X^{\times }$, denoted ${\mathcal{M}}_X^{f\times }$, formed of the germs of regular meromorphic functions on an open of $X$ whose inverse image by $f$ exists and is regular on the inverse image open (20.1.11). Then if $f=(\psi ,\theta )$, the canonical homomorphism (20.1.11) ${\psi }^{\ast }({\mathcal{M}}_X)\to {\mathcal{M}}_{X^{\prime }}$ gives by restriction a homomorphism of sheaves of groups ${\psi }^{\ast }({\mathcal{M}}_X^{f\times })\to {\mathcal{M}}_{X^{\prime }}^{\times }$. Setting $\mathcal{D}i{v}_X^f={\mathcal{M}}_X^{f\times }/{\mathcal{O}}_X^{\times }$, one has ${\mathrm{Div}}_f(X)=\Gamma (X,\mathcal{D}i{v}_X^f)$, and the map $D\mapsto f^{\ast }(D)$ corresponds to the homomorphism of sheaves of groups ${\psi }^{\ast }({\mathcal{M}}_X^{f\times })/{\psi }^{\ast }({\mathcal{O}}_X^{\times })\to {\mathcal{M}}_{X^{\prime }}^{\times }/{\mathcal{O}}_{X^{\prime }}^{\times }=\mathcal{D}i{v}_{X^{\prime }}$ deduced from the preceding one by passage to the quotients.

(21.4.4). It follows at once from the preceding definitions that if $f^{\prime }:X^{\prime \prime }\to X^{\prime }$ is a second morphism of ringed spaces, $D$ a divisor on $X$ such that the inverse images $f^{\ast }(D)$ and $f^{\prime \ast }(f^{\ast }(D))$ exist, then $(f\circ f^{\prime }{)}^{\ast }(D)$ exists and is equal to $f^{\prime \ast }(f^{\ast }(D))$.

Proposition (21.4.5).

Let $f:X^{\prime }\to X$ be a morphism of ringed spaces. In each of the three following cases, the inverse image by $f$ of every divisor on $X$ is defined:

(i) $f$ is flat.

(ii) $X$ and $X^{\prime }$ are locally Noetherian preschemes and one has $f(Ass({\mathcal{O}}_{X^{\prime }}))\subset Ass({\mathcal{O}}_X)$.

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

It suffices to show that in the three cases one has ${\mathcal{M}}_X^{f\times }={\mathcal{M}}_X^{\times }$. In case (i), this follows from (20.1.12). In case (iii), one may restrict to the case where $X=\mathrm{Spec}(A)$ and $X^{\prime }=\mathrm{Spec}(A^{\prime })$ are affine; if $s\in A$ is regular, it does not belong to any minimal prime ideal of $A$ (20.1.3.1), hence the hypothesis implies that its image in $A^{\prime }$ does not belong to any minimal prime ideal of $A^{\prime }$, and is consequently a regular element of $A^{\prime }$ (20.1.3.1). In case (ii) the meromorphic functions on $X^{\prime }$ are identified with the pseudo-functions on $X^{\prime }$ (20.2.11), and the hypothesis, joined to (20.2.13, (iv)), ensures that the inverse image

by $f$ of every schematically dense open of $X$ is a schematically dense open of $X^{\prime }$; one therefore concludes by (20.3.12).

Corollary (21.4.6).

Let $X$ be a prescheme having one of the following properties:

(i) $X$ is locally Noetherian.

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

Then, for every $x\in X$, one has a canonical isomorphism

$$(\mathrm{21.4.6.1})(\mathcal{D}i{v}_X{)}_x\stackrel{\sim }{\to }\mathrm{Div}({\mathcal{O}}_{X,x}).$$

This follows from (20.2.11), (20.3.7) and from the fact that ${\mathcal{O}}_X^{\times }$ identifies with the group of invertible elements of the ring ${\mathcal{O}}_{X,x}$.

(21.4.7). Let $X$, $X^{\prime }$ be two preschemes, $f:X^{\prime }\to X$ a morphism. If $D$ is a positive divisor on $X$ such that the inverse image $f^{\ast }(D)$ is defined (21.4.2), then the closed sub-prescheme $Y(f^{\ast }(D))$ of $X^{\prime }$ is none other than the inverse image $f^{-1}(Y(D))$; this follows at once from the definitions (21.4.2) and (21.2.12).

Proposition (21.4.8).

Let $X$, $Y$ be two preschemes; $f:X\to Y$ a faithfully flat morphism. Then, if a divisor $D$ on $Y$ is such that $f^{\ast }(D)\ge 0$ (the existence of $f^{\ast }(D)$ following from (21.4.5)), one has $D\ge 0$. In particular, the map $D\mapsto f^{\ast }(D)$ from $\mathrm{Div}(Y)$ to $\mathrm{Div}(X)$ is injective.

The question being local on $Y$, one may restrict to the case where $D=div(w)$, with $w=u{v}^{-1}$, $u$ and $v$ being two regular sections of ${\mathcal{O}}_Y$ over $Y$. By hypothesis one has $u{\mathcal{O}}_X\subset v{\mathcal{O}}_X$, hence, for every $x\in X$, if one sets $y=f(x)$, one has $u_y{\mathcal{O}}_x\subset v_y{\mathcal{O}}_x$; one concludes that $u_y{\mathcal{O}}_y\subset v_y{\mathcal{O}}_y$ by virtue of the hypothesis that ${\mathcal{O}}_x$ is a faithfully flat ${\mathcal{O}}_y$-module and of Bourbaki, Alg. comm., chap. I, §3, n° 5, prop. 10, (ii); whence $u{\mathcal{O}}_Y\subset v{\mathcal{O}}_Y$ since $f$ is surjective, and consequently $D\ge 0$.

21.5. Direct images of divisors

(21.5.1). Let $X$, $X^{\prime }$ be two preschemes, $f:X^{\prime }\to X$ a morphism. We shall, in this n°, give sufficient conditions to be able to associate with every divisor $D^{\prime }$ on $X^{\prime }$ a divisor $D$ on $X$, direct image of $D^{\prime }$ by $f$. We shall restrict to the case where $f$ is a finite morphism (for more general conditions, see the chapter of this Treatise devoted to intersection theory).

Lemma (21.5.2).

Let $A$ be a ring, $E$ a free $A$-module of finite rank. For an endomorphism $u$ of $E$ to be injective, it is necessary and sufficient that $det(u)$ be a regular element of $A$.

This is proved in Bourbaki, Alg., chap. III, 3rd ed., §8, n° 2, prop. 3.

(21.5.3). Suppose now that the morphism $f:X^{\prime }\to X$ is finite, and moreover that $f$ verifies one of the two following properties:

I) $f$ is a finite locally free morphism, in other words (18.2.7) the quasi-coherent ${\mathcal{O}}_X$-Module of finite type $f_{\ast }({\mathcal{O}}_{X^{\prime }})$ is locally free.

II) $X$ is a reduced locally Noetherian prescheme, the $\mathcal{O}(X)$-Module $f_{\ast }({\mathcal{O}}_{X^{\prime }}){\otimes }_{\mathcal{O}}\mathcal{K}(X)$ is locally free, and for every section $s^{\prime }\in \Gamma (f^{-1}(U),{\mathcal{O}}_{X^{\prime }})$ ($U$ open

in $X$), $N_{X^{\prime }/X}(s^{\prime })$ is a section of ${\mathcal{O}}_X$ over $U$ (cf. (II, 6.5.1)). (One recalls that condition (II) is verified for every finite morphism $f$ when $X$ is a locally Noetherian normal prescheme (loc. cit.).)

One then knows (II, 6.5.5) that for every invertible ${\mathcal{O}}_{X^{\prime }}$-Module ${\mathcal{L}}^{\prime }$ one defines the norm $\mathcal{L}=N_{X^{\prime }/X}({\mathcal{L}}^{\prime })$ (which we shall also write $N({\mathcal{L}}^{\prime })$), which is an invertible ${\mathcal{O}}_X$-Module. Moreover, for every open $U$ of $X$ and every regular section $s^{\prime }\in \Gamma (f^{-1}(U),{\mathcal{O}}_{X^{\prime }})$, the norm $N_{X^{\prime }/X}(s^{\prime })=N_{f^{-1}(U)/U}(s^{\prime })$ (which we shall also write $N(s^{\prime })$) is a regular element of $\Gamma (U,{\mathcal{O}}_X)$; one is indeed at once reduced to the case where $U=X$ is affine, and then the conclusion follows from (21.5.2) under hypothesis (I); on the other hand, under hypothesis (II), the fact that $\mathcal{K}(X)$ is a flat ${\mathcal{O}}_X$-Module entails that the section $s^{\prime }\otimes 1$ of $\Gamma (U,f_{\ast }({\mathcal{O}}_{X^{\prime }}){\otimes }_{\mathcal{O}}\mathcal{K}(X))$ is also regular $(0_I,\mathrm{6.1.2})$, and the conclusion follows again from (21.5.2) applied to the ring $\Gamma (U,\mathcal{K}(X))$, taking into account the definition of the norm of a section (II, 6.5.3). This being so, let $u^{\prime }$ be a meromorphic section of ${\mathcal{L}}^{\prime }$ over $X^{\prime }$; the morphism $f$ being affine, every point $x\in X$ admits an open neighbourhood $U$ such that $u^{\prime }|f^{-1}(U)$ is of the form $t^{\prime }/s^{\prime }$, where $t^{\prime }\in \Gamma (f^{-1}(U),{\mathcal{L}}^{\prime })$ and $s^{\prime }$ is a regular section in $\Gamma (f^{-1}(U),{\mathcal{O}}_{X^{\prime }})$; the element $N(t^{\prime })/N(s^{\prime })$ (where $N(t^{\prime })$ is the section of $\mathcal{L}$ defined in (II, 6.5.3)) is then a meromorphic section of $\mathcal{L}$ over $U$ by virtue of what precedes, and it follows from the multiplicative properties of the norm (II, 6.5.3.1) that this section depends only on $u^{\prime }|f^{-1}(U)$ and not on its expression in the form $t^{\prime }/s^{\prime }$; for the same reason, when $U$ varies, the meromorphic sections of $\mathcal{L}$ over $U$ thus defined are the restrictions of one and the same meromorphic section of $\mathcal{L}$ over $X$, which one calls the norm of $u^{\prime }$ and denotes $N_{X^{\prime }/X}(u^{\prime })$ (or simply $N(u^{\prime })$). The map thus defined

  (21.5.3.1)             N_{X'/X} : Γ(X', 𝓜_{X'}(ℒ')) → Γ(X, 𝓜_X(N_{X'/X}(ℒ')))

extends the norm defined in (II, 6.5.3); if $u^{\prime }$ is a regular meromorphic section of ${\mathcal{L}}^{\prime }$ over $X^{\prime }$, it follows at once from what precedes that $N(u^{\prime })$ is a regular meromorphic section of $\mathcal{L}$ over $X$, for (with the same notations) $N(t^{\prime })$ is regular if $t^{\prime }$ is. Finally, if ${\mathcal{L}}_1^{\prime }$, ${\mathcal{L}}_2^{\prime }$ are two invertible ${\mathcal{O}}_{X^{\prime }}$-Modules, $s_1^{\prime }$ (resp. $s_2^{\prime }$) a meromorphic section of ${\mathcal{L}}_1^{\prime }$ (resp. ${\mathcal{L}}_2^{\prime }$) over $X^{\prime }$, one has, by virtue of what precedes and of (II, 6.5.3.1),

  (21.5.3.2)             N(s_1' ⊗ s_2') = N(s_1') ⊗ N(s_2').

(21.5.4). Suppose still that $f$ verifies one of the hypotheses I), II) of (21.5.3). If $({\mathcal{L}}_1^{\prime },s_1^{\prime })$, $({\mathcal{L}}_2^{\prime },s_2^{\prime })$ are two pairs each formed of an invertible ${\mathcal{O}}_{X^{\prime }}$-Module and a regular meromorphic section of that Module over $X^{\prime }$, which are moreover equivalent in the sense of (21.2.10), then the pairs $(N({\mathcal{L}}_1^{\prime }),N(s_1^{\prime }))$ and $(N({\mathcal{L}}_2^{\prime }),N(s_2^{\prime }))$ are also equivalent, for an isomorphism of invertible ${\mathcal{O}}_{X^{\prime }}$-Modules has for norm an isomorphism of their norms (II, 6.5.3), and one has seen above that $N_{X^{\prime }/X}$ transforms sections of $\Gamma (X^{\prime },{\mathcal{O}}_{X^{\prime }}^{\times })$ into those of $\Gamma (X,{\mathcal{O}}_X^{\times })$. One can therefore lay down the following definition: