Cartier and Weil Divisors Line Bundles and Picard Groups

Let $k$ be a field. A proper curve over $k$ is an integral $k$-scheme $C$ (Integral schemes) whose structure morphism $C\to \mathrm{Spec}k$ is proper (Proper morphisms) and whose underlying topological space has chain dimension one (Chain dimension and the empty-space convention). Thus $C$ is of finite type over $k$. No normality, regularity, projectivity, or smoothness is assumed.

For a closed point $x\in C$, its residue field $\kappa (x)$ (The residue field at a point of an affine scheme) is finite over $k$. Indeed, choose an affine open neighborhood $U=\mathrm{Spec}A$ of $x$ (Affine schemes and their coordinate rings). Since $C\to \mathrm{Spec}k$ is of finite type, $A$ is a finite-type $k$-algebra; the closed point $x$ corresponds to a maximal ideal $\mathfrak{m}\subset A$, so $\kappa (x)\cong A\mathrm{/}\mathfrak{m}$ is finite over $k$ (A maximal ideal of an affine algebra has finite residue field over the base field). Write $[\kappa (x):k]={\dim }_k\kappa (x)$.

We use divisor on $C$ to mean a finite formal integral linear combination of closed points, $D={\sum }_{x\in C\text{ closed}}{n}_x[x],n_x\in \mathbb{Z},$ where only finitely many $n_x$ are nonzero. These divisors form the free abelian group $\mathrm{Div}(C)$ on the closed points. Define the $k$-degree by ${\mathrm{deg}}_{k}D={\sum }_{x\in C\text{ closed}}{n}_x[\kappa (x):k]\in \mathbb{Z}\mathrm{.}$ This is well-defined because the support is finite, and coefficientwise addition makes ${\mathrm{deg}}_k:\mathrm{Div}(C)\to \mathbb{Z}$ a group homomorphism.

A scheme $X$ is locally factorial if every local ring ${\mathcal{O}}_{X,x}$, for $x\in X$, is a unique factorisation domain (A local ring is a nonzero commutative ring with a unique maximal ideal, Unique factorisation domain). This is a condition on the stalks; it does not assert that $X$ has an affine open cover whose coordinate rings are unique factorisation domains. The empty scheme is locally factorial vacuously.

Here is the componentwise form, with its hypotheses made explicit. Assume the Axiom of Choice (The Axiom of Choice), and suppose $X$ is a Noetherian normal scheme: Noetherian means that $X$ has a finite affine open cover by spectra of Noetherian rings (Locally Noetherian and Noetherian schemes), and normal means that each local ring is a normal Noetherian ring in the sense of normal noetherian ring. Then $X$ is reduced, and its irreducible components, with their reduced induced scheme structures, are integral, pairwise disjoint, and open. Consequently $X$ is locally factorial if and only if each of these components is locally factorial.

For the local-domain and reducedness inputs, fix a point $x$ and choose a chart $U=\mathrm{Spec}A$ from the finite Noetherian affine cover, with $x$ corresponding to $\mathfrak{p}\subset A$. The stalk is ${\mathcal{O}}_{X,x}\cong A_{\mathfrak{p}}$ by Affine open subschemes, Affine schemes and their coordinate rings, and The stalk of the affine structure sheaf at a prime is A_p. It is Noetherian by Every quotient and every localisation of a Noetherian ring is Noetherian. It is a local normal ring by the stated hypothesis, so the normal-ring condition at its unique maximal ideal makes ${\mathcal{O}}_{X,x}$ an integrally closed domain (A local ring is a nonzero commutative ring with a unique maximal ideal, normal noetherian ring). Thus every stalk is a domain. The nilpotent ideal sheaf has these stalkwise nilpotent elements as its germs (The reduction of a scheme), so it is zero and $X$ is reduced.

To see why the components are disjoint, let distinct irreducible components $C,D$ meet at $x$, and choose such a chart $U=\mathrm{Spec}A$ containing $x$. The nonempty intersections $C\cap U$ and $D\cap U$ are irreducible closed subsets of $U$. They are maximal there: if an irreducible closed subset of $U$ contains $C\cap U$, its closure in $X$ is irreducible, contains the dense open subset $C\cap U$ of $C$, and hence equals $C$ by maximality; since the original subset is closed in $U$, intersecting back with $U$ gives exactly $C\cap U$. The same holds for $D$. (Irreducible components as schemes, Irreducibility via nonempty open subsets, connectedness and open subspaces, Existence and basic properties of irreducible components) The affine components therefore correspond to distinct minimal primes ${\mathfrak{q}}_C$ and ${\mathfrak{q}}_D$ of $A$ contained in $\mathfrak{p}$ (Irreducible components of the spectrum correspond to minimal prime ideals). Under the prime-localisation correspondence, each remains minimal after extending to $A_{\mathfrak{p}}$: a prime below an extension contracts to a prime below the original minimal prime. The extensions remain distinct by injectivity of that correspondence, so $A_{\mathfrak{p}}\cong {\mathcal{O}}_{X,x}$ would have two distinct minimal primes (Prime ideals of a localization are exactly the primes disjoint from the denominator set). This is impossible for a domain, which has only the minimal prime $(0)$. Thus distinct components do not meet.

Finally, each chart in the finite cover has only finitely many irreducible components by A Noetherian ring has only finitely many irreducible components in its spectrum. Every irreducible component of $X$ meets a chart, and its intersection with that chart is an affine component as above; distinct global components give distinct such intersections because each is dense in its global component. There are therefore only finitely many global components. They are closed and cover $X$ by Existence and basic properties of irreducible components; since they are disjoint and finite in number, each is also open. Its reduced induced structure is integral by Irreducible components as schemes and Integral schemes. Restriction to an open subscheme preserves the stalks (Affine open subschemes), so the locally factorial condition holds on $X$ exactly when it holds on every component.

This component conclusion is asserted for the Noetherian normal case above. For arbitrary schemes outside the locally Noetherian setting, no openness or componentwise conclusion is being asserted.

Let $X$ be a scheme. The Picard group $\mathrm{Pic}(X)$ is the set of isomorphism classes $[\mathcal{L}]$ of invertible ${\mathcal{O}}_X$-modules (Invertible sheaves), with product $[\mathcal{L}][\mathcal{M}]:=[\mathcal{L}{\otimes }_{{\mathcal{O}}_X}\mathcal{M}]\mathrm{.}$ Its identity is $[{\mathcal{O}}_X]$, and its inverse operation is $[\mathcal{L}{]}^{-1}=[{\mathcal{L}}^{\vee }],{\mathcal{L}}^{\vee }=\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{L},{\mathcal{O}}_X)\mathrm{.}$ This is an abelian group. The source states this construction as the Picard group definition and leaves the group-law verification as an exercise (Vakil, §14.1.G, PDF p. 308); the proof below supplies that verification.

Let $X$ be a scheme. For each open $U\subseteq X$, put $S_X(U)=\{s\in {\mathcal{O}}_X(U):({\mathcal{O}}_{X,x}\stackrel{m_{s_x}}{\to }{\mathcal{O}}_{X,x})\text{ is injective for every }x\in U\}\mathrm{.}$ These are the regular sections of ${\mathcal{O}}_X$ on $U$. Equivalently, their germs are nonzerodivisors in the sense that multiplication by each germ is injective. Restriction preserves this property, and the product of two such sections has the property because the corresponding multiplication map is a composite of injective maps. The identity section belongs to $S_X(U)$, so $S_X(U)$ is a multiplicative subset of ${\mathcal{O}}_X(U)$. Define the presheaf of rings $P_X(U)=S_X(U{)}^{-1}{\mathcal{O}}_X(U),$ with restrictions induced by those of ${\mathcal{O}}_X$. The sheaf of meromorphic functions, also called the sheaf of total quotient rings, is the sheafification ${\mathcal{K}}_X:=a{P}_X\mathrm{.}$ The canonical localization maps give a morphism of presheaves of rings ${\mathcal{O}}_X\to P_X$; composing with the sheafification map gives a morphism of sheaves of rings ${\mathcal{O}}_X\to {\mathcal{K}}_X$. A meromorphic function on $X$ is a global section of ${\mathcal{K}}_X$.

If $X$ is integral, let $\eta$ be its generic point and set $K(X):={\mathcal{O}}_{X,\eta }\mathrm{.}$ Then ${\mathcal{K}}_X$ is canonically isomorphic to the constant sheaf $\underset{‾}{K(X)}$.

Let $X$ be a Noetherian normal scheme (Schemes, Locally Noetherian and Noetherian schemes). Noetherian means that $X$ is locally Noetherian and quasi-compact, equivalently that it has a finite affine open cover by spectra of Noetherian rings. Normal means every local ring ${\mathcal{O}}_{X,x}$ is an integrally closed domain; on an affine chart this is the local condition of normal noetherian ring. In particular $X$ is reduced (The reduction of a scheme).

An integral closed subscheme $Z\subseteq X$ has a generic point $\xi$ (Closed immersions of schemes, Integral schemes, Generic points of irreducible closed subsets). It is a prime divisor if it has codimension one, meaning $\dim {\mathcal{O}}_{X,\xi }=1.$ This is the Krull dimension of the local ring at $\xi$ (A local ring is a nonzero commutative ring with a unique maximal ideal, The height of a prime ideal).

A Weil divisor on $X$ is a formal sum $D={\sum }_Z{n}_Z[Z],n_Z\in \mathbb{Z},$ indexed by the prime divisors of $X$, with locally finite support: every point has an open neighbourhood meeting only finitely many of the closed subsets whose coefficients are nonzero. Addition is coefficientwise; these sums form an abelian group $\mathrm{Div}(X)$. Since $X$ is quasi-compact, a locally finite support on $X$ is in fact finite.

If $X$ is integral, this is the usual group of codimension-one cycles: its generators are the integral closed subschemes whose generic point has local-ring dimension one. This is the integral case of the definition in the Stacks Project, Divisors, Definition 31.27.2. For an integral closed subscheme the reduced induced structure is understood.

For a nonirreducible $X$, the same definition applies component by component. Under the Axiom of Choice (The Axiom of Choice), a Noetherian normal scheme has finitely many irreducible components, which are pairwise disjoint and open. Thus each integral closed subscheme lies in exactly one component. This component description is asserted here for Noetherian normal schemes; no component-openness claim is made for arbitrary normal schemes. The structural claim is verified below, with the Axiom of Choice used only for the published existence and finiteness inputs about irreducible components.

Let $X$ be a scheme and let $K_X$ be its sheaf of meromorphic functions, with the injective structure map ${\mathcal{O}}_X\to K_X$ (Sheaf total quotient rings, A sheaf on a topological space). The presheaf $U⟼K_X(U{)}^{\times }\mathrm{/}{\mathcal{O}}_X(U{)}^{\times }$ of abelian groups (units of the two sheaves of rings, the second embedded in the first through the injective structure map) has a sheafification in the sense of Sheafification of a presheaf; the resulting sheaf of abelian groups is denoted $K_X^{\times }\mathrm{/}{\mathcal{O}}_X^{\times }\mathrm{.}$

A Cartier divisor on $X$ is a global section of $K_X^{\times }\mathrm{/}{\mathcal{O}}_X^{\times }$. The group of Cartier divisors is denoted $\mathrm{CaDiv}(X)$; its law is induced by the group law of the quotient sheaf, so that the sum of two Cartier divisors is represented by the product of their local meromorphic equations, the zero element $0$ is the class of the constant equation $1$, and the inverse of a divisor is represented by the inverted local equations.

Concretely, a Cartier divisor can be described as follows. Let $\mathcal{U}=\{U_i{\}}_{i\in I}$ be an open cover of $X$ and let $f_i\in K_X(U_i{)}^{\times }$ be a meromorphic unit on $U_i$ for each $i$, with $f_i\mathrm{/}{f}_j\in {\mathcal{O}}_X(U_i\cap U_j{)}^{\times }\text{for all }i,j\mathrm{.}$ The images of the $f_i$ in $(K_X^{\times }\mathrm{/}{\mathcal{O}}_X^{\times })(U_i)$ agree on the overlaps $U_i\cap U_j$ (their quotient becomes $1$ in the quotient group), so the sheaf axiom glues them to a global section, and a different choice of cover or of representatives $f_i$ (that is, passing to a refinement and multiplying $f_i$ by units of ${\mathcal{O}}_X$ over the pieces) yields the same class. Conversely every global section of the quotient sheaf is locally represented in this way, as the local-equation description of Cartier divisors on this page records.

The principal Cartier divisors are the Cartier divisors that are the images of a global meromorphic unit under the canonical map $\mathrm{\Gamma }(X,K_X^{\times })\to \mathrm{\Gamma }(X,K_X^{\times }\mathrm{/}{\mathcal{O}}_X^{\times })$. They form a subgroup of $\mathrm{CaDiv}(X)$; a Cartier divisor is principal exactly when it admits a representation by a single global equation on $X$. The sign convention used on this page is that a Cartier divisor with local equation $f$ records a zero of $f$ with positive coefficient and a pole with negative coefficient; the convention is fixed once and for all in the definition of the principal Cartier divisor of a meromorphic unit.

If $X=\mathrm{\varnothing }$ then ${\mathcal{O}}_X=K_X=0$, the quotient sheaf is the zero sheaf and $\mathrm{CaDiv}(\mathrm{\varnothing })=0$; if $X$ is the spectrum of a field, then $K_X={\mathcal{O}}_X=\kappa$ and again $\mathrm{CaDiv}(X)=0$, consistently with the fact that a Cartier divisor measures the failure of a meromorphic unit to be a global unit.

Let $k$ be a field and let $C$ be a proper curve over $k$, so that $C$ is an integral proper $k$-scheme of dimension one and a divisor on $C$ is a finite integral sum $D={\sum }_x{n}_x[x]$ over the closed points (Degree divisor proper curve). For such a divisor define

1. the support $\mathrm{Supp}(D)=\{x:n_x\mathrm{\ne }0\}$, a finite set of closed points of $C$;
2. the positive part $D^{+}={\sum }_x\max (n_x,0)[x]$;
3. the negative part $D^{-}={\sum }_x\max (-n_x,0)[x]$, so that all coefficients of $D^{-}$ are nonnegative and $D=D^{+}-D^{-}$.

The supports of $D^{+}$ and $D^{-}$ are disjoint subsets of $\mathrm{Supp}(D)$: if $n_x>0$ then the coefficient of $x$ in $D^{-}$ is $\max (-n_x,0)=0$, and if $n_x<0$ then the coefficient of $x$ in $D^{+}$ is $\max (n_x,0)=0$. Both parts are effective divisors in the sense that all their coefficients are nonnegative, and $D$ is effective if and only if $D^{-}=0$. The same definitions apply verbatim to a Weil divisor on any integral normal locally Noetherian scheme, using prime divisors in place of closed points (Weil divisor normal noetherian scheme), and they are used on this page only for divisors on a curve, where the finite-support convention makes all three sums finite without further hypotheses.

Normality of $C$ is not required for the construction: the closed points of $C$ and the integers $n_x$ are the only data used, and the identity $D=D^{+}-D^{-}$ together with the disjointness of the two supports is a coefficientwise statement.

Let $X$ be a normal locally Noetherian scheme (normal noetherian ring) and let $Z\subseteq X$ be an integral closed subscheme with generic point $\xi$ and $\dim {\mathcal{O}}_{X,\xi }=1$ (Integral schemes, Generic points of irreducible closed subsets). We call such a $Z$ a prime divisor also in this locally Noetherian setting. This extends the same codimension-one definition in Weil divisor normal noetherian scheme; it requires no quasi-compactness of $X$.

The local ring ${\mathcal{O}}_{X,\xi }$ is a discrete valuation ring. It is a Noetherian local ring, because $X$ is locally Noetherian; it is a domain with fraction field equal to the function field of the irreducible component containing $Z$, because $\xi$ lies in a unique irreducible component of the normal scheme $X$; it is integrally closed, by normality; and it has dimension equal to one, by the definition of a prime divisor. A one-dimensional Noetherian local integrally closed domain is a discrete valuation ring by the characterisation of discrete valuation rings, and Height-one localizations of normal Noetherian domains are DVRs is exactly this statement in global form (Equivalent characterizations of a DVR, Discrete valuation rings).

Let $v_{\xi }$ denote the discrete valuation of the fraction field $K=\mathrm{Frac}({\mathcal{O}}_{X,\xi })$ whose valuation ring is ${\mathcal{O}}_{X,\xi }$, normalised so that $v_{\xi }(\pi )=1$ for a uniformiser $\pi$ of ${\mathcal{O}}_{X,\xi }$ (Discrete valuations, The field of fractions $\mathrm{Frac}(D)=(D\setminus \{0\}{)}^{-1}D$ of an integral domain). Now let $f\in \mathrm{\Gamma }(X,K_X^{\times })$ be a global meromorphic unit (Sheaf total quotient rings). The generic point $\xi$ lies in a unique irreducible component $X_i$ of $X$; the sheaf $K_X$ restricts on the integral scheme $X_i$ to the constant sheaf with value the function field $K(X_i)$, and ${\mathcal{O}}_{X,\xi }={\mathcal{O}}_{X_i,\xi }$ has fraction field $K(X_i)$. The restriction of $f$ to $X_i$ is therefore an element of $K(X_i{)}^{\times }=K^{\times }$, written $f_{\xi }$, and the order of vanishing of $f$ along $Z$ is the integer ${\mathrm{ord}}_Z(f):=v_{\xi }(f_{\xi })\in \mathbb{Z}\mathrm{.}$ Since $v_{\xi }$ is a group homomorphism $K^{\times }\to \mathbb{Z}$ and $f_{\xi }$ depends only on the restriction of $f$ to $X_i$, this is well defined: ${\mathrm{ord}}_Z(1)=0$, ${\mathrm{ord}}_Z(fg)={\mathrm{ord}}_Z(f)+{\mathrm{ord}}_Z(g)$ and ${\mathrm{ord}}_Z(f^{-1})=-{\mathrm{ord}}_Z(f)$.

If $X$ is integral (Integral schemes) then $K_X$ is the constant sheaf with value $K(X)$, so a meromorphic unit is simply an element of $K(X{)}^{\times }$, and ${\mathrm{ord}}_Z(f)=v_{\xi }(f)$ for the element $f\in K(X{)}^{\times }=\mathrm{Frac}({\mathcal{O}}_{X,\xi }{)}^{\times }$. In this case ${\mathrm{ord}}_Z(f)\ge 0$ if and only if $f$ lies in ${\mathcal{O}}_{X,\xi }$, and ${\mathrm{ord}}_Z(f)=0$ if and only if $f$ is a unit of ${\mathcal{O}}_{X,\xi }$, since $v_{\xi }$ is the normalised valuation of a discrete valuation ring.

For $X=\mathrm{\varnothing }$ there are no prime divisors, so the order domain is empty. The meromorphic-unit group is trivial, with its unique identity; this does not give an order without a prime divisor.

Let $X$ be an integral scheme (Integral schemes) with generic point $\eta$ (Generic points of irreducible closed subsets), let $K_X$ be its sheaf of meromorphic functions (Sheaf total quotient rings), and let $\mathcal{L}$ be an invertible ${\mathcal{O}}_X$-module (Invertible sheaves).

The sheaf of meromorphic sections of $\mathcal{L}$ is the ${\mathcal{O}}_X$-module $K_X(\mathcal{L})=\mathcal{L}{\otimes }_{{\mathcal{O}}_X}{K}_X,$ the tensor product of sheaves of modules (Tensor product of sheaves of modules). A meromorphic section of $\mathcal{L}$ is a global section of $K_X(\mathcal{L})$; it is regular, or a rational section, when it is nonzero.

On an integral scheme the sheaf $K_X$ is the constant sheaf with value the function field $K(X)={\mathcal{O}}_{X,\eta }$, so $K_X(\mathcal{L})$ is the constant sheaf with value the stalk ${\mathcal{L}}_{\eta }$. This stalk is a one-dimensional vector space over the function field: fixing a ${\mathcal{O}}_{X,\eta }$-basis of identity $1_{\eta }$ of the field $K(X)$, the vector space is $K(X){\otimes }_{{\mathcal{O}}_{X,\eta }}{\mathcal{L}}_{\eta }\cong K(X),$ and a rational section is a nonzero element of the one-dimensional $K(X)$-vector space ${\mathcal{L}}_{\eta }$. The stalk ${\mathcal{L}}_{\eta }$ is one-dimensional over $K(X)$ because $\mathcal{L}$ is locally free of rank one (Locally free sheaves of finite rank): on a neighbourhood of $\eta$ a generator identifies $\mathcal{L}$ with ${\mathcal{O}}_X$, and passing to stalks gives ${\mathcal{L}}_{\eta }\cong {\mathcal{O}}_{X,\eta }=K(X)$. A rational section is therefore the same thing as a $K(X)$-multiple of any chosen local generator of $\mathcal{L}$ near $\eta$, and two rational sections $s,s^{\mathrm{\prime }}$ satisfy $s^{\mathrm{\prime }}=gs$ for a unique $g\in K(X{)}^{\times }$ when both are nonzero.

On the empty scheme there is no generic point and no invertible module with a nonzero stalk, so the notation is not used there; on a nonempty integral scheme the generic point exists and the construction is never vacuous. The definition imposes no properness, finiteness or normality assumption on $X$; those enter only when one wants to associate divisors to the sections.

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field and let $C$ be a proper curve over $k$ (Degree divisor proper curve) that is normal, meaning that every local ring ${\mathcal{O}}_{C,x}$ is an integrally closed domain. Let $K=k(C)={\mathcal{O}}_{C,\eta }$ be the function field of $C$ at its generic point (Function field of an integral finite-type scheme) and let $f\in K^{\times }$.

1. Algebraic case. If $f$ is algebraic over $k$, then $f$ and $f^{-1}$ are global units: $f\in \mathrm{\Gamma }(C,{\mathcal{O}}_C{)}^{\times }$.
2. Transcendental case. If $f$ is transcendental over $k$, then $d=[K:k(f)]$ is finite and there is a finite locally free dominant morphism ${\phi }_f:C⟶{\mathbb{P}}_k^1$ of degree $d$. For the standard chart coordinates $t=x_1^{(0)}$ and $s=x_0^{(1)}$ on ${\mathbb{P}}_k^1$ (Relative projective space from standard charts), its pullbacks are ${\phi }_f^{\mathrm{\#}}(t)=f$ and ${\phi }_f^{\mathrm{\#}}(s)=f^{-1}$. The images in $K$ of the target chart coordinate rings $k[t]$ and $k[s]$ are respectively $k[f]$ and $k[f^{-1}]$. The coordinate rings of the affine preimages of these charts may be larger; each is finite free of rank $d$ over its target chart coordinate ring.

Thus the finite dominant morphism conclusion applies in the transcendental case. Over a general field, being outside $k$ does not imply transcendence: for a finite extension $L\mathrm{/}k$, an element $a\in L\setminus k$ on the normal proper curve ${\mathbb{P}}_L^1$ is algebraic over $k$ and a global unit. Its constant map to ${\mathbb{P}}_k^1$ has closed image, not a dominant image.