Divisor (algebraic geometry)

For other meanings of the term 'Divisor' see the page Divisor (disambiguation)

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

In algebraic geometry, the term divisor is used as a generalization of the concept of a divisor of an element of a commutative ring. First introduced by E.E. Kummer [Ku] under the name of "ideal divisor" in his studies on cyclotomic fields.

The theory of divisors for an integral commutative ring $A$ with a unit element consists in constructing a homomorphism $\phi$ from the multiplicative semi-group $A^{\ast }$ of non-zero elements of $A$ into some semi-group $D_0$ with unique factorization, the elements of which are known as (integral) divisors of the ring $A$. The theory of divisors makes it possible to reduce a series of problems connected with prime factorization in $A$, where this factorization may be not unique, to the problem of prime factorization in $D_0$. The image $\phi (a)\in D_0$ of an element $a\in A^{\ast }$ is denoted by $(a)$ and is called the principal divisor of the element $a$. One says that $a\in A^{\ast }$ is divisible by the divisor $\mathfrak{a}\in D_0$ if $\mathfrak{a}$ divides $(a)$ in $D_0$.

More exactly, let $D_0$ be a free Abelian semi-group with a unit element, the free generators of which are known as prime divisors, and let a homomorphism $\phi :A^{\ast }\to D_0$ be given. The homomorphism $\phi$ defines a theory of divisors of the ring $A$ if it satisfies the following conditions.

1) For $a,b\in A^{\ast }$ the element $a$ divides $b$ in $A$ if and only if $(a)$ divides $(b)$ in $D_0$.

2) For any $\mathfrak{a}\in D_0$,

$$\{a\in A|\mathfrak{a}\text{ divides }(a)\}\cup \{0\}$$ is an ideal of $A$.

3) If $\mathfrak{a},{\mathfrak{a}}^{\prime }\in D_0$ and if, for any $a\in A^{\ast }$, $(a)$ is divisible by $\mathfrak{a}$ if and only if $(a)$ is divisible by ${\mathfrak{a}}^{\prime }$, then $\mathfrak{a}={\mathfrak{a}}^{\prime }$.

If a homomorphism $\phi$ exists, it is uniquely determined, up to an isomorphism, by the conditions just listed. The kernel $\ker \phi$ coincides with the group of unit elements of $A$. The elements of $D_0$ are called positive divisors of $A$. Let $K$ be the field of quotients of $A$, and let $D\supset D_0$ be the free Abelian group generated by the set of prime divisors. Then for any $c\in K^{\ast }$, $K^{\ast }=K\setminus 0$, it is possible to define a principal divisor $(c)\in D$. If $c=a/b$ where $a,b\in A^{\ast }$, then $(c)=(a)/(b)$. The elements of the group $D$ are known as fractional divisors (or, simply, divisors) of $A$ (or of $K$). Any divisor $\mathfrak{a}\in D$ may be written in the form

$$\mathfrak{a}={\mathfrak{p}}_1^{n_1}\cdots {\mathfrak{p}}_r^{n_r},$$ where ${\mathfrak{p}}_i$ is a prime divisor. In additive notation: $\mathfrak{a}=n_1{\mathfrak{p}}_1+\cdots +n_r{\mathfrak{p}}_r$. If $a\in K^{\ast }$ and $(a)=\sum n_i{\mathfrak{p}}_i$, the mapping $a\mapsto \sum n_i$ is a discrete valuation on $K$, and is known as the essential valuation of $K$. The homomorphism $\phi$ is extended to a homomorphism $\psi :K^{\ast }\to D$, where $\psi (c)=(c)$, contained in the exact sequence

$$1\to U(A)\to K^{\ast }\stackrel{\psi }{\to }D\to C(A)\to 1.$$ Here $U(A)$ is the group of invertible elements of $A$, while the group $C(A)$ is called the divisor class group of $A$ (or of $K$). Two divisors which belong to the same equivalence class by the subgroup of principal divisors are called equivalent (in algebraic geometry, where a large number of other divisor equivalences are considered, this equivalence is known as linear).

The theory of divisors is valid for any Dedekind ring, in particular for rings of integral elements in algebraic number fields, and the elements of $D_0$ are in one-to-one correspondence with the non-zero ideals of the ring $A$ (to the divisor $\mathfrak{a}$ corresponds the ideal of all elements of $A$ that are divisible by $\mathfrak{a}$). This is why, in a Dedekind ring, the group of divisors is also called the group of ideals, while the divisor class group is called the ideal class group.

The divisor class group of an algebraic number field is finite, and many problems in algebraic number theory involve the computation of its order (the number of classes) and structure [BoSh].

More generally, the theory of divisors is valid for Krull rings (cf. Krull ring, [Bo]). In such a case the role of $D_0$ is played by the semi-group of divisorial ideals (cf. Divisorial ideal) of the ring, while the part of $D$ is played by the group of fractional divisorial ideals.

The concept of a Weil divisor is a generalization of the concept of a fractional divisorial ideal of a commutative ring to algebraic varieties or analytic spaces $X$. The name Weil divisor is given to integral formal finite linear combinations $\sum n_{W}W$ of irreducible closed subspaces $W$ in $X$ of codimension 1. A Weil divisor is called positive, or effective, if all $n_W\ge 0$. All Weil divisors form a group $Z^1(X)$ (the group of Weil divisors). If $X$ is a smooth algebraic variety, the concept of a Weil divisor coincides with that of an algebraic cycle of codimension 1.

If $A$ is a Noetherian Krull ring, each prime divisorial ideal $\mathfrak{p}$ in $A$ defines a subspace $V(\mathfrak{p})$ of codimension 1 in the scheme $X=\text{Spec}(A)$, while each divisor $\mathfrak{a}={\mathfrak{p}}_1^{n_1}\cdots {\mathfrak{p}}_k^{n_k}$ may thus be identified with the Weil divisor $\sum n_{i}V({\mathfrak{p}}_i)$.

Let $X$ be a normal scheme and let $f$ be a rational (meromorphic in the analytic case) function on $X$. A principal Weil divisor is defined canonically:

$$(f)=\sum n_{W}W.$$ Here $n_W$ is the value of the discrete valuation of the ring ${\mathcal{O}}_{X,W}$ of the subvariety $W$ on the representative of $f$ in ${\mathcal{O}}_{X,W}$. If

$$(f)=\sum n_W^{+}W+\sum n_W^{-}W,$$ where $n_W^{+}>0$ and $n_W^{-}<0$, the Weil divisor $(f{)}_0=\sum n_W^{+}W$ is known as the divisor of the zeros, while $\sum n_W^{-}W$ is known as the divisor of the poles of the function $f$. The set of principal Weil divisors is a subgroup $Z_p^1(X)$ of the group $Z^1(X)$. The quotient group $Z^1(X)/Z_p^1(X)$ is denoted by $C(X)$ and is known as the divisor class group of the scheme $X$. If $X=\text{Spec}A$, where $A$ is a Noetherian Krull ring, $C(X)$ coincides with the divisor class group of the ring $A$.

Let $K$ be an algebraic function field. A divisor of $K$ is sometimes defined as a formal integral combination of discrete valuations of rank 1 of $K$. If $K$ is a field of algebraic functions in one variable, each such divisor may be identified with the Weil divisor of its complete non-singular model.

Let $X$ be a regular scheme or a complex variety and let $D=\sum n_{W}W$ be a Weil divisor. For any point $x\in X$ there exists an open neighbourhood $U$ such that the restriction of $D$ on $U$,

$$D{|}_U=\sum n_W(W\cap U)$$ is the principal divisor $(f_U)$ for a certain meromorphic function $f_U$ on $U$. The function $f_U$ is uniquely defined, up to an invertible function on $U$, and is known as the local equation of the divisor $D$ in the neighbourhood $U$, while the correspondence $U\mapsto f_U$ defines a section of the sheaf $M_X^{\ast }/{\mathcal{O}}_X^{\ast }$. In general, a Cartier divisor on a ringed space $(X,{\mathcal{O}}_X)$ is defined as a global section of the sheaf $M_X^{\ast }/{\mathcal{O}}_X^{\ast }$ of germs of divisors. Here $M_X$ denotes the sheaf of germs of meromorphic (or rational) functions on $X$, i.e. the sheaf which brings into correspondence each open $U\subset X$ with the total quotient ring of the ring $\mathrm{\Gamma }(U,{\mathcal{O}}_X)$, while $M_X^{\ast }$ and ${\mathcal{O}}_X^{\ast }$ are the sheaves of invertible elements in $M_X$ and ${\mathcal{O}}_X$, respectively. A Cartier divisor may be defined by a selection of local equations

$$f_i\in \mathrm{\Gamma }(U_i,M_X^{\ast }),$$ where $\{U_i\}$ is an open covering of $X$, and the functions $f_i/f_j$ should be a section of the sheaf ${\mathcal{O}}_X^{\ast }$ over $U_i\cap U_j$. In particular, a meromorphic function $f$ defines a divisor $\text{div}(f)$ known as a principal divisor. The set of $x\in X$ such that $(f_i{)}_x\notin {\mathcal{O}}_{X,x}^{\ast }$ is called the support of the divisor. The Cartier divisors form an Abelian group $\text{Div}(X)$, while the principal divisors form a subgroup of it, ${\text{Div}}_l(X)$. Each divisor $D\in \text{Div}(X)$ defines an invertible sheaf ${\mathcal{O}}_X(D)$ contained in $M_X$: If $D$ is represented by the local equations $f_i$ on the covering $\{U_i\}$, then

$${\mathcal{O}}_X(D){|}_{U_i}=f_i^{-1}\mathcal{O}{|}_{U_i}\subset M_X{|}_{U_i}$$ The correspondence $D\mapsto {\mathcal{O}}_X(D)$ is a homomorphism of the group $\text{Div}(X)$ into the Picard group $\text{Pic}(X)=H^1(X,{\mathcal{O}}_X^{\ast })$. This homomorphism is included in the exact sequence

$$\mathrm{\Gamma }(X,M_X^{\ast })\to \text{Div}(X)\stackrel{\delta }{\to }\text{Pic}(X)\to H_1(X,M_X^{\ast }),$$ which is obtained from the exact sequence of sheaves

$$0\to {\mathcal{O}}_X^{\ast }\to M_X^{\ast }\to M_X^{\ast }/{\mathcal{O}}_X^{\ast }\to 0.$$ Thus, $\ker \delta ={\text{Div}}_l(X)$. If $D-D_1$ is a principal divisor, $D$ and $D_1$ are said to be linearly equivalent. If $X$ is a quasi-projective algebraic variety or a complex Stein space, the homomorphism $\delta :\text{Div}(X)\to \text{Pic}(X)$ is surjective and induces an isomorphism of the group of classes of linearly equivalent divisors $\text{Div}(X)/{\text{Div}}_l(X)$ onto the Picard group $\text{Pic}(X)$.

If $X$ is a complex space, the problem arises as to when a given divisor is a principal divisor; this is the so-called second Cousin problem (cf. Cousin problems). For example, the divisor class group on a complex Stein space $(X,\mathcal{O})$ is trivial if and only if $H^2(X,\mathbf{Z})=0$.

A divisor $D$ is said to be effective (or positive) if ${\mathcal{O}}_X\subset {\mathcal{O}}_X(D)$. In such a case ${\mathcal{O}}_X(-D)$ is a sheaf of ideals in ${\mathcal{O}}_X$; the support of a divisor $D$ with structure sheaf ${\mathcal{O}}_X/{\mathcal{O}}_X(-D)$ forms a subspace in $X$, which is also denoted by $D$.

For a normal Noetherian scheme or a normal analytic space $X$ there is a natural homomorphism:

$$\text{cyc}:\text{Div}(X)\to Z^1(X),$$ mapping $D\in \text{Div}(X)$ into $\sum n_{W}W$, where $n_W={\nu }_W(f)$ and where $f$ is a local equation of $D$ in the neighbourhood $W$, while ${\nu }_W$ is the discrete valuation corresponding to $W$ [We]. The homomorphism $\text{cyc}$ is injective and maps effective divisors to effective cycles; $\text{cyc}$ is bijective if and only if $X$ is locally factorial (e.g. when $X$ is a non-singular scheme or an analytic manifold). If $\text{cyc}$ is bijective, Weil and Cartier divisors coincide.

Let $f:X^{\prime }\to X$ be a morphism of schemes which is flat in codimension 1. Then, for any Cartier or Weil divisor $D$ on $X$ the inverse image $f^{\ast }(D)$ is defined; also, $\text{cyc}(f^{\ast }(D))=f^{\ast }(\text{cyc}(D))$. The mapping $D\to f^{\ast }(D)$ is a homomorphism of groups which maps principal divisors to principal ones, and thus defines a homomorphism of groups

$$f^{\ast }:\text{Pic}(X)\to \text{Pic}(X^{\prime })$$ (respectively,

$$f^{\ast }:C(X)\to C(X^{\prime })).$$ If $X^{\prime }$ is an open set in $X$ whose codimension of the complement is at least 2 and if $f$ is the imbedding of $X^{\prime }$ into $X$, then $f^{\ast }:C(X)\to C(X^{\prime })$ is an isomorphism, while $f^{\ast }:\text{Pic}(X)\to \text{Pic}(X^{\prime })$ is an isomorphism if the scheme $X$ is locally factorial.

Let $X$ be a smooth projective variety over $\mathbf{C}$. Any divisor $D$ on $X$ defines a homology class

$$[D]\in H_{2\dim X-2}(X,\mathbf{Z}).$$ The cohomology class which is Poincaré dual to $[D]$ is identical with the Chern class $c_1({\mathcal{O}}_X(D))\in H^2(X,\mathbf{Z})$ of the invertible sheaf ${\mathcal{O}}_X(D)$. Thus there appears a homological equivalence on $\text{Div}(X)$. There exists a theory of intersections of divisors [Sh], leading to the concept of algebraic equivalence of divisors (cf. Algebraic cycle). The group

$${\text{Pic}}^0(X)={\text{Div}}_{\alpha }(X)/{\text{Div}}_l(X),$$ where ${\text{Div}}_{\alpha }(X)$ denotes the group of divisors which are algebraically equivalent to zero, is naturally provided with the structure of an Abelian variety (the Picard variety; if $X$ is a curve, it is also called the Jacobi variety of $X$). The group $\text{Div}(X)/{\text{Div}}_{\alpha }(X)$, known as the Néron–Severi group, has a finite number of generators. The last two facts also apply to algebraic varieties over an arbitrary field.

If $X$ is a one-dimensional complex manifold (a Riemann surface), a divisor on $X$ is a finite linear combination

$$D=\sum _i{k}_i{x}_i,$$ where $k_i\in \mathbf{Z}$, $x_i\in X$. The number $\sum k_i$ is called the degree of the divisor $D$. For a compact Riemann surface $X$ of genus $g$ the group of divisor classes of degree zero is a $g$-dimensional Abelian variety and is identical with the Picard variety (or with the Jacobi variety). If $f$ is a meromorphic function on $X$, a principal divisor is

$$\text{div}(f)=\sum _i{m}_i{x}_i-\sum _j{n}_j{y}_j,$$ where $x_i$ are the zeros and $y_j$ are the poles of $f$ and $m_i,$, $n_j$ are their multiplicities. Then $\sum _i{m}_i=\sum _j{n}_j,$, i.e. a principal divisor has degree 0. A divisor of degree 0 on $X$ is principal if and only if there exists a singular one-dimensional chain $C$ such that

$$\partial C=D\text{ and }{\int }_C\omega =0$$ for all holomorphic forms $\omega$ of degree 1 on $X$ (Abel's theorem). See also Abelian differential.

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